Updated on 2022/05/21

OZAWA, Tohru

Affiliation
Faculty of Science and Engineering, School of Advanced Science and Engineering
Job title
Professor
Homepage URL

### Concurrent Post

• Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

• Faculty of Science and Engineering   Graduate School of Advanced Science and Engineering

### Research Institute

• 2020
-
2022

理工学術院総合研究所   兼任研究員

### Education

• 1988.03
-

Kyoto University   Graduate School of Science

• 1986.04
-

京都大学大学院理学研究科数理解析専攻博士課程 進学

• 1986.04
-

京都大学大学院理学研究科数理解析専攻博士課程 進学

• 1986.03
-

Kyoto University   Graduate School of Science

• 1986.03
-

Kyoto University   Graduate School of Science

• 1984.04
-

京都大学大学院理学研究科数理解析専攻修士課程 入学

• 1984.03
-

早稲田大学理工学部物理学科 卒業 (応用物理学科 飯野理一・堤正義研究室)

• 1984.03
-

早稲田大学理工学部物理学科 卒業 (応用物理学科 飯野理一・堤正義研究室)

• 1980.04
-

Waseda University   School of Science and Engineering

• 1980.03
-

Waseda University   Senior High School

### Degree

• Kyoto University   Doctor of Science

### Research Experience

• 2008.09
-

Waseda University   Faculty of Science and Engineering

• 1995.04
-

北海道大学大学院理学研究科 教授（数学専攻数理解析学講座）

• 1995.04
-

北海道大学大学院理学研究科 教授（数学専攻数理解析学講座）

• 1993.04
-

Hokkaido University   School of Science

• 1993.04
-

Hokkaido University   School of Science

• 1992.04
-

Hokkaido University   School of Science

• 1990.06
-

Kyoto University   Research Institute for Mathematical Sciences

• 1988.04
-

Nagoya University   School of Science

•

大学院基幹理工学研究科数学応用数理専攻 兼担 現在に至る

•

大学院先進理工学研究科物理学及応用物理学専攻 担当

### Professional Memberships

•

International Society for Analysis, its Applications and Computation (ISAAC)

•

American Mathematical Society (AMS)

•

Mathematical Society of Japan (MSJ)

### Research Areas

• Mathematical analysis

• Basic analysis

### Research Interests

• Mathematical Physics

• Harmonic Analysis

• Nonlinear Partial Differential Equations

### Papers

• Small data scattering of 2d Hartree type Dirac equations

Yonggeun Cho, Kiyeon Lee, Tohru Ozawa

Journal of Mathematical Analysis and Applications   506 ( 1 ) 125549 - 125549  2022.02  [Refereed]

• Zakharov system in two space dimensions

Tohru Ozawa, Kenta Tomioka

Nonlinear Analysis   214   112532 - 112532  2022.01  [Refereed]

• Jishan Fan, Tohru Ozawa

Partial Differential Equations and Applications   2 ( 6 )  2021.12  [Refereed]

View Summary

<title>Abstract</title>In this note, we prove a new <inline-formula><alternatives><tex-math>$$L^4$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>L</mml:mi>
<mml:mn>4</mml:mn>
</mml:msup>
</mml:math></alternatives></inline-formula>-estimate of the velocity by the technique of Hardy space <inline-formula><alternatives><tex-math>$${\mathcal {H } }^1$$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>H</mml:mi>
</mml:mrow>
<mml:mn>1</mml:mn>
</mml:msup>
</mml:math></alternatives></inline-formula> and <italic>BMO</italic>.

• Regularity criteria for a Ginzburg-Landau-Navier-Stokes system

J. Fan, T.Ozawa

Funkcialaj Ekvacioj   64   349 - 360  2021.11  [Refereed]

• Yonggeun Cho, Tohru Ozawa, Changhun Yang

Funkcialaj Ekvacioj   64 ( 1 ) 1 - 15  2021.04  [Refereed]

• Jishan Fan, Tohru Ozawa

Nonlinear Partial Differential Equations for Future Applications   346   95 - 102  2021.04  [Refereed]

• Poincaré inequalities with exact missing terms on homogeneous groups

T. Ozawa, D. Suragan

Journal of the Mathematical Society of Japan   73 ( 2 ) 497 - 503  2021.04  [Refereed]

• A note on bilinear estimates in the homogeneous Triebel-Lizorkin spaces

J. Fan, T.Ozawa

RIMS Kokyuroku Bessatsu   B88   147 - 150  2021  [Refereed]

• Well-posedness of a 2D time-dependent Ginzburg-Landau superconductivity model

J. Fan, T. Ozawa

Nonlinear Analysis and Differential Equations   8 ( 1 ) 89 - 97  2021.01  [Refereed]

• Sharp remainder of the Poincaré inequality

T. Ozawa D. Suragan

Proceedings of the American Mathematical Society   148 ( 10 ) 4235 - 4239  2020.10  [Refereed]

• Global solutions to the Maxwell-Navier-Stokes system in a bounded domain in 2D

J. Fan, T. Ozawa

Zeitschrift für angewandte Mathematik und Physik   71 ( 136 )  2020.07  [Refereed]

• Existence and uniqueness of classical paths under quadratic potentials

K. Narita, T. Ozawa

Calculus of Variations and PDE's   59 ( 128 )  2020.07  [Refereed]

• Self-similar solutions to the derivative nonlinear Schrödinger equation

K. Fujiwara, V. Georgiev, T. Ozawa

Journal of Differential Equations   268 ( 12 ) 7940 - 7961  2020.06  [Refereed]

• Multidimensional inverse Cauchy problems for evolution equations

M. Karazym, T.Ozawa, D. Suragan

Inverse Problems in Science and Engineering   28 ( 11 ) 1582 - 1590  2020.05  [Refereed]

• A blow-up criterion for the modified Navier-Stokes-Fourier equations

J. Fan, T. Ozawa

Journal of Mathematical Fluid Mechanics   22, Article number 16  2020.02  [Refereed]

• Cauchy problem and vanishing dispersion limit for Schr&ouml;dinger-improved Boussinesq equations

J. Fan, T. Ozawa

Journal of Mathematical Analysis and Applications   485, Issue 2   123857  2020.01  [Refereed]

• Hardy type inequalities with spherical derivatives

N. Bez, S. Machihara, T. Ozawa

SN Partial Differential Equations and Applications   1, Issue 1, Article 5.  2020.01  [Refereed]

• On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases

K. Fujiwara, V. Georgiev, T. Ozawa

J. Math. Pures Appl.,   136   239 - 256  2020  [Refereed]

• A note on bilinear estimates in the Sobolev spaces

J. Fan, T. Ozawa

International Journal of Mathematical Analysis   13 ( 12 ) 551 - 554  2019.11  [Refereed]

• Remarks on the Hardy type inequalities with remainder terms in the framework of equalities

Adv. Studies Pure Math.   81   247 - 258  2019.11  [Refereed]

• Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential

L. Forcella, K. Fujiwara, V. Georgiev, T. Ozawa

Discrete and Continuous Dynamical Systems A   39 ( 5 ) 2661 - 2678  2019.05  [Refereed]

• Lp-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups

T. Ozawa, M. Ruzhansky, D. Suragan

Quaterly J. Math.   70 ( 1 ) 305 - 318  2019.03  [Refereed]

• Note for global existence of semilinear heat equation in weighted $L^\infty$ space

K. Fujiwara, V. Georgiev, T. Ozawa

Pliska Stud. Math.   30   7 - 20  2019  [Refereed]

• Dynamical behavior for the solutions of the Navier-Stokes equation

Kuijie Li, Tohru Ozawa, Baoxiang Wang

Communications on Pure and Applied Analysis   17 ( 4 ) 1511 - 1560  2018.07  [Refereed]

View Summary

We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions: (Equation Presented) More precisely, for the blow up mild solutions with initial data in L∞(ℝRd) and Hd/2-1(ℝd), we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form supp u0 ⊂ {ξ ∈ ℝn : ξ1 ≥ L} and ||u0||∞ ≪ L for some L &gt
0, then (1) has a unique global solution u ∈ C(ℝR+
L∞). In 3D, we show the compactness of the set consisting of minimal-Lp singularity-generating initial data with 3 &lt
p &lt
1, furthermore, if the mild solution with data in Lp(ℝ3) blows up in a Type-I manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces B-1+6/p p/2,∞ (ℝ3).

• Higher Order Fractional Leibniz Rule

Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa

Journal of Fourier Analysis and Applications   24 ( 3 ) 650 - 665  2018.06  [Refereed]

View Summary

The fractional Leibniz rule is generalized by the Coifman–Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.

• ON THE FOCUSING ENERGY-CRITICAL FRACTIONAL NONLINEAR SCHRODNGER EQUATIONS

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa

ADVANCES IN DIFFERENTIAL EQUATIONS   23 ( 3-4 ) 161 - 192  2018.03  [Refereed]

View Summary

We consider the fractional nonlinear Schrodinger equation (FNLS) with non-local dispersion vertical bar V vertical bar(alpha) and focusing energy-critical Hartree type nonlinearity [-(vertical bar x vertical bar(-2 alpha) *vertical bar u vertical bar(2))u]. We first establish a global well-posedness of radial case in energy space by adopting Kenig-Merle arguments [20] when the initial energy and initial kinetic energy are less than those of ground state, respectively. We revisit and highlight long time perturbation, profile decomposition and localized virial inequality. As an application of the localized virial inequality, we provide a proof for finite time blowup for energy critical Hartree equations via commutator technique introduced in [2].

• Global well-posedness of weak solutions to the time-dependent Ginzburg-Landau model for superconductivity

J. Fan, T. Ozawa

Taiwanese J. Math.   22 ( 4 ) 851 - 858  2018  [Refereed]

• Blow-up for self-interacting fractional Ginzburg-Landau equation

Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa

Dynamics of Partial Differential Equations   15 ( 3 ) 175 - 182  2018  [Refereed]

View Summary

The blow-up of solutions for the Cauchy problem of fractional Ginzburg-Landau equation with non-positive nonlinearity is shown by an ODE argument. Moreover, in one dimensional case, the optimal lifespan estimate for size of initial data is obtained.

• Lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation

K. Fujiwara, T. Ozawa

Evolution Equations and Control Theory   7 ( 2 ) 275 - 280  2018  [Refereed]

• Small data scattering of hartree type fractional schrödinger equations in dimension 2 and 3

Yonggeun Cho, Tohru Ozawa

Journal of the Korean Mathematical Society   55 ( 2 ) 373 - 390  2018  [Refereed]

View Summary

In this paper we study the small-data scattering of the d dimensional fractional Schrödinger equations with d = 2, 3, Lévy index 1 &lt
α &lt
2 and Hartree type nonlinearity F (u) = µ(|x|−γ ∗ |u|2)u with max (Formula presented) &lt
γ ≤ 2, γ &lt
d. This equation is scaling-critical in Ḣsc, (Formula presented). We show that the solution scatters in Hs,1 for any s &gt
sc, where Hs,1 is a space of Sobolev type taking in angular regularity with norm defined by (Formula presented). For this purpose we use the recently developed Strichartz estimate which is L2 -averaged on the unit sphere Sd−1 and utilize Up -Vp space argument.

• Ground States for Semi-Relativistic Schrodinger-Poisson-Slater Energy

Jacopo Bellazzini, Tohru Ozawa, Nicola Visciglia

FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA   60 ( 3 ) 353 - 369  2017.12  [Refereed]

View Summary

We prove the existence of ground states for the semi-relativistic Schrodinger-Poisson-Slater energy
I-alpha,I- beta(rho) = inf (u is an element of H 1/2 (R3) integral R3 vertical bar u vertical bar 2dx=rho) 1/2 parallel to u parallel to(2)(H1/2(R3)) + alpha integral integral(R3xR3) vertical bar u(x)vertical bar(2)vertical bar u(y)vertical bar(2)/vertical bar x - y vertical bar dxdy - beta integral(R3) vertical bar u vertical bar(8/3)dx alpha, beta &gt; 0 and rho &gt; 0 is small enough. The minimization problem is L-2 critical and in order to characterize the values alpha, beta &gt; 0 such that I-alpha,I- beta(rho) &gt; -infinity for every rho &gt; 0, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant S &gt; 0 such that
1/S parallel to phi parallel to(L8/3(R3))/parallel to phi parallel to(1/2)(H1/2(R3)) &lt;= (integral integral(R3 x R3) vertical bar phi(x)vertical bar(2)vertical bar phi(y)vertical bar(2)/vertical bar x - y vertical bar dxdy)(1/8)
for all phi is an element of C-0(infinity)(R-3). Besides, we show that similar compactness property fails if we replace the inhomogeneous Sobolev norm parallel to u parallel to(2)(H1/2(R3)) by the homogeneous one parallel to u parallel to(H1/2(R3)) in the energy above.

• Hardy type inequalities in Lp with sharp remainders

Norisuke Ioku, Michinori Ishiwata, Tohru Ozawa

Journal of Inequalities and Applications   2017 ( 1 )  2017.12  [Refereed]

View Summary

Sharp remainder terms are explicitly given on the standard Hardy inequalities in Lp(Rn) with 1 &lt
p&lt
n. Those remainder terms provide a direct and exact understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals.

• Lifespan of strong solutions to the periodic nonlinear Schrodinger equation without gauge invariance

Kazumasa Fujiwara, Tohru Ozawa

JOURNAL OF EVOLUTION EQUATIONS   17 ( 3 ) 1023 - 1030  2017.09  [Refereed]

View Summary

A lifespan estimate and sharp condition of the initial data for finite time blowup for the periodic nonlinear Schrodinger equation are presented from a viewpoint of the total signed densities of initial data.

• LOCAL WELL-POSEDNESS FOR AN ERICKSEN-LESLIE'S PARABOLIC-HYPERBOLIC COMPRESSIBLE NON-ISOTHERMAL MODEL FOR LIQUID CRYSTALS

Jishan Fan, Tohru Ozawa

ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS   2017 ( 232 ) 1 - 8  2017.09  [Refereed]

View Summary

In this article we prove the local well-posedness for an Ericksen-Leslie's parabolic-hyperbolic compressible non-isothermal model for nematic liquid crystals with positive initial density.

• Remarks on the Rellich inequality

MATHEMATISCHE ZEITSCHRIFT   286 ( 3-4 ) 1367 - 1373  2017.08  [Refereed]

View Summary

We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers.

• Short-range scattering of Hartree type fractional NLS II

Yonggeun Cho, Tohru Ozawa

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   157   62 - 75  2017.07  [Refereed]

View Summary

We prove the small data scattering for Hartree type fractional Schrodinger equation with inverse square potential. This is the border line problem between Strichartz range and weighted space range in view of the method of approach. To show this we carry out a subtle trilinear estimate via fractional Leibniz rule and Balakrishnan's formula. This paper is a sequel of the previous result (Cho, 2017). (C) 2017 Elsevier Ltd. All rights reserved.

• Stability of Trace Theorems on the Sphere

Neal Bez, Chris Jeavons, Tohru Ozawa, Mitsuru Sugimoto

Journal of Geometric Analysis   28   1 - 21  2017.06  [Refereed]

View Summary

We prove stable versions of trace theorems on the sphere in (Formula presented.) with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the trace theorem into (Formula presented.) for (Formula presented.), by combining a refined Hardy–Littlewood–Sobolev inequality on the sphere with a duality–stability result proved very recently by Carlen. Finally, we extend a local version of Carlen’s duality theorem to establish local stability of certain Strichartz estimates for the kinetic transport equation.

• ON LANDAU-KOLMOGOROV INEQUALITIES FOR DISSIPATIVE OPERATORS

Masayuki Hayashi, Tohru Ozawa

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   145 ( 2 ) 847 - 852  2017.02  [Refereed]

View Summary

We revisit Kato's theory on Landau-Kolmogorov (or Kallman-Rota) inequalities for dissipative operators in an algebraic framework in a scalar product space.

• Hardy type inequalities in L-p with sharp remainders

Norisuke Ioku, Michinori Ishiwata, Tohru Ozawa

JOURNAL OF INEQUALITIES AND APPLICATIONS    2017.01  [Refereed]

View Summary

Sharp remainder terms are explicitly given on the standard Hardy inequalities in L-p(R-n) with 1 &lt; p &lt; n. Those remainder terms provide a direct and exact understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals.

• A note on regularity criteria for Navier-Stokes system: Note on Navier-Stokes

Jishan Fan, Tohru Ozawa

Springer Proceedings in Mathematics and Statistics   215   47 - 50  2017  [Refereed]

View Summary

We use some interpolation inequalities on Besov spaces to show a regularity criterion for n-dimensional Navier-Stokes system.

• Uniform regularity for the time-dependent Ginzburg-Landau-Maxwell equations

J. Fan, T. Ozawa

“New Trends in Analysis and Interdisciplinary Applications,” Trends in Mathematics, Springer     301 - 306  2017  [Refereed]

• A conjecture regarding optimal Strichartz estimates for the wave equation

N. Bez, C. Jeavons, T. Ozawa, H. Saito

“New Trends in Analysis and Interdisciplinary Applications,” Trends in Mathematics, Springer     293 - 300  2017  [Refereed]

• Weak solutions to the time-dependent Ginzburg-Landau-Maxwell equations

J. Fan, T. Ozawa

RIMS Kokyuroku Bessatsu   63   23 - 30  2017  [Refereed]

• Uniform existence and uniqueness for a time-dependent Ginzburg-Landau model for superconductivity

J. Fan, T. Ozawa

Nonlinear Analysis and Differential Equations   5 ( 6 ) 249 - 259  2017  [Refereed]

• Global well-posedness of weak solutions to the time-dependent Ginzburg-Landau model for superconductivity in R2

J. Fan, T. Ozawa

International Journal of Mathematical Analysis   11 ( 4 ) 199 - 207  2017  [Refereed]

• REGULARITY CRITERIA FOR NAVIER-STOKES AND RELATED SYSTEM

Jishan Fan, Tohru Ozawa

DIFFERENTIAL AND INTEGRAL EQUATIONS   30 ( 1-2 ) 101 - 114  2017.01  [Refereed]

View Summary

We show some regularity criteria for Navier-Stokes equations, the harmonic heat flow, two liquid crystals models, and a model for magneto-elastic materials. The method of proof depends on a systematic use of interpolation inequalities in Besov spaces and is independent on logarithmic inequalities.

• Uncertainty relations in the framework of equalities

Tohru Ozawa, Kazuya Yuasa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   445 ( 1 ) 998 - 1012  2017.01  [Refereed]

View Summary

We study the Schrodinger Robertson uncertainty relations in an algebraic framework. Moreover, we show that some specific commutation relations imply new equalities, which are regarded as equality versions of well-known inequalities such as Hardy's inequality. (C) 2016 Elsevier Inc. All rights reserved.

• Well-posedness for a generalized derivative nonlinear Schrodinger equation

Masayuki Hayashi, Tohru Ozawa

JOURNAL OF DIFFERENTIAL EQUATIONS   261 ( 10 ) 5424 - 5445  2016.11  [Refereed]

View Summary

We study the Cauchy problem for a generalized derivative nonlinear Schrodinger equation with the Dirichkt boundary condition. We establish the local well-posedness results in the Sobolev spaces H-1 and H-2. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space H1. (C) 2016 Elsevier Inc. All rights reserved.

• ON SMALL DATA SCATTERING OF HARTREE EQUATIONS WITH SHORT-RANGE INTERACTION

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS   15 ( 5 ) 1809 - 1823  2016.09  [Refereed]

View Summary

In this note we study Hartree type equations with vertical bar del vertical bar(alpha) (1 &lt; alpha &lt;= 2) and potential whose Fourier transform behaves like vertical bar gamma vertical bar(-(d-gamma 1)) at the origin and vertical bar xi vertical bar(-(d-gamma 1)) at infinity. We show non-existence of scattering when 0 &lt; gamma(1) &lt;= 1 and small data scattering in H-S for s &gt; 2 2-alpha/2 when 2 &lt; gamma(1) &lt;= d and 0 &lt; gamma(2) &lt;= 2. For this we use U-P-V-P space argument and Strichartz estimates.

• Finite time blowup of solutions to the nonlinear Schrodinger equation without gauge invariance

Kazumasa Fujiwara, Tohru Ozawa

JOURNAL OF MATHEMATICAL PHYSICS   57 ( 8 )  2016.08  [Refereed]

View Summary

A lifespan estimate and a condition of the initial data for finite time blowup for the nonlinear Schrodinger equation are presented from a view point of ordinary differential equation (ODE) mechanism. Published by AIP Publishing.

• Normal form and global solutions for the Klein-Gordon-Zakharov equations

T. Ozawa, K. Tsutaya, Y. Tsutsumi

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire   12 ( 4 ) 459 - 503  2016.07  [Refereed]

View Summary

In this paper we study the global existence and asymptotic behavior of solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations in three space dimensions. We prove that for small initial data, there exist the unique global solutions of the Klein-Gordon-Zakharov equations. We also show that these solutions approach asymptotically the free solutions as t → ∞. Our proof is based on the method of normal forms introduced by Shatah [12], which transforms the original system with quadratic nonlinearity into a new system with cubic nonlinearity.

• An improvement on the Brezis-Gallouet technique for 2D NLS and 1D half-wave equation

Tohru Ozawa, Nicola Visciglia

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   33 ( 4 ) 1069 - 1079  2016.07  [Refereed]

View Summary

We revise the classical approach by Brezis-Gallouet to prove global well-posedness for nonlinear evolution equations. In particular we prove global well-posedness for the quartic NLS on general domains Omega in R-2 with initial data in H-2(Omega) boolean AND H-0(1)(Omega), and for the quartic nonlinear half-wave equation on R with initial data in H-1(R). (C) 2015 Elsevier Masson SAS. All rights reserved.

• Space-time analytic smoothing effect for the pseudo-conformally invariant Schrödinger equations

Gaku Hoshino, Tohru Ozawa

Nonlinear Differential Equations and Applications   23 ( 1 ) 1 - 10  2016.02  [Refereed]

View Summary

© 2016, Springer International Publishing. We study the global Cauchy problem for the mass critical nonlinear Schrödinger equations. We prove the global existence of analytic solutions in both space and time variables for sufficiently small and exponentially decaying Cauchy data. The method of proof depends on the Leibniz rule for the generator of pseudo-conformal transforms at the L2 critical level.

• Remarks on bilinear estimates in the Sobolev spaces

K. Fujiwara, T. Ozawa

RIMS Kokyuroku Bessatsu   56   1 - 9  2016  [Refereed]

• Blow-up criterion for 3D navier-stokes equations and Landau-Lifshitz system in a bounded domain

Jishan Fan, Tohru Ozawa

Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday   none   175 - 182  2016  [Refereed]

View Summary

In this paper we prove a blow-up criterion for the 3D Navier-Stokes equations in a bounded domain in terms of a BMO norm of vorticity. We will also prove a regularity criterion for the Landau-Lifshitz system in a bounded domain.

• Some remarks on gauge choice and Navier-Stokes equations

J. Fan, T. Ozawa

Nonlinear Analysis and Differential Equations   4 ( 14 ) 659 - 667  2016  [Refereed]

• Global strong solutions to the time-dependent Ginzburg-Landau model in superconductivity with four new gauges

J. Fan, T. Ozawa

Nonlinear Analysis and Differential Equations   4 ( 11 ) 513 - 519  2016  [Refereed]

• Remarks on regularity criteria for harmonic heat flow and related system

Jishan Fan, Tohru Ozawa

International Journal of Mathematical Analysis   10 ( 13-16 ) 749 - 755  2016  [Refereed]

View Summary

We prove regularity criteria for harmonic heat flow and a liquid crystals model.

• Weighted L-P-boundedness of convolution type integral operators associated with bilinear estimates in the Sobolev spaces

Kazumasa Fujiwara, Tohru Ozawa

JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   68 ( 1 ) 169 - 191  2016.01  [Refereed]

• Sharp lower bounds for Coulomb energy

Jacopo Bellazzini, Marco Ghimenti, Tohru Ozawa

MATHEMATICAL RESEARCH LETTERS   23 ( 3 ) 621 - 632  2016  [Refereed]

View Summary

We prove L-p lower bounds for Coulomb energy for radially symmetric functions in H(over dot)(s)(R-3) with 1/2 &lt; s &lt; 3/2. In case 1/2 &lt; s &lt;= 1 we show that the lower bounds are sharp.

• FINITE TIME BLOWUP FOR THE FOURTH-ORDER NLS

Yonggeun Cho, Tohru Ozawa, Chengbo Wang

BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY   53 ( 2 ) 615 - 640  2016  [Refereed]

View Summary

We consider the fourth-order Schrodinger equation with focusing inhomogeneous nonlinearity (-vertical bar x vertical bar(-2)vertical bar u vertical bar(4/n) u) in high space dimensions. Extending Glassey's virial argument, we show the finite time blow-up of solutions when the energy is negative.

• SOME SHARP BILINEAR SPACE-TIME ESTIMATES FOR THE WAVE EQUATION

Neal Bez, Chris Jeavons, Tohru Ozawa

MATHEMATIKA   62 ( 3 ) 719 - 737  2016  [Refereed]

View Summary

We prove a family of sharp bilinear space-time estimates for the halfwave propagator e(it root-Delta). As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

• ANALYTIC SMOOTHING EFFECT FOR THE CUBIC HYPERBOLIC SCHRODINGER EQUATION IN TWO SPACE DIMENSIONS

Gaku Hoshino, Tohru Ozawa

ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS   2016   1 - 8  2016.01  [Refereed]

View Summary

We study the Cauchy problem for the cubic hyperbolic Schrodinger equation in two space dimensions. We prove existence of analytic global solutions for sufficiently small and exponential decaying data. The method of proof depends on the generalized Leibniz rule for the generator of pseudo-conformal transform acting on pseudo-conformally invariant nonlinearity.

• Regularity criteria for harmonic heat flow and related system

Jishan Fan, Tohru Ozawa

MATHEMATISCHE NACHRICHTEN   289 ( 1 ) 28 - 33  2016.01  [Refereed]

View Summary

We will prove some regularity criteria for harmonic heat flow, biharmonic heat flow and a surface growth model. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

• Sharp remainder of a critical Hardy inequality

Norisuke Ioku, Michinori Ishiwata, Tohru Ozawa

ARCHIV DER MATHEMATIK   106 ( 1 ) 65 - 71  2016.01  [Refereed]

View Summary

An explicit representation is given to the remainder of a critical Hardy inequality in L-n (R-n) with n &gt;= 2.

• Weak solutions to the Ginzburg-Landau model in superconductivity with the temporal gauge

Jishan Fan, Tohru Ozawa

APPLICABLE ANALYSIS   95 ( 9 ) 2029 - 2038  2016  [Refereed]

View Summary

We first prove the uniqueness of weak solutions (psi, A) to the 3-D Ginzburg-Landau system in superconductivity with the temporal gauge if (psi, A). W := {(psi, A)vertical bar psi is an element of L-2(0, T; L-infinity), del psi is an element of L-2(0, T; L-3), A is an element of C([0, T]; L-3)}, which is a critical space for some positive constant T. We also prove the global existence of solutions for the Ginzburg-Landau system with magnetic diffusivity mu &gt; 0 or mu = 0. Finally, we prove the uniform bounds with respect to mu of strong solutions in space dimensions d = 2. Consequently, the existence of the limit as mu -&gt; 0 can be established.

• Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space

Journal of Inequalities and Applications   2015 ( 1 )  2015.12  [Refereed]

View Summary

In this paper, we establish Hardy inequalities of logarithmic type involving singularities on spheres in R&lt
sup&gt
n&lt
/sup&gt
in terms of the Sobolev-Lorentz-Zygmund spaces. We prove it by absorbing singularities of functions on the spheres by subtracting the corresponding limiting values.

• Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space

JOURNAL OF INEQUALITIES AND APPLICATIONS    2015.09  [Refereed]

View Summary

In this paper, we establish Hardy inequalities of logarithmic type involving singularities on spheres in R-n in terms of the Sobolev-Lorentz-Zygmund spaces. We prove it by absorbing singularities of functions on the spheres by subtracting the corresponding limiting values.

• Well-Posedness for the Cauchy Problem for a System of Semirelativistic Equations

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa

COMMUNICATIONS IN MATHEMATICAL PHYSICS   338 ( 1 ) 367 - 391  2015.08  [Refereed]

View Summary

The local well-posedness for the Cauchy problem of a system of semirelativistic equations in one space dimension is shown in the Sobolev space H-s of order s &gt;= 0. We apply the standard contraction mapping theorem by using Bourgain type spaces X-s,X-b. We also use an auxiliary space for the solution in L-2 = H-0. We give the global well-posedness by this conservation law and the argument of the persistence of regularity.

• ON A SYSTEM OF SEMIRELATIVISTIC EQUATIONS IN THE ENERGY SPACE

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS   14 ( 4 ) 1343 - 1355  2015.07  [Refereed]

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Well-posedness of the Cauchy problem for a system of semirelativistic equations is shown in the energy space. Solutions are constructed as a limit of an approximate solutions. A Yudovitch type argument plays an important role for the convergence arguments.

• Analytic smoothing effect for a system of schrödinger equations with two wave interaction

Gaku Hoshino, Tohru Ozawa, Gustavo Ponce

Advances in Differential Equations   20 ( 7-8 ) 697 - 716  2015.07  [Refereed]

View Summary

We study the global Cauchy problem for a system of Schrödinger equations with two wave interaction of quadratic, cubic and quintic degrees. For suciently small data with exponential decay at innity we prove the existence and uniqueness of global solutions which are analytic with respect to Galilei and/or pseudo-conformal generators for suciently small data with exponential decay at innity. This paper is a sequel to our paper [22], where three wave interaction is studied. We also discuss the associated Lagrange structure.

• Finite Time Extinction for Nonlinear Schrodinger Equation in 1D and 2D

Remi Carles, Tohru Ozawa

COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS   40 ( 5 ) 897 - 917  2015.05  [Refereed]

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We consider a nonlinear Schrodinger equation with power nonlinearity, either on a compact manifold without boundary, or on the whole space in the presence of harmonic confinement, in space dimension one and two. Up to introducing an extra superlinear damping to prevent finite time blow up, we show that the presence of a sublinear damping always leads to finite time extinction of the solution in 1D, and that the same phenomenon is present in the case of small mass initial data in 2D.

• Remark on a semirelativistic equation in the energy space

K. Fujiwara, S. Machihara, T. Ozawa

Discrete and Continuous Dynamical Systems, Suppl.     473 - 478  2015  [Refereed]

• A regularity criterion for 3D density-dependent MHD system with zero viscocity

J. Fan, T. Ozawa

Discrete and Continuous Dynamical Systems, Suppl.     395 - 399  2015  [Refereed]

• A regularity criterion for the Schrödinger map

Jishan Fan, Tohru Ozawa

Trends in Mathematics   2   217 - 223  2015  [Refereed]

View Summary

We prove a regularity criterion ∇u ∈ L2(0,T
BMO(ℝn)) with 2 ≤ n ≤ 5 for the Schrödinger map. Here BMO is the space of functions with bounded mean oscillations.

• Remarks on global solutions to the cauchy problem for semirelativistic equations with power type nonlinearity

Kazumasa Fujiwara, Tohru Ozawa

International Journal of Mathematical Analysis   9 ( 53-56 ) 2599 - 2610  2015  [Refereed]

View Summary

Existence and nonexistence results on global solutions to the Cauchy problem for semirelativistic equations are shown by a simple compact- ness argument and a test function method, respectively. To obtain the nonexistence of global solutions, semirelativistic equations are trans- formed into a new equation without nonlocal operators in linear part but with a time derivative in nonlinear part, which is shown to be under control of special choice of test functions.

• Analytic smoothing effect for a system of Schrödinger equations with three wave interaction

Gaku Hoshino, Tohru Ozawa

Journal of Mathematical Physics   56 ( 9 )  2015  [Refereed]

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© 2015 AIP Publishing LLC. We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝn with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ4. We also discuss the associated Lagrange structure.

• Regularity criteria for the incompressible MHD with the hall or ion-slip effects

Jishan Fan, Tohru Ozawa

International Journal of Mathematical Analysis   9 ( 21-24 ) 1173 - 1186  2015  [Refereed]

View Summary

This paper proves some regularity criteria for the 3D incompressible MHD with the Hall or ion-slip effects.

• On the semilinear Schrodinger equation with time dependent coefficients

Takuya Gonda, Shuji Machihara, Tohru Ozawa

MATHEMATISCHE NACHRICHTEN   287 ( 17-18 ) 1986 - 2001  2014.12  [Refereed]

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We consider nonlinear Schrodinger equation with time dependent coefficients. Fanelli [5] found a transformation between solutions of the original equation and of the usual Schrodinger equation with power nonlinearity involving time dependent coefficients in some Lorentz spaces. In this paper we extend the results in [5] in space-time integrability properties of solutions. Particularly, we prove that the existence and uniqueness of solutions can be described exclusively in terms of Lebesgue spaces (not Lorentz spaces as in [5]) as far as the space integrability of solutions. We also discuss the equation with coefficient of an explicit homogeneous function and describe the associated Strichartz estimate and contraction mapping argument.

• Analytic smoothing effect for nonlinear Schrödinger equation with quintic nonlinearity

Gaku Hoshino, Tohru Ozawa

Journal of Mathematical Analysis and Applications   419 ( 1 ) 285 - 297  2014.11  [Refereed]

View Summary

We prove the global existence of analytic solutions to nonlinear Schrödinger equation with quintic nonlinearity in n space dimensions for sufficiently small Cauchy data with exponential decay. The smallness assumption on the data is imposed in terms of the critical Sobolev space Ḣn/2-1/2. Moreover, a characterization of some class of analytic functions is given. © 2014 Elsevier Inc.

• Regularity criteria for the density-dependent Hall-magnetohydrodynamics

Jishan Fan, Tohru Ozawa

APPLIED MATHEMATICS LETTERS   36   14 - 18  2014.10  [Refereed]

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This paper proves two regularity criteria for the density-dependent Hall-MHD system with positive initial density. We also prove a global nonexistence result for initial density with a high decrease at infinity. (C) 2014 Elsevier Ltd. All rights reserved.

• Notes on the paper entitled 'Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces'

JOURNAL OF INEQUALITIES AND APPLICATIONS   2014:253  2014.07  [Refereed]

• ANALYTIC SMOOTHING EFFECT FOR NONLINEAR SCHRODINGER EQUATION IN TWO SPACE DIMENSIONS

Gaku Hoshino, Tohru Ozawa

OSAKA JOURNAL OF MATHEMATICS   51 ( 3 ) 609 - 618  2014.07  [Refereed]

View Summary

We prove the global existence of analytic solutions to the Cauchy problem for nonlinear Schrodinger equations in two dimensions, where the nonlinearity behaves as a cubic power at the origin and the Cauchy data are small and decay exponentially at infinity.

• Stability of the Young and Holder inequalities

Kazumasa Fujiwara, Tohru Ozawa

JOURNAL OF INEQUALITIES AND APPLICATIONS   2014:162  2014.05  [Refereed]

View Summary

We give a simple proof of the Aldaz stability version of the Young and Holder inequalities and further refinements of available stability versions of those inequalities.

• ON THE ORBITAL STABILITY OF FRACTIONAL SCHRODINGER EQUATIONS

Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS   13 ( 3 ) 1267 - 1282  2014.05  [Refereed]

View Summary

We show the existence of ground state and orbital stability of standing waves of fractional Schrodinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.

• REGULARITY CRITERIA FOR THE 2D MHD SYSTEM WITH HORIZONTAL DISSIPATION AND HORIZONTAL MAGNETIC DIFFUSION

Jishan Fan, Tohru Ozawa

KINETIC AND RELATED MODELS   7 ( 1 ) 45 - 56  2014.03  [Refereed]

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This paper proves some regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. We also prove the global existence of strong solutions of its regularized MHD-alpha system.

• On the Hardy type inequality in critical Sobolev-Lorentz spaces

RIMS Kokyuroku Bessatsu   49   103 - 118  2014  [Refereed]

• Analytic smoothing effect for nonlinear Schrödinger equations with quadratic interaction

G. Hoshino, T. Ozawa

RIMS Kokyuroku Bessatsu   49   1 - 12  2014  [Refereed]

• Regularity criteria for Hall-magnetohydrodynamics and the space-time Monopole equation in Lorenz gauge

Jishan Fan, Tohru Ozawa

HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS   612   81 - +  2014  [Refereed]

View Summary

We prove some regularity criteria of local smooth solutions for viscous or inviscid Hall-MHD model and the space-time Monopole equation in the Lorenz gauge. We also prove time decay estimates for weak solutions of the viscous Hall-MHD model by the Fourier splitting device.

• Uniform regularity for the Landau-Lifshitz-Maxwell system without Dissipation

Jishan Fan, Tohru Ozawa

Applied Mathematical Sciences   8 ( 169-172 ) 8547 - 8557  2014  [Refereed]

View Summary

In this paper, we prove the existence of local solutions to the Cauchy problem for the 3D Landau-Lifshitz-Maxwell system without dissipation, where the local existence time and the corresponding Sobolev estimates are independent of the dielectric constant e{open} with 0 &lt
ε &lt
1. Consequently, the limit as ε → 0 can be established.

• Identities for the difference between the arithmetic and geometric means

Kazumasa Fujiwara, Tohru Ozawa

International Journal of Mathematical Analysis   8 ( 29-32 ) 1525 - 1542  2014  [Refereed]

View Summary

We prove a formula which expresses the difference between the arithmetic mean of variables of odd numbers and the corresponding geometric mean in the form of a linear combination of independent variables with coefficients given by sums of squares of polynomials.

• A Sharp Bilinear Estimate for the Klein-Gordon Equation in R1+1

Tohru Ozawa, Keith M. Rogers

INTERNATIONAL MATHEMATICS RESEARCH NOTICES   ( 5 ) 1367 - 1378  2014  [Refereed]

View Summary

We prove a sharp bilinear estimate for the one-dimensional Klein-Gordon equation. The proof involves an unlikely combination of five trigonometric identities. We also prove new estimates for the restriction of the Fourier transform to the hyperbola, where the pullback measure is not assumed to be compactly supported.

• A blow-up criterion for the 3d full magnetohydrodynamic equations

Jishan Fan, Tohru Ozawa

International Journal of Mathematical Analysis   8 ( 1-4 ) 101 - 108  2014  [Refereed]

View Summary

In this paper we establish a regularity criterion for the 3D incompressible full MHD equations with variable viscosity. © 2014 Jishan Fan and Tohru Ozawa.

• Global existence of strong solutions to a time-dependent 3D Ginzburg-Landau model for superconductivity with partial viscous terms

Jishan Fan, Tohru Ozawa

MATHEMATISCHE NACHRICHTEN   286 ( 17-18 ) 1792 - 1796  2013.12  [Refereed]

View Summary

We study an initial boundary value problem for a time-dependent 3D Ginzburg-Landau model of superconductivity with partial viscous terms. We prove the global existence of strong solutions. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

• Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces

Journal of Inequalities and Applications   2013  2013.12  [Refereed]

View Summary

In this paper, we establish the Hardy inequality of the logarithmic type in the critical Sobolev-Lorentz spaces. More precisely, we generalize the Hardy type inequality obtained in Edmunds and Triebel (Math. Nachr. 207:79-92, 1999). The generalized inequality allows us to take the exponents appearing in the inequality more flexibly, and its optimality is discussed in detail. O'Neil's inequality and its reverse play an essential role for the proof. © 2013 Machihara et al.

• SHARP MORAWETZ ESTIMATES

Tohru Ozawa, Keith M. Rogers

JOURNAL D ANALYSE MATHEMATIQUE   121   163 - 175  2013.10  [Refereed]

View Summary

We prove sharp Morawetz estimates - global in time with a singular weight in the spatial variables - for the linear wave, Klein-Gordon, and Schrodinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.

• HARDY TYPE INEQUALITIES ON BALLS

TOHOKU MATHEMATICAL JOURNAL   65 ( 3 ) 321 - 330  2013.09  [Refereed]

View Summary

Hardy type inequalities are presented on balls with radius R at the origin in R-n with n = 2 at least. A special attention is paid on the behavior of functions on the boundary.

• On the Cauchy Problem of Fractional Schrodinger Equation with Hartree Type Nonlinearity

Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa

FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA   56 ( 2 ) 193 - 224  2013.08  [Refereed]

View Summary

We study the Cauchy problem for the fractional Schrodinger equation i partial derivative(t)u = (m(2) - Delta)(alpha/2) u + F (u) in R1+n, where n &gt;= 1, m &gt;= 0, 1 &lt; alpha &lt; 2, and F stands for the nonlinearity of Hartree type F(u) = lambda(psi(center dot)vertical bar center dot vertical bar(-gamma) * vertical bar u vertical bar(2))u with lambda = +/-1, 0 &lt; gamma &lt; n, and 0 &lt;= psi is an element of L-infinity (R-n). We prove the existence and uniqueness of local and global solutions for certain alpha, gamma, lambda, psi. We also remark on finite time blowup of solutions when lambda = -1.

• On a system of nonlinear Schrodinger equations with quadratic interaction

Nakao Hayashi, Tohru Ozawa, Kazunaga Tanaka

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   30 ( 4 ) 661 - 690  2013.07  [Refereed]

View Summary

We study a system of nonlinear Schrodinger equations with quadratic interaction in space dimension n &lt;= 6. The Cauchy problem is studied in L-2, in H-1, and in the weighted L-2 space &lt; x &gt; L--1(2) = F(H-1) under mass resonance condition, where &lt; x &gt; = (1 + vertical bar x vertical bar(2))(1/2) and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations. (c) 2012 Elsevier Masson SAS. All rights reserved.

• GLOBAL WELL-POSEDNESS OF CRITICAL NONLINEAR SCHRODINGER EQUATIONS BELOW L-2

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   33 ( 4 ) 1389 - 1405  2013.04  [Refereed]

View Summary

The global well-posedness on the Cauchy problem of nonlinear Schrodinger equations (NLS) is studied for a class of critical nonlinearity below L-2 in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index s(c) is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

• Small data blow-up for a system of nonlinear Schrodinger equations

Tohru Ozawa, Hideaki Sunagawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   399 ( 1 ) 147 - 155  2013.03  [Refereed]

View Summary

We give examples of small data blow-up for a three-component system of quadratic nonlinear Schrodinger equations in one space dimension. Our construction of the blowing-up solution is based on the Hopf-Cole transformation, which allows us to reduce the problem to getting suitable growth estimates for a solution to the transformed system. Amplification in the reduced system is shown to have a close connection with the mass resonance. (C) 2012 Elsevier Inc. All rights reserved.

• FINITE CHARGE SOLUTIONS TO CUBIC SCHRODINGER EQUATIONS WITH A NONLOCAL NONLINEARITY IN ONE SPACE DIMENSION

Kei Nakamura, Tohru Ozawa

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   33 ( 2 ) 789 - 801  2013.02  [Refereed]

View Summary

We study the Cauchy problem for cubic Schrodinger equations modelling ultra-short laser pulses propagating along the line. The global existence, blow-up, and scattering of solutions is described exclusively in the charge space L-2(R) without any approximating arguments.

• A regularity criterion for compressible nematic liquid crystal flows

J. Fan, T. Ozawa

ISRN Mathematical Analysis   2013(2013), Article ID 271324, 4pages  2013  [Refereed]

• An approximation model for the density-dependent magnetohydrodynamic equations

J. Fan, T. Ozawa

Discrete and Continuous Dynamical Systems, Suppl.     207 - 216  2013  [Refereed]

• Exact remainder formula for the young inequality and applications

Kazumasa Fujiwara, Tohru Ozawa

International Journal of Mathematical Analysis   7 ( 53-56 ) 2723 - 2735  2013  [Refereed]

View Summary

We present explicit formulae for the remainder arising in the Young, Hölder, and Clarkson inequalities. © 2013 Kazumasa Fujiwara and Tohru Ozawa.

• Analytic smoothing effect for a system of nonlinear Schr&#246;dinger equations

G. Hoshino, T. Ozawa

Differential Equations and Applications - DEA   5   395 - 408  2013  [Refereed]

• Some inequalities related to the lorentz spaces

Shuji Machihara, Tohru Ozawa

Hokkaido Mathematical Journal   42 ( 2 ) 247 - 267  2013  [Refereed]

View Summary

In this paper we introduce three types of inequalities related to the Lorentz spaces on a measure space (M
m).

• Regularity criteria for a coupled Navier-Stokes and Q-tensor system

J. Fan, T. Ozawa

International Journal of Analysis   2013(2013), Article ID 718173, 5pages  2013  [Refereed]

• A note on the existence of a ground state solution to a fractional Schrödinger equation

Y. Cho, T. Ozawa

Kyushu J. Math.   67 ( 1 ) 227 - 236  2013  [Refereed]

• A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions

Soichiro Katayama, Tohru Ozawa, Hideaki Sunagawa

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS   65 ( 9 ) 1285 - 1302  2012.09  [Refereed]

View Summary

We consider the Cauchy problem for quadratic nonlinear Klein-Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.-M. Delort, D. Fang, and R. Xue (J. Funct. Anal. 211 (2004), no. 2, 288323), we show the global existence of asymptotically free solutions if the initial data are sufficiently small in some weighted Sobolev space. Our proof is based on an algebraic characterization of nonlinearities satisfying the null condition. (c) 2012 Wiley Periodicals, Inc.

• Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity

Jishan Fan, Tohru Ozawa

ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK   63 ( 3 ) 453 - 459  2012.06  [Refereed]

View Summary

We prove the uniqueness for weak solutions of the time-dependent 2-D Ginzburg-Landau model for superconductivity with L (2) initial data in the case of Coulomb gauge. This question was left open in Tang and Wang (Physica D, 88:139-166, 1995). We also prove the uniqueness of the 3-D radially symmetric solution in bounded annular domain with the choice of Lorentz gauge and L (2) initial data.

• Regularity criteria for hyperbolic Navier-Stokes and related system

J. Fan, T. Ozawa

ISRN Mathematical Analysis   2012(2012), Article ID 796368, 7pages  2012  [Refereed]

• Global strong solutions of the time-dependent Ginzburg-Landau model for superconductivity with a new gauge

J. Fan, T. Ozawa

International Journal of Mathematical Analysis   6   1679 - 1684  2012  [Refereed]

• Continuation criterion for the 2D liquid crystal flows

J. Fan, T. Ozawa

ISRN Mathematical Analysis   2012(2012), Article ID 248473, 7pages  2012  [Refereed]

• Uniqueness of weak solutions to the 3D Ginzburg-Landau model for superconductivity

J. Fan, T. Ozawa

International Journal of Mathematical Analysis   6   1095 - 1104  2012  [Refereed]

• LOCAL CAUCHY PROBLEM FOR THE MHD EQUATIONS WITH MASS DIFFUSION

Jishan Fan, Tohru Ozawa

DIFFERENTIAL AND INTEGRAL EQUATIONS   24 ( 11-12 ) 1037 - 1046  2011.11  [Refereed]

View Summary

This paper studies the Cauchy problem for the MHD equations with mass diffusion in a bounded domain in R(3). We use Tikhonov&apos;s fixed-point theorem to prove the existence and uniqueness of local solutions.

• Invariant elliptic estimates

Yonggeun Cho, Tohru Ozawa, Yong-Sun Shim

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   382 ( 1 ) 162 - 171  2011.10  [Refereed]

View Summary

In this paper, we revisit elliptic estimates invariant under domain expansion. We improve the invariant elliptic estimates in the previous paper [Y. Cho, T. Ozawa, Y. Shim, Calc. Var. PDE 34 (2009) 321-339] via gradient estimate and discuss an application to the Lame system. (C) 2011 Elsevier Inc. All rights reserved.

• GLOBAL CAUCHY PROBLEM OF AN IDEAL DENSITY-DEPENDENT MHD-alpha MODEL

Jishan Fan, Tohru Ozawa

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   31   400 - 409  2011.09  [Refereed]

View Summary

The global Cauchy problem for an approximation model for the ideal density-dependent MHD-alpha model is studied. The vanishing limit on alpha is also discussed.

• Life span of positive solutions for a semilinear heat equation with general non-decaying initial data

Tohru Ozawa, Yusuke Yamauchi

Journal of Mathematical Analysis and Applications   379 ( 2 ) 518 - 523  2011.07  [Refereed]

View Summary

We prove upper bounds on the life span of positive solutions for a semilinear heat equation. For non-decaying initial data, it is well known that the solutions blow up in finite time. We give two types estimates of the life span in terms of the limiting values of the initial data in space. © 2011 Elsevier Inc.

• REMARKS ON SOME DISPERSIVE ESTIMATES

Yonggeun Cho, Tohru Ozawa, Suxia Xia

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS   10 ( 4 ) 1121 - 1128  2011.07  [Refereed]

View Summary

In this paper we consider the initial value problem for i partial derivative(t)u + omega(vertical bar del vertical bar)u = 0. Under suitable smoothness and growth conditions on omega, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].

• On Hartree equations with derivatives

Yonggeun Cho, Sanghyuk Lee, Tohru Ozawa

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   74 ( 6 ) 2094 - 2108  2011.03  [Refereed]

View Summary

We consider the Cauchy problem of two types of Hartree equations with exchange-correlation correction terms:
{iu(t) - Delta u = V(k)(u)u in R(1+n), k = 1, 2,
u(0) = phi in R(n), n &gt;= 1,
where
V(1)(u) = vertical bar x vertical bar(-gamma) * (lambda(1)vertical bar u vertical bar(2) + lambda(2)vertical bar del u vertical bar(2)), V(2)(u) = vertical bar x vertical bar(-gamma) * (lambda vertical bar vertical bar del vertical bar(delta) u vertical bar(2)).
We establish the well- posedness of Cauchy problems and show the smoothing effect of solutions for each 0 &lt; gamma &lt; n and 0 &lt;= delta &lt;= 1. (C) 2010 Elsevier Ltd. All rights reserved.

• Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier- Stokes and generalized Boson equations

H. Hajaiej, L. Molinet, T. Ozawa, B. Wang

RIMS Kokyuroku Bessatsu   26   159 - 175  2011  [Refereed]

• Small data scattering for a system of nonlinear Schr&#246;dinger equations

N. Hayashi, C. Li, T. Ozawa

Differential Equations and Applications - DEA   3   415 - 426  2011  [Refereed]

• REGULARITY CRITERION FOR THE INCOMPRESSIBLE VISCOELASTIC FLUID SYSTEM

Jishan Fan, Tohru Ozawa

HOUSTON JOURNAL OF MATHEMATICS   37 ( 2 ) 627 - 636  2011  [Refereed]

View Summary

We consider the incompressible viscoelastic fluid system of the Oldroyd-B model. We prove a regularity criterion del v is an element of L(1)(0, T; L(infinity)) for the 3-D Oldroyd-B system with the initial data (v(0), H(0) - I) is an element of H(2) x W(1,q) with 3 &lt; q &lt;= 6. Global well-posedness of smooth solution is also proven for a regularization model of this Oldroyd-B system in two space dimension.

• Endpoint Strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications

Jun Kato, Tohru Ozawa

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES   95 ( 1 ) 48 - 71  2011.01  [Refereed]

View Summary

We prove the endpoint Strichartz estimates for the Klein-Gordon equation in mixed norms on the polar coordinates in two space dimensions. As an application, similar endpoint estimates for the Schrodinger equation in two space dimensions are shown by using the non-relativistic limit. The existence of global solutions for the cubic nonlinear Klein-Gordon equation in two space dimensions for small data is also shown. (C) 2010 Elsevier Masson SAS. All rights reserved.

• Global Cauchy Problem for the 2-D Magnetohydrodynamic-alpha Models with Partial Viscous Terms

Jishan Fan, Tohru Ozawa

JOURNAL OF MATHEMATICAL FLUID MECHANICS   12 ( 2 ) 306 - 319  2010.05  [Refereed]

View Summary

The global Cauchy problem for the 2-D magnetohydrodynamic-alpha models with partial viscous terms is studied. The vanishing limit on alpha is also considered in this paper.

• Analytic smoothing effect for global solutions to nonlinear Schrodinger equations

T. Ozawa, K. Yamauchi

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   364 ( 2 ) 492 - 497  2010.04  [Refereed]

View Summary

We prove the global existence of analytic solutions to the Cauchy problem for the cubic Schrodinger equation in space dimension n &gt;= 3 for sufficiently small data with exponential decay at infinity. Minimal regularity assumption regarding scaling invariance is imposed on the Cauchy data. (C) 2009 Elsevier Inc. All rights reserved.

• Global Cauchy problems of certain magnetohydrodynamic-&#945; models

J. Fan, T. Ozawa

Advances Appl. Math. Sci.   6   169 - 190  2010  [Refereed]

• On regularity criterion for the 2D wave maps and the 4D biharmonic wave maps

J. Fan, T. Ozawa

GAKUTO International Series, Math. Sci. Appl.   32   69 - 83  2010  [Refereed]

• Logarithmically improved regularity criteria for Navier-Stokes and related equations

Jishan Fan, Tohru Ozawa

MATHEMATICAL METHODS IN THE APPLIED SCIENCES   32 ( 17 ) 2309 - 2318  2009.11  [Refereed]

View Summary

We use an interpolation inequality on Besov spaces to show some logarithmically improved regularity criteria for Navier-Stokes equations, the harmonic heat flow, the Landau-Lifshitz equations, and the Landau-Lifshitz-Maxwell system. Copyright (C) 2009 John Wiley & Sons, Ltd.

• REGULARITY CRITERIA FOR A SIMPLIFIED ERICKSEN-LESLIE SYSTEM MODELING THE FLOW OF LIQUID CRYSTALS

Jishan Fan, Tohru Ozawa

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   25 ( 3 ) 859 - 867  2009.11  [Refereed]

View Summary

We consider the hydrodynamic theory of liquid crystals. We prove some regularity criteria for a simplified Ericksen-Leslie system. The existence and uniqueness of global smooth solutions is also proved for a regularization model of this simplified system.

• Regularity criterion for a Bona-Colin-Lannes system

Jishan Fan, Tohru Ozawa

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS   71 ( 7-8 ) 2634 - 2639  2009.10  [Refereed]

View Summary

We prove a regularity criterion of the strong solutions for a Bona-Colin-Lannes system in Besov space. As a byproduct, we show the existence of globally smooth solutions for a symmetric Bona-Colin-Lannes system. (c) 2009 Elsevier Ltd. All rights reserved.

• SOBOLEV INEQUALITIES WITH SYMMETRY

Yonggeun Cho, Tohru Ozawa

COMMUNICATIONS IN CONTEMPORARY MATHEMATICS   11 ( 3 ) 355 - 365  2009.06  [Refereed]

View Summary

In this paper, we derive some Sobolev inequalities for radially symmetric functions in. (H)over dot(s) with 1/2 &lt; s &lt; n/2. We show the end point case s = 1/2 on the homogeneous Besov space. (B)over dor(2,1)(1/2). These results are extensions of the well-known Strauss&apos; inequality [13]. Also non-radial extensions are given, which show that compact embeddings are possible in some L(p) spaces if a suitable angular regularity is imposed.

• REGULARITY CRITERIA FOR THE MAGNETOHYDRODYNAMIC EQUATIONS WITH PARTIAL VISCOUS TERMS AND THE LERAY-alpha-MHD MODEL

Jishan Fan, Tohru Ozawa

KINETIC AND RELATED MODELS   2 ( 2 ) 293 - 305  2009.06  [Refereed]

View Summary

We prove some regularity conditions for the MHD equations with partial viscous terms and the Leray-alpha-MHD model. Since the solutions to the Leray-alpha-MHD model are smoother than that of the original MHD equations, we are able to obtain better regularity conditions in terms of the magnetic field B only.

• INEQUALITIES ASSOCIATED WITH DILATIONS

Tohru Ozawa, Hironobu Sasaki

COMMUNICATIONS IN CONTEMPORARY MATHEMATICS   11 ( 2 ) 265 - 277  2009.04  [Refereed]

View Summary

Some properties of distributions f satisfying x . del f is an element of L(p)(R(n)), 1 &lt;= p &lt; infinity, are studied. The operator x . del is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x . del. Using the inequalities, we also show that if f is an element of L(loc)(p)(R(n)), x . del f is an element of L(p)(R(n)) and vertical bar x vertical bar(n/p)vertical bar f(x)vertical bar vanishes at infinity, then f belongs to L(p)( R(n)). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L(2)(R(n)).

• REMARKS ON THE SEMIRELATIVISTIC HARTREE EQUATIONS

Yonggeun Cho, Tohru Ozawa, Hironobu Sasaki, Yongsun Shim

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   23 ( 4 ) 1277 - 1294  2009.04  [Refereed]

View Summary

We study the global well-posedness (GWP) and small data scattering of radial solutions of the semirelativistic Hartree type equations with nonlocal nonlinearity F(u) = lambda(vertical bar u vertical bar(-gamma) * vertical bar u vertical bar(2)) u, lambda epsilon R \ {0}, 0 &lt; gamma &lt; n, n &gt;= 3. We establish a weighted L(2) Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.

• Regularity criteria for the 3D density-dependent Boussinesq equations

Jishan Fan, Tohru Ozawa

NONLINEARITY   22 ( 3 ) 553 - 568  2009.03  [Refereed]

View Summary

We consider the 3D density-dependent Boussinesq equations and the classical Boussinesq equations with partial viscosity terms. We prove some regularity criteria of strong solutions for the Boussinesq equations.

• Elliptic estimates independent of domain expansion

Yonggeun Cho, Tohru Ozawa, Yong-Sun Shim

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   34 ( 3 ) 321 - 339  2009.03  [Refereed]

View Summary

In this paper, we consider elliptic estimates for a system with smooth variable coefficients on a domain Omega subset of R(n), n &gt;= 2 containing the origin. We first show the invariance of the estimates under a domain expansion defined by the scale that y = Rx, x, y is an element of R(n) with parameter R &gt; 1, provided that the coefficients are in a homogeneous Sobolev space. Then we apply these invariant estimates to the global existence of unique strong solutions to a parabolic system defined on an unbounded domain.

• A remark on global solutions to nonlinear Klein-Gordon equation with a special quadratic nonlinearity in two space dimensions

J. Kato, T. Ozawa

RIMS Kokyuroku Bessatsu   14   17 - 25  2009  [Refereed]

• Remarks on analytic smoothing effect for the Schr&#246;dinger equation

T. Ozawa, K. Yamauchi

Math. Z.   261   511 - 524  2009  [Refereed]

• UNIQUENESS OF WEAK SOLUTIONS TO THE CAUCHY PROBLEM FOR THE 3-D TIME-DEPENDENT GINZBURG-LANDAU MODEL FOR SUPERCONDUCTIVITY

Jishan Fan, Tohru Ozawa

DIFFERENTIAL AND INTEGRAL EQUATIONS   22 ( 1-2 ) 27 - 34  2009.01  [Refereed]

View Summary

We prove some uniqueness results for the Cauchy problem for the 3-D time-dependent Ginzburg-Landau (TDGL) model for superconductivity with the choice of the Lorentz gauge in the multiplier spaces (Morrey spaces) and in the inhomogeneous Besov spaces, respectively.

• Nonlinear Schrodinger equation with a point defect

Reika Fukuizumi, Masahito Ohta, Tohru Ozawa

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   25 ( 5 ) 837 - 845  2008.09  [Refereed]

View Summary

We study the nonlinear Schrodinger equation with a delta-function impurity in one space dimension. Local well-posedness is verified for the Cauchy problem in H-1(R). In case of attractive delta-function, orbital stability and instability of the ground state is proved in H-1(R). (C) 2007 Elsevier Masson SAS. All fights reserved.

• REGULARITY CRITERIA FOR THE GENERALIZED NAVIER-STOKES AND RELATED EQUATIONS

Jishan Fan, Tohru Ozawa

DIFFERENTIAL AND INTEGRAL EQUATIONS   21 ( 7-8 ) 681 - 691  2008.07  [Refereed]

View Summary

We use the maximum principle type estimate and interpolation inequality on Besov spaces to show some regularity criteria for the generalized Navier-Stokes equations, the quasi-geostrophic equations, and the harmonic heat flow.

• Asymptotic stability for the Navier-Stokes equations

Jishan Fan, Tohru Ozawa

JOURNAL OF EVOLUTION EQUATIONS   8 ( 2 ) 379 - 389  2008.05  [Refereed]

View Summary

We prove the asymptotic stability for weak solutions to the 3-D Navier-Stokes equations in the class
[GRAPHICS]
with arbitrary initial and external perturbations. This solves a problem due to Yong Zhou (Proc. Roy. Soc. Edinburgh, 136A ( 2006), 1099-1109).

• ON THE REGULARITY CRITERIA FOR THE GENERALIZED NAVIER-STOKES EQUATIONS AND LAGRANGIAN AVERAGED EULER EQUATIONS

Jishan Fan, Tohru Ozawa

DIFFERENTIAL AND INTEGRAL EQUATIONS   21 ( 5-6 ) 443 - 457  2008.05  [Refereed]

View Summary

We obtain some regularity conditions for solutions of the 3D generalized Navier-Stokes equations with fractional powers of the Laplacian, in terms of the velocity, the vorticity, and the pressure in Besov space, Triebel-Lizorkin space, and Lorentz space, respectively. We also present a regularity condition for the 3D Lagrangian averaged Euler equations.

• On radial solutions of semi-relativistic Hartree equations

Yonggeun Cho, Tohru Ozawa

Discrete and Continuous Dynamical Systems - Series S   1 ( 1 ) 71 - 82  2008.03  [Refereed]

View Summary

We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity F(u) = λ(|x| -λ * |u| 2)u,0 &lt
γ &lt
n,n ≥ 1. In [2, 3], the global well-posedness (GWP) was shown for the value of γ ∈ (0, 2n/n+1),n ≥ 2 with large data and γ ∈ (2, n), n ≥ 3 with small data. In this paper" we extend the previous GWP result to the case for γ ∈ (1, 2n-1/n),n ≥ 2 with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.

• Regularity Criterion for Weak Solutions to the Navier-Stokes Equations in Terms of the Gradient of the Pressure

Jishan Fan, Tohru Ozawa

JOURNAL OF INEQUALITIES AND APPLICATIONS   2008(2008), Article ID 412678, 6 pages  2008  [Refereed]

View Summary

We prove a regularity criterion. del pi is an element of L(2/3) (0,T; BMO) for weak solutions to the Navier-Stokes equations in three-space dimensions. This improves the available result with L(2/3) (0,T; L(infinity)). Copyright (C) 2008 J. Fan and T. Ozawa.

• A Poisson formula for the Schrodinger operator

Remi Carles, Tohru Ozawa

JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS   14 ( 3 ) 475 - 483  2008  [Refereed]

View Summary

We prove a Poisson type formula for the Schrodinger group. Such a formula had been derived in a previous article by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H-1-critical nonlinearities are allowed.

• On the wave operators for the critical nonlinear Schrodinger equation

Remi Carles, Tohru Ozawa

MATHEMATICAL RESEARCH LETTERS   15 ( 1 ) 185 - 195  2008.01  [Refereed]

View Summary

We prove that for the L-2-critical nonlinear Schrodinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one-dimensional case, we show a precise similarity between the L-2-critical nonlinear Schrodinger equation and a nonlinear Schrodinger equation of derivative type.

• Global existence of small classical solutions to nonlinear Schrodinger equations

Tohru Ozawa, Jian Zhai

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE   25 ( 2 ) 303 - 311  2008  [Refereed]

View Summary

We study the global Cauchy problem for nonlinear Schrodinger equations with cubic interactions of derivative type in space dimension n &gt;= 3. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates. (C) 2007 Elsevier Masson SAS. All rights reserved.

• Global solutions of semirelativistic hartree type equations

Yonggeun Cho, Tohru Ozawa

JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY   44 ( 5 ) 1065 - 1078  2007.09  [Refereed]

View Summary

We consider initial value problems for the semirelativistic Hartree type equations with cubic convolution nonlinearity F(u) = (V (*) vertical bar U vertical bar(2))u. Here V is a sum of two Coulomb type potentials. Under a specified decay condition and a symmetric condition for the potential V we show the global existence and scattering of solutions.

• On small amplitude solutions to the generalized Boussinesq equations

Yonggeun Cho, Tohru Ozawa

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS   17 ( 4 ) 691 - 711  2007.04  [Refereed]

View Summary

We study the existence and scattering of global small amplitude solutions to generalized Boussinesq (Bq) and improved modified Boussinesq (imBq) equations with nonlinear term f(u) behaving as a power u(p) as u --&gt; 0 in R-n, n &gt;= 1.

• A generalization of the weighted Strichartz estimates for wave equations and an application to self-similar solutions

Jun Kato, Makoto Nakamura, Tohru Ozawa

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS   60 ( 2 ) 164 - 186  2007.02  [Refereed]

View Summary

Weighted Strichartz estimates with homogeneous weights with critical exponents are proved for the wave equation without a support restriction on the forcing term. The method of proof is based on expansion by spherical harmonics and on the Sobolev space over the unit sphere, by which the required estimates are reduced to the radial case. As an application of the weighted Strichartz estimates, the existence and uniqueness of self-similar solutions to nonlinear wave equations are proved on up to five space dimensions. (c) 2006 Wiley Periodicals, Inc.

• On the Cauchy problem for Schr&#246;dinger improved Boussinesq equations

K. Tsutaya, T. Ozawa

Advanced Studies in Pure Math.   47   291 - 301  2007  [Refereed]

• Remarks on modified improved Boussinesq equations in one space dimension

Yonggeun Cho, Tohru Ozawa

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES   462 ( 2071 ) 1949 - 1963  2006.07  [Refereed]

View Summary

We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term f(u) behaving as a power u(p) as u -&gt; 0. Solutions are considered in H-s space for all s &gt; 1/2. According to the value of s, the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J. 45, 797-816) obtained the minimum value of p greater than 8 at s = 3/2 for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than 9/2 at s &gt; 17/10 and the corresponding solution u has the time decay, such as parallel to u(t)parallel to(infinity)(L) = O(t(-2/5)) as t -&gt; infinity. We also prove non-existence of non-trivial asymptotically free solutions for 1 &lt; p &lt;= 2 under vanishing condition near zero frequency on asymptotic states.

• Remarks on proofs of conservation laws for nonlinear Schrodinger equations - Dedicated to professor Nakao Hayashi on his fiftieth birthday

T Ozawa

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS   25 ( 3 ) 403 - 408  2006.03  [Refereed]

View Summary

Conservation laws of the charge and of the energy are proved for nonlinear Schrodinger equations with nonlinearities of gauge invariance in a way independent of approximate solutions.

• On a decay property of solutions to the Haraux-Weissler equation

R Fukuizumi, T Ozawa

JOURNAL OF DIFFERENTIAL EQUATIONS   221 ( 1 ) 134 - 142  2006.02  [Refereed]

View Summary

We give a sufficient condition that non-radial H-1-Solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space H-rho(1)(R-n), where rho is the weight function exp(vertical bar x vertical bar(2)/4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space H-rho(1)(R-n). (c) 2005 Elsevier Inc. All rights reserved.

• Global existence on nonlinear Schrodinger-IMBq equations

Yonggeun Cho, Tohru Ozawa

JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   46 ( 3 ) 535 - 552  2006  [Refereed]

View Summary

In this paper, we consider the Cauchy problem of Schrodinger-IMBq equations in R-n, n &gt;= 1. We first show the global existence and blowup criterion of solutions in the energy space for the 3 and 4 dimensional system without power nonlinearity under suitable smallness assumption. Secondly the global existence is established to the system with p-powered nonlinearity in H-s(R-n), n = 1, 2 for some n/2 &lt; s &lt; min (2, p) and some p &gt; n/2. We also provide a blowup criterion for n = 3 in Triebel-Lizorkin space containing BMO space naturally.

• On the semirelativistic Hartree-type equation

Yonggeun Cho, Tohru Ozawa

SIAM JOURNAL ON MATHEMATICAL ANALYSIS   38 ( 4 ) 1060 - 1074  2006  [Refereed]

View Summary

We study the global Cauchy problem and scattering problem for the semirelativistic Hartree-type equation in R-n, n = 1, with nonlocal nonlinearity F(u) = lambda(vertical bar x vertical bar(-gamma) * vertical bar u vertical bar(2)) u, 0 &lt; gamma &lt; n. We prove the existence and uniqueness of global solutions for 0 &lt; gamma &lt; (2n)/(n + 1), n &gt;= 2 or gamma &gt; 2, n &gt;= 3, and the nonexistence of asymptotically-free solutions for 0 &lt; gamma &lt;= 1, n &gt;= 3. We also specify asymptotic behavior of solutions as the mass tends to zero and in. nity.

• Analytic smoothing effect for solutions to Schrodinger equations with nonlinearity of integral type

T Ozawa, K Yamauchi, Y Yamazaki

OSAKA JOURNAL OF MATHEMATICS   42 ( 4 ) 737 - 750  2005.12  [Refereed]

View Summary

We study analytic smoothing effects for Solutions to the Cauchy problem for the Schrodinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. The only assumption oil the Cauchy data is the weight condition of exponential type and no regularity assumption is imposed.

• Exponential decay of solutions to nonlinear elliptic equations with potentials

R Fukuizumi, T Ozawa

ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK   56 ( 6 ) 1000 - 1011  2005.11  [Refereed]

View Summary

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in R-n, where the linear term is given by Schrodinger operators H = -Delta + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V (x) = |x|(2).

• Sharp asymptotic behavior of solutions to nonlinear Schrodinger equations with repulsive interactions

N Kita, T Ozawa

COMMUNICATIONS IN CONTEMPORARY MATHEMATICS   7 ( 2 ) 167 - 176  2005.04  [Refereed]

View Summary

A detailed description is given on the large time behavior of scattering solutions to the Cauchy problem for nonlinear Schrodinger equations with repulsive interactions in the short-range case without smallness condition on the data.

• Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation

S Machihara, M Nakamura, K Nakanishi, T Ozawa

JOURNAL OF FUNCTIONAL ANALYSIS   219 ( 1 ) 1 - 20  2005.02  [Refereed]

View Summary

We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposed-ness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable. (C) 2004 Elsevier Inc. All fights reserved.

• STRUCTURE OF DIRAC MATRICES AND INVARIANTS FOR NONLINEAR DIRAC EQUATIONS

Tohru Ozawa, Kazuyuki Yamauchi

DIFFERENTIAL AND INTEGRAL EQUATIONS   17 ( 9-10 ) 971 - 982  2004.09  [Refereed]

View Summary

We present invariants for nonlinear Dirac equations in space-time Rn+1, by which we prove that a special choice of the Cauchy data yields free solutions. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. Related problems on the structure of Dirac matrices are studied.

• Smoothing effect and large time behavior of solutions to Schrodinger equations with nonlinearity of integral type

T Ozawa, Y Yamazaki

COMMUNICATIONS IN CONTEMPORARY MATHEMATICS   6 ( 4 ) 681 - 703  2004.08  [Refereed]

View Summary

We study the smoothing effect in space and asymptotic behavior in time of solutions to the Cauchy problem for the nonlinear Schrodinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. A detailed description is given on the phase modification of scattering solutions by taking into account the long range effect of the interaction.

• Weighted Strichartz estimates for the wave equation in even space dimensions

J Kato, T Ozawa

MATHEMATISCHE ZEITSCHRIFT   247 ( 4 ) 747 - 764  2004.08  [Refereed]

View Summary

We prove the weighted Strichartz estimates for the wave equation in even space dimensions with radial symmetry in space. Although the odd space dimensional cases have been treated in our previous paper [5], the lack of the Huygens principle prevents us from a similar treatment in even space dimensions. The proof is based on the two explicit representations of solutions due to Rammaha [11] and Takamura [14] and to Kubo-Kubota [6]. As in the odd space dimensional cases [5], we are also able to construct self-similar solutions to semilinear wave equations on the basis of the weighted Strichartz estimates.

• SMALL GLOBAL SOLUTIONS FOR NONLINEAR DIRAC EQUATIONS

Shuji Machihara, Makoto Nakamura, Tohru Ozawa

DIFFERENTIAL AND INTEGRAL EQUATIONS   17 ( 5-6 ) 623 - 636  2004.05  [Refereed]

View Summary

The global Cauchy problem for nonlinear Dirac and Klein-Gordon equations in space-time Rn+1 is studied in Sobolev and Besov spaces. Global existence of small solutions is proved under a scale-invariant setting when reduced to the corresponding massless case.

• On solutions of the wave equation with homogeneous Cauchy data

J Kato, T Ozawa

ASYMPTOTIC ANALYSIS   37 ( 2 ) 93 - 107  2004.02  [Refereed]

View Summary

In this article, the behavior of solutions to the free wave equation with homogeneous Cauchy data are considered. In particular, the propagation of singularities are observed explicitly. Such Cauchy data are of special interest in view of applications to self-similar solutions to nonlinear wave equations.

• Life-span of smooth solutions to the complex Ginzburg-Landau type equation on a torus

T Ozawa, Y Yamazaki

NONLINEARITY   16 ( 6 ) 2029 - 2034  2003.11  [Refereed]

View Summary

An upper bound of the life-span of smooth solutions to the complex Ginzburg-Landau equation with periodic boundary condition in one space dimension is given explicitly in terms of an integral mean of the Cauchy data in the case where the interaction is focusing.

• Weighted Strichartz estimates and existence of self-similar solutions for semilinear wave equations

J Kato, T Ozawa

INDIANA UNIVERSITY MATHEMATICS JOURNAL   52 ( 6 ) 1615 - 1630  2003  [Refereed]

View Summary

We study the existence of self-similar solutions to the Cauchy problem for semilinear wave equations with power type nonlinearity. Radially symmetric self-similar solutions are obtained in odd space dimensions when the power is greater than the critical one that are widely referred to in other existence problems of global solutions to nonlinear wave equations with small data. This result is a partial generalization of [ 11] to odd space dimensions. To construct self-similar solutions, we prove the weighted Strichartz estimates in terms of weak Lebesgue spaces over space-time.

• Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation

S Machihara, K Nakanishi, T Ozawa

REVISTA MATEMATICA IBEROAMERICANA   19 ( 1 ) 179 - 194  2003  [Refereed]

View Summary

In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space H-s. We prove the existence and uniqueness of global solutions for small data in H-s with s &gt; 1. The method of proof is based on the Strichartz estimate of L-t(2) type for Dirac and Klein-Gordon equations. We also prove that the solutions of the nonlinear Dirac equation after modulation of phase converge to the corresponding solutions of the nonlinear Schrodinger equation as the speed of light tends to infinity.

• Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations

S Machihara, K Nakanishi, T Ozawa

MATHEMATISCHE ANNALEN   322 ( 3 ) 603 - 621  2002.03  [Refereed]

View Summary

We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schrodinger equation in the energy space, after the infinite oscillation in time is removed. We also derive the optimal rate of convergence in L-2.

• Interpolation inequalities in Besov spaces

S. Machihara, T. Ozawa

Proc. AMS   131   1553 - 1556  2002  [Refereed]

• Small Data Scattering for Nonlinear Schrodinger Wave and Klein-Gordon Equations

Makoto Nakamura, Tohru Ozawa

ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE   1 ( 2 ) 435 - 460  2002  [Refereed]

View Summary

Small data scattering for nonlinear Schrodinger equations (NLS), nonlinear wave equations (NLW), nonlinear Klein-Gordon equations (NLKG) with power type nonlinearities is studied in the scheme of Sobolev spaces on the whole space R(n) with order s &lt; n/2. The assumptions on the nonlinearities are described in terms of power behavior p(1) at zero and p(2) at infinity such as 1 + 4/n &lt;= p(1) &lt;= p(2) &lt; 1 + 4/(n - 2s) for NLS and NLKG, and 1 + 4/(n - 1) &lt;= p1 &lt;= p(2) &lt;= 1 + 4/(n - 2s) for NLW.

• Global solutions for nonlinear schrödinger equations with arbitrarily growing nonlinearity and contracted initial data

Kenji Nakanishi, Tohru Ozawa

Kyushu Journal of Mathematics   56 ( 1 ) 221 - 224  2002  [Refereed]

• Remarks on scattering for nonlinear Schrodinger equations

K Nakanishi, T Ozawa

NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS   9 ( 1 ) 45 - 68  2002  [Refereed]

View Summary

We unify two distinct methods of the global analysis for the nonlinear Schrodinger equations, namely those in the Sobolev spaces and in the weighted spaces. Thus we can deal with various sums of power nonlinearies \u$$p-1)u for 1 + 2/n &lt; p &lt; infinity, since the former works for p greater than or equal to 1 + 4/n, while the latter for 1 + 2/n &lt; p less than or equal to 1 + 4/n. Even for a single power, our result is much simpler and slightly better than the previous ones as to restriction on the initial data. Moreover, we extend the result to the maximal regularity, thereby obtaining scattering at the lower critical value p = 1 + 8/(rootn(2) + 4n + 36 + n + 2) for n greater than or equal to 4. We also show the asymptotic completeness in FH1 without smallness for p greater than or equal to 1+8/(rootn(2) + 12n + 4 + n - 2) and any n is an element of N. • The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces M Nakamura, T Ozawa PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 37 ( 3 ) 255 - 293 2001.11 [Refereed] View Summary The local and global well-posedness for the Cauchy problem for a class of non-linear Klein-Gordon equations is studied in the Sobolev space H-s = H-s(R-n) with s greater than or equal to n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f (u) behaves as a power u(1+4/n) near zero. At infinity f (u) has an exponential growth rate such as exp(kappa\u\(v)) with kappa &gt; 0 and 0 &lt; v &lt; 2 if s = n/2, and has an arbitrary growth rate if s &gt; n/2. (2) Concerning the Cauchy data (phi, psi) is an element of H-s + Hs-1, parallel to(phi, omega); H(1/2)parallel to is relatively small with respect to parallel to(phi, psi); (H) over dot (s*) parallel to, where s* is a number with s* = n/2 if s = n/2, n/2 &lt; s* &LE; s if s &gt; n/2, and the smallness of parallel to(phi, psi); (H) over dot (n/2)parallel to is also needed when s = n/2 and v = 2. • Small solutions to nonlinear wave equations in the Sobolev spaces M Nakamura, T Ozawa HOUSTON JOURNAL OF MATHEMATICS 27 ( 3 ) 613 - 632 2001 [Refereed] View Summary The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space (H) over dot(s) = (H) over dot(s) (R-n) of fractional order s &gt; n/2 under the following assumptions: (1) Concerning the Cauchy data (phi, psi) is an element of (H) over dot(s) = (H) over dot(s) + (H) over dot(s-1), parallel to(phi, psi); (H) over dot(1/2)parallel to is relatively small with respect to parallel to(phi, psi); (H)over dot(sigma)parallel to for any fixed sigma with n/2 &lt; &sigma; &LE; s. (2) Concerning the nonlinearity f, f(u) behaves as a power u(1+4/(n-1)) near zero and has an arbitrary growth rate at infinity. • On the coupled system of nonlinear wave equations with different propagation speeds T. Ozawa, K. Tsutaya, Y. Tsutsumi Banach Center Publications 52 181 - 188 2000 [Refereed] • Small solutions to nonlinear Schrodinger equations in the Sobolev spaces M Nakamura, T Ozawa JOURNAL D ANALYSE MATHEMATIQUE 81 305 - 329 2000 [Refereed] View Summary The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrodinger equations is studied. The global well-posedness of the problem is proved in the Sobolev space H-s = H-s(R-n) of fractional order s &gt; n/2 under the following assumptions. (1) Concerning the Cauchy data phi is an element of H-s: parallel to phi; L(2)parallel to is relatively small with respect to parallel to phi; (H)over dot(sigma) for any fixed sigma with n/2 &lt; sigma &lt; s. (2) Concerning the nonlinearity f: f(u) behaves as a conformal power u(1+4/n) near zero and has an arbitrary growth rate at infinity. • The Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space M Nakamura, T Ozawa ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE 71 ( 2 ) 199 - 215 1999.08 [Refereed] View Summary We consider the Cauchy problem for nonlinear wave equations in the homogeneous Sobolev space (H) over dot (mu)(R-n), where n greater than or equal to 2 and 0 less than or equal to mu &lt; n/2 using the generalized Strichartz estimates given by J. Ginibre and G. Velo (1995). (C) Elsevier, Paris. • Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth M Nakamura, T Ozawa MATHEMATISCHE ZEITSCHRIFT 231 ( 3 ) 479 - 487 1999.07 [Refereed] • Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions T Ozawa, K Tsutaya, Y Tsutsumi MATHEMATISCHE ANNALEN 313 ( 1 ) 127 - 140 1999.01 [Refereed] • The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order M Nakamura, T Ozawa DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 5 ( 1 ) 215 - 231 1999.01 [Refereed] View Summary We show the local in time solvability of the Cauchy problem for nonlinear wave equations in the Sobolev space of critical order with nonlinear term of exponential type. • Scattering theory for the Hartree equation N Hayashi, PI Naumkin, T Ozawa SIAM JOURNAL ON MATHEMATICAL ANALYSIS 29 ( 5 ) 1256 - 1267 1998.09 [Refereed] View Summary We study the scattering problem for the Hartree equation i partial derivative(t)u = ?1/2 Delta u + f(\u\(2))u, (t, x) is an element of R x R-n with initial data u(0, x) = u(0)(x), x is an element of R-n, where f(\u\(2)) = V * \u\(2), V(x) = lambda\x\(?1), lambda is an element of R, n greater than or equal to 2. We prove that for any u(0) is an element of H-0,H- (gamma) boolean AND H-gamma,H- 0, with 1/2 &lt; gamma &lt; n/2, such that the value epsilon = \\u(0)\\(0, gamma) + \\u(0)\\(gamma,) (0) is sufficiently small, there exist unique u(+/-) is an element of H-sigma,H- (0) boolean AND H-0,H- sigma with 1/2 &lt; sigma &lt; gamma such that for all \t\ greater than or equal to 1 \\u(t) ? exp (-/+ if (\(u) over cap(+/-)\(2)) (x/t) log \t$$ U(t)u(+/-)\$$L2) less than or equal to C epsilon\t\(-mu+7 nu)(,) where mu = min(1, gamma/2), 0 &lt; nu &lt; min(1, gamma-sigma/12), &lt;(phi)over cap&gt; denotes the Fourier transform of phi, U(t) is the free Schrodinger evolution group, and H-m,H- s is the weighted Sobolev space defined by H-m,H- (s) = {phi is an element of S'; \\phi\\(m,) (s) = \\(1 + \x\(2))(s/2) (1 ? Delta)(m/2) phi\\(L2) &lt; infinity}. • Nonlinear Schrodinger equations in the Sobolev space of critical order M Nakamura, T Ozawa JOURNAL OF FUNCTIONAL ANALYSIS 155 ( 2 ) 364 - 380 1998.06 [Refereed] View Summary The Cauchy problem for the nonlinear Schrodinger equations is considered in the Sobolev space H-n/2(R-n) of critical order n/2, where the embedding into L-infinity(R-n) breaks down and any power behavior of interaction works as a subcritical nonlinearity. Under the interaction of exponential type the existence and uniqueness is proved far global H-n/2-solutions with small Cauchy data. (C) 1998 Academic Press. • Finite energy solutions for the Schr&#246;dinger equations with quadratic nonlinearity in one space dimension Tohru Ozawa Funkcialaj Ekvacioj 41 451 - 468 1998 [Refereed] • Space-time estimates for null gauge forms and nonlinear Schr&#246;dinger equations T. Ozawa, Y. Tsutsumi Differential and Integral Eqs. 11 201 - 222 1998 [Refereed] • Low energy scattering for nonlinear Schrodinger equations in fractional order Sobolev spaces M Nakamura, T Ozawa REVIEWS IN MATHEMATICAL PHYSICS 9 ( 3 ) 397 - 410 1997.04 [Refereed] View Summary We consider the scattering problem for the nonlinear Schrodinger equations with interactions behaving as a power p at zero. In the critical and subcritical cases (s greater than or equal to n/2-2/(p-1) greater than or equal to 0). we prove the existence and asymptotic completeness of wave operators in the sense of Sobolev norm of order s on a set of asymptotic states with small homogeneous norm of order n/2 - 2/(p - 1) in space dimension n greater than or equal to 1. • Characterization of Trudinger's inequality T Ozawa JOURNAL OF INEQUALITIES AND APPLICATIONS 1 ( 4 ) 369 - 374 1997 [Refereed] View Summary A characterization of a sharp form of Trudinger's inequality is established in terms of the Gagliardo-Nirenberg inequality in the limiting case for Sobolev's imbeddings. • Remarks on the Klein-Gordon equation with quadratic nonlinearity in two space dimensions T. Ozawa, K. Tsutaya, Y. Tsutsumi GAKUTO International Series, Math. Sci. Appl. 10 383 - 392 1997 [Refereed] • Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions T Ozawa, K Tsutaya, Y Tsutsumi MATHEMATISCHE ZEITSCHRIFT 222 ( 3 ) 341 - 362 1996.07 [Refereed] • On the nonlinear Schr&#246;dinger equations of derivative type Tohru Ozawa Indiana Univ. Math. J. 45 137 - 163 1996 [Refereed] • Dilation method and smoothing effects of solutions to the Benjamin-One equation N Hayashi, K Kato, T Ozawa PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 126 273 - 285 1996 [Refereed] View Summary In this paper we study smoothing effects of solutions to the Benjamin-One equation [GRAPHICS] where H is the Hilbert transform defined by Hf)(x) = p.v. 1/pi integral f(y)/x-y dy. We prove that if phi is an element of H-4 and (x partial derivative(x))(4) phi, then the solution u of(BO) belongs to L(loc)(infinity)(R\{0}; H-8,H--4), where H-m,H-s = {f is an element of L(2.),parallel to(1+x(2))(s/2)(1 - partial derivative(x)(2))(m/2) f parallel to L(2) &lt; infinity}. • DILATION METHOD AND SMOOTHING EFFECT OF THE SCHRODINGER EVOLUTION GROUP N HAYASHI, K KATO, T OZAWA REVIEWS IN MATHEMATICAL PHYSICS 7 ( 7 ) 1123 - 1132 1995.10 [Refereed] View Summary We reexamine the mechanism of smoothing effects of the Schrodinger evolution group in the weighted Sobolev spaces by using the generator of space-time dilations instead of Galilei transformations. • ON CRITICAL CASES OF SOBOLEV INEQUALITIES T OZAWA JOURNAL OF FUNCTIONAL ANALYSIS 127 ( 2 ) 259 - 269 1995.02 [Refereed] View Summary We present a new form of the Trudinger-type inequality, which shows an explicit dependence. Moreover, we give an alternative proof of the Brezis-Gallouet-Wainger inequality. (C) 1995 Academic Press, Inc. • Schr&#246;dinger equations with nonlinearity of integral type, N. Hayashi, T. Ozawa Discrete and Continuous Dynamical Systems 1 475 - 484 1995 [Refereed] • Global, small radially symmetric solutions to nonlinear Schr&#246;dinger equations and a gauge transformation N. Hayashi, T. Ozawa Differential and Integral Eqs. 8 1061 - 1072 1995 [Refereed] • NORMAL-FORM AND GLOBAL-SOLUTIONS FOR THE KLEIN-GORDON-ZAKHAROV EQUATIONS T OZAWA, K TSUTAYA, Y TSUTSUMI ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE 12 ( 4 ) 459 - 503 1995 [Refereed] View Summary In this paper we study the global existence and asymptotic behavior of solutions for the Cauchy problem of the Klein-Gordon-Zakharov equations in three space dimensions. We prove that for small initial data, there exist the unique global solutions of the Klein- Gordon-Zakharov equations. We also show that these solutions approach asymptotically the free solutions as t --&gt; infinity. Our proof is based on the method of normal forms introduced by Shatah [12], which transforms the original system with quadratic nonlinearity into a new system with cubic nonlinearity. • Remarks on quadratic nonlinear Schr&#246;dinger equations Tohru Ozawa Funkcialaj Ekvacioj 38 217 - 232 1995 [Refereed] • FINITE-ENERGY SOLUTIONS OF NONLINEAR SCHRODINGER-EQUATIONS OF DERIVATIVE TYPE N HAYASHI, T OZAWA SIAM JOURNAL ON MATHEMATICAL ANALYSIS 25 ( 6 ) 1488 - 1503 1994.11 [Refereed] View Summary This paper is concerned with the initial value problem for nonlinear Schrodinger equations of the form [GRAPHICS] where partial-derivative = partial-derivative(x) = partial-derivative/partial-derivativex, lambda, lambda1, lambda2, is-an-element-of R and 2 less-than-or-equal-to p1 &lt; p2 &lt; 5. It is shown that if phi is-an-element-of H1(R) and parallel-tophiparallel-to2(2) &lt; 2pi/\lambda\, then there exists a unique global solution psi of (dagger) such that psi is-an-element-of C(R; H1(R)). This paper introduces a new method to obtain the result. • MODIFIED WAVE-OPERATORS FOR THE DERIVATIVE NONLINEAR SCHRODINGER-EQUATION N HAYASHI, T OZAWA MATHEMATISCHE ANNALEN 298 ( 3 ) 557 - 576 1994.03 [Refereed] • NORMAL-FORM AND GLOBAL-SOLUTIONS FOR THE KLEIN-GORDON-ZAKHAROV EQUATIONS T OZAWA, K TSUTAYA, Y TSUTSUMI SPECTRAL AND SCATTERING THEORY 161 153 - 179 1994 [Refereed] • LOCAL DECAY-ESTIMATES FOR SCHRODINGER-OPERATORS WITH LONG-RANGE POTENTIALS T OZAWA ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE 61 ( 2 ) 135 - 151 1994 [Refereed] View Summary For a class of long range potentials, sharp propagation estimates of the corresponding Schrodinger evolution groups are obtained without low-energy cut-off technique. Instead of low-energy cut-off, an explicit condition is given on the vanishing order in the L(2) sense at zero energy of initial states. • Wave propagation in even-dimensional spaces T. Ozawa Asymptotic Analysis 9 ( 2 ) 163 - 176 1994 [Refereed] View Summary Asymptotic expansions of solutions of the wave equations in even dimensional spaces are obtained with the initial data of non-compact support. A relationship is proved between the vanishing order at the origin of the Fourier transform of the data and the decay rate of the corresponding solutions in semi-infinite cylinders or along rays inside the forward light cone. © 1994 IOS Press and the authors. • ON THE EXISTENCE OF THE WAVE-OPERATORS FOR A CLASS OF NONLINEAR SCHRODINGER-EQUATIONS J GINIBRE, T OZAWA, G VELO ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE 60 ( 2 ) 211 - 239 1994 [Refereed] View Summary We study the wave operators for nonlinear Schrodinger equations with interactions behaving as a power p at zero. We extend the existence proof of those operators from the previously known range p - 1 &gt; 4/(n + 2) to the optimal range p - 1 &gt; 2/n in space dimension n = 3 and to an intermediate range in space dimension n greater-than-or-equal-to 4. • Remarks on nonlinear Schr&#246;dinger equations in one space dimension N. Hayashi, T. Ozawa Differential and Integral Eqs. 7 453 - 461 1994 [Refereed] • Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions T. Ozawa, Y. Tsutsumi Adv. Math. Sci. Appl. 3 301 - 334 1994 [Refereed] • EXISTENCE AND NONEXISTENCE RESULTS FOR WAVE-OPERATORS FOR PERTURBATIONS OF THE LAPLACIAN A JENSEN, T OZAWA REVIEWS IN MATHEMATICAL PHYSICS 5 ( 3 ) 601 - 629 1993.09 [Refereed] View Summary Schrodinger operators with time-dependent potentials are studied. Necessary and sufficient conditions for existence of ordinary and Dollard-type modified wave operators are obtained. Sharp results for potentials with a specified leading term are obtained. Applications are given to the surfboard Schrodinger equation and to Stark Hamiltonians. In the latter case the discrepancy between classical and quantum scattering in dimension one is resolved. • LONG-RANGE SCATTERING FOR NONLINEAR SCHRODINGER AND HARTREE-EQUATIONS IN SPACE DIMENSION N-GREATER-THAN-OR-EQUAL-TO-2 J GINIBRE, T OZAWA COMMUNICATIONS IN MATHEMATICAL PHYSICS 151 ( 3 ) 619 - 645 1993.02 [Refereed] View Summary We consider the scattering problem for the non-linear Schrodinger (NLS) equation with a power interaction with critical power p = 1 + 2/n in space dimensions n = 2 and 3 and for the Hartree equation with potential \x\-1 in space dimension n greater-than-or-equal-to 2. We prove the existence of modified wave operators in the L2 sense on a dense set of small and sufficiently regular asymptotic states. • Asymptotic behavior of solutions for the coupled Klein-Gordon- Schr&#246;dinger equations T. Ozawa, Y. Tsutsumi Advanced Studies in Pure Math. 23 295 - 305 1993 [Refereed] • EXISTENCE AND SMOOTHING EFFECT OF SOLUTIONS FOR THE ZAKHAROV EQUATIONS T OZAWA, Y TSUTSUMI PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 28 ( 3 ) 329 - 361 1992.10 [Refereed] • NONLINEAR SCATTERING WITH NONLOCAL INTERACTION H NAWA, T OZAWA COMMUNICATIONS IN MATHEMATICAL PHYSICS 146 ( 2 ) 259 - 275 1992.05 [Refereed] View Summary We consider the scattering problem for the Hartree type equation in R(n) with n greater-than-or-equal-to 2: i partial derivative u/partial derivative t + 1/2 DELTA-u = (V*\u\2)u, where V(x) = SIGMA(j = 1)2 lambda(j)\x\-gamma-j, (lambda-1, lambda-2) not-equal (0, 0), lambda(j) is-an-element-of R, gamma(j) &gt; 0, and * denotes the convolution in R(n). We prove the existence of wave operators in H0,k = {psi is-an-element-of L2(R(n)); \x\(k)psi is-an-element-of L2(R(n))} for any positive integer k under the assumption 1 &lt; gamma-1, gamma-2 &lt; 2. This is an optimal result in the sense that the existence of wave operators breaks down if min (gamma-1, gamma-2) less-than-or-equal-to 1. The case where 1 &lt; gamma-1 &lt; gamma-2 = 2 is also treated according to the sign of lambda-2. • EXACT BLOW-UP SOLUTIONS TO THE CAUCHY-PROBLEM FOR THE DAVEY-STEWARTSON SYSTEMS T OZAWA PROCEEDINGS OF THE ROYAL SOCIETY-MATHEMATICAL AND PHYSICAL SCIENCES 436 ( 1897 ) 345 - 349 1992.02 [Refereed] View Summary We present exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. It is shown that for any prescribed blow-up time there is an exact solution whose mass density converges to the Dirac measure as time goes to the blow-up time and that the solution extends beyond the blow-up time and behaves like the free solution as time tends to infinity. • ON THE DERIVATIVE NONLINEAR SCHRODINGER-EQUATION N HAYASHI, T OZAWA PHYSICA D 55 ( 1-2 ) 14 - 36 1992.02 [Refereed] View Summary In this paper we discuss the Cauchy problem for the derivative nonlinear Schrodinger equation: i partial derivative(t)psi + partial derivative(x)2-psi + 2i-delta-partial derivative(x)(\psi\2-psi) = 0, psi(0, x) = phi(x), where delta not-equal 0. Under an explicit smallness condition of the initial data, we prove the unique global existence of solutions to this problem in the usual Sobolev spaces, in the weighted Sobolev spaces, and in the Schwartz class. We describe the smoothing effect in detail. Furthermore, for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction. • On the initial value problem for the Zakharov equations T. Ozawa, Y. Tsutsumi Matematica Contemporanea 3 149 - 164 1992 [Refereed] • The nonlinear Schr&#246;dinger limit and the initial layer of the Zakharov equations T. Ozawa, Y. Tsutsumi Differential and Integral Eqs. 5 721 - 745 1992 [Refereed] • SPACE-TIME BEHAVIOR OF PROPAGATORS FOR SCHRODINGER EVOLUTION-EQUATIONS WITH STARK-EFFECT T OZAWA JOURNAL OF FUNCTIONAL ANALYSIS 97 ( 2 ) 264 - 292 1991.05 [Refereed] • THE NONLINEAR SCHRODINGER LIMIT AND THE INITIAL LAYER OF THE ZAKHAROV EQUATIONS T OZAWA, Y TSUTSUMI PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 67 ( 4 ) 113 - 116 1991.04 [Refereed] • TRUDINGER TYPE INEQUALITIES AND UNIQUENESS OF WEAK SOLUTIONS FOR THE NONLINEAR SCHRODINGER MIXED PROBLEM T OGAWA, T OZAWA JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 155 ( 2 ) 531 - 540 1991.03 [Refereed] • LONG-RANGE SCATTERING FOR NONLINEAR SCHRODINGER-EQUATIONS IN ONE SPACE DIMENSION T OZAWA COMMUNICATIONS IN MATHEMATICAL PHYSICS 139 ( 3 ) 479 - 493 1991 [Refereed] View Summary We consider the scattering problem for the nonlinear Schrodinger equation in 1 + 1 dimensions: i(partial)t(u) + (1/2)partial2u = lambda\u\2u + mu\u\p-1u, (t,x)epsilon-R x R, (*) where partial = partial/partial(x), lambda-epsilon-R\{0}, mu-epsilon-R, p &gt; 3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue space L2(R) or in the Sobolev space H-1(R). The modified wave operators are introduced in order to control the long range nonlinearity lambda\u\2u. • CLASSICAL AND QUANTUM SCATTERING FOR STARK-HAMILTONIANS WITH SLOWLY DECAYING POTENTIALS A JENSEN, T OZAWA ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE 54 ( 3 ) 229 - 243 1991 [Refereed] View Summary A discrepancy between classical and quantum scattering for Stark Hamiltonians is shown to exist for slowly decaying potentials. Let H0 = -(1/2) DELTA + x1 and H = H0 + V (x) on L2 (R(n)). For V (x) approximately c\x\-gamma, 0 &lt; gamma less-than-or-equal-to 1/2, as \x\ --&gt; infinity, the usual quantum wave operators between H0 and H do not exist. In classical one-dimensional scattering the classical wave operators exist and are asymptotically complete for the corresponding classical problem for V (x) = O (log (1 + \x$$)-alpha), alpha &gt; 1, as x --&gt; - infinity.

• NONEXISTENCE OF WAVE-OPERATORS FOR STARK-EFFECT HAMILTONIANS

T OZAWA

MATHEMATISCHE ZEITSCHRIFT   207 ( 3 ) 335 - 339  1991  [Refereed]

• INVARIANT SUBSPACES FOR THE SCHRODINGER EVOLUTION GROUP

T OZAWA

ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE   54 ( 1 ) 43 - 57  1991  [Refereed]

View Summary

The formation of dispersion with finite velocity of quantum states is described in detail. To be more specific, we prove the invariance of the domains D(\x\m) intersection D(\p\m), m is-an-element-of N, and of their topologies under the Schrodinger evolution group {e(-itH)}, where we denote by x and p the position and momentum operator, respectively. Moreover, we give a characterization of invariant subspaces under unitary groups in a rather general setting.

• STABILITY IN LR FOR THE NAVIER-STOKES FLOW IN AN N-DIMENSIONAL BOUNDED DOMAIN

H KOZONO, T OZAWA

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS   152 ( 1 ) 35 - 45  1990.10  [Refereed]

• SMOOTHING EFFECT FOR THE SCHRODINGER EVOLUTION-EQUATIONS WITH ELECTRIC-FIELDS

T OZAWA

FUNCTIONAL-ANALYTIC METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS   1450   226 - 235  1990  [Refereed]

• Non-existence of positive commutators

Tohru Ozawa

Hiroshima Math. J.   20   209 - 211  1990  [Refereed]

• Relative bounds of closable operators in non-reflexive Banach spaces

Hideo Kozono, Tohru Ozawa

Hokkaido Mathematical Journal   19 ( 2 ) 241 - 248  1990  [Refereed]

• SMOOTHING EFFECTS AND DISPERSION OF SINGULARITIES FOR THE SCHRODINGER-EVOLUTION-GROUP

T OZAWA

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS   110 ( 2 ) 165 - 186  1990  [Refereed]

• LOWER BOUNDS FOR ORDER OF DECAY OR OF GROWTH IN TIME FOR SOLUTIONS TO LINEAR AND NONLINEAR SCHRODINGER-EQUATIONS

T OZAWA, N HAYASHI

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES   25 ( 6 ) 847 - 859  1989.12  [Refereed]

• LOWER LP-BOUNDS FOR SCATTERING SOLUTIONS OF THE SCHRODINGER-EQUATIONS

T OZAWA

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES   25 ( 4 ) 579 - 586  1989.10  [Refereed]

• NEW LP-ESTIMATES FOR SOLUTIONS TO THE SCHRODINGER-EQUATIONS AND TIME ASYMPTOTIC-BEHAVIOR OF OBSERVABLES

T OZAWA

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES   25 ( 4 ) 521 - 577  1989.10  [Refereed]

• SMOOTHING EFFECT FOR SOME SCHRODINGER-EQUATIONS

N HAYASHI, T OZAWA

JOURNAL OF FUNCTIONAL ANALYSIS   85 ( 2 ) 307 - 348  1989.08  [Refereed]

• TIME DECAY FOR SOME SCHRODINGER-EQUATIONS

N HAYASHI, T OZAWA

MATHEMATISCHE ZEITSCHRIFT   200 ( 4 ) 467 - 483  1989  [Refereed]

• SCATTERING-THEORY IN THE WEIGHTED L2(RN) SPACES FOR SOME SCHRODINGER-EQUATIONS

N HAYASHI, T OZAWA

ANNALES DE L INSTITUT HENRI POINCARE-PHYSIQUE THEORIQUE   48 ( 1 ) 17 - 37  1988  [Refereed]

• REMARKS ON THE SPACE-TIME BEHAVIOR OF SCATTERING SOLUTIONS TO THE SCHRODINGER-EQUATIONS

T OZAWA

PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES   23 ( 3 ) 479 - 486  1987.09  [Refereed]

• TIME DECAY OF SOLUTIONS TO THE CAUCHY-PROBLEM FOR TIME-DEPENDENT SCHRODINGER-HARTREE EQUATIONS

N HAYASHI, T OZAWA

COMMUNICATIONS IN MATHEMATICAL PHYSICS   110 ( 3 ) 467 - 478  1987  [Refereed]

### Books and Other Publications

• The role of metrics in the theory of partial differential equations, Advanced Studies in Pure Mathematics, 85

Y. Giga, N. Hamamuki, H. Kubo, H. Kuroda, T. Ozawa( Part： Joint editor)

Mathematical Society of Japan  2020 ISBN: 9784864970907  [Refereed]

• Advances in harmonic analysis and partial differential equations, Trends in Mathematics

V. Georgiev, T. Ozawa, M. Ruzhansky, J. Wirth( Part： Joint editor)

Birkhäuser,Springer  2020 ISBN: 9783030582142  [Refereed]

• Asymptotic Analysis of Nonlinear Dispersive and Wave Equations, Advanced Studies in Pure Mathematics, 81

K. Kato, T. Ogawa, T. Ozawa( Part： Joint editor)

Mathematical Society of Japan  2019 ISBN: 9784864970815

• New Tools for Nonlinear PDEs and Application, Trends in Mathematics

M. D’Abbicco, M. R. Ebert, V. Georgiev, T. Ozawa( Part： Joint editor)

Birkhäuser  2019 ISBN: 9783030109363

• 小澤徹( Part： Sole author)

サイエンス社  2016.11

View Summary

本書は，主として物理現象を例に取り，現象の本質を記述する言葉である数学の機能が埋め込まれた対象としての微分方程式を論じたものである．本誌の連載「微分方程式を考える－数学は現象を如何に記述しているか」（2014年7月～2016年7月）の待望の一冊化．

• “International Conference for the 25th Anniversary of Viscosity Solutions”, GAKUTO International Series, Mathematical Sciences and Applications, 30

Y. Giga, K. Ishii, S. Koike, T. Ozawa, N. Yamada( Part： Joint editor)

2008 ISBN: 9784762504396

• “Lectures on Nonlinear Dispersive Equations”, GAKUTO International Series, Mathematical Sciences and Applications, 27

T. Ozawa, F. Planchon, P. Raphaël, Y. Tsutsumi, N. Tzvetkov( Part： Joint editor)

2006 ISBN: 9784762504365

• “Nonlinear Dispersive Equations”, GAKUTO International Series, Mathematical Sciences and Applications, 26,

T. Ozawa, Y. Tsutsumi( Part： Joint editor)

2006 ISBN: 4762504351

• “Nonlinear Waves,” Proceedings of the Fourth MSJ International Research Institute, GAKUTO International Series, Mathematical Sciences and Applications 10

R. Agemi, Y. Giga, T. Ozawa( Part： Joint editor)

GAKUTO International Series  1997 ISBN: 9784762504198

### Awards

• Spring Prize (Mathematical Society of Japan)

1998.03

• Furukawa Sansui Prize (Furukawa Group)

1984.03

### Research Projects

• Project Year :

2018.10
-
2023.03

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(挑戦的萌芽研究)

Project Year :

2016.04
-
2019.03

View Summary

零条件(null condition)を満たす特別な相互作用に付随する零形式の時空大域的性質(global properties of null forms in space-time)を振動(oscillation)と特異性(singularity)と直交性 (orthogonality)をキーワードとして、多重フーリエ解析、熱流解析、運動論、変分解析の4つの立場から研究し、既存の理論を等式の枠組に書き換える事により、多重線型ストリッカーツ評価等の不等式を剰余項付きで与え、最良定数と最適化函数のクラスを直接的に特徴付け、一層深い理解を得るとともに、剰余項に潜む高次の零構造を見出す手掛りを探求するのが本研究の目的である。これにより、高次の零条件を満たす特別な相互作用およびそのラグランジアン形式を見出し、スカラー場とゲージ場の新たな相互作用モデルの提案を目指す。

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(A))

Project Year :

2014
-
2018

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古典場の理論に現れる非線型偏微分方程式の連立系に関して、 (i) 相互作用の（高次）零構造 (ii) 付随する多重線型積分作用素 (iii) 定在波の三つの対象に焦点を当て、 (a) 漸近解析的方法 (b) 調和解析・実解析的方法 (c) 変分解析的方法 の三つの方法論から総合的に研究し、臨界的相互作用の分類及び特徴付けを通して、解の特異性などの空間局所的理論や解の長時間的挙動の記述などの時空大域的理論に新たな切り口を見出すとともに、厖大な既存の理論を再構築し大きな立場から理解するための全体像を提示する事が本研究の目的である。臨界的相互作用の把握と記述において現れる、調和解析や変分解析における臨界問題も重要な課題として取り組み、さまざまな最良定数の特徴付けや埋蔵定理の臨界問題への波及効果も視野に入れて研究を行う。

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(B))

Project Year :

2013
-
2016

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非線形楕円型方程式 (系) に対する特異摂動問題を中心に研究を実施した. 非線形シュレディンガー方程式に対する特異摂動問題に関しては, 局所的な変分法によるアプローチを研究代表者田中は J. Byeon 氏と共に開発し, ポテンシャル関数の極大点, 鞍点に凝集する解の構成に成功した. この構成法は非常に広いクラスの非線形項に対して適用可能であり, 従来の Lyapunov-Schmidt
法による極限方程式の解の一意性, 非退化性を要求する存在結果を大きく拡張するものである. なお, 本年度の研究において 1 点に与えられた数のピークが凝集する multi-peak 解の存在の構成にも成功している. このようなmulti-peak 解の存在は非退化条件なしには証明されていなかったものである.
常微分方程式の Lagrange 系に対する特異摂動問題に関しては, 高振動解の adiabatic invariant を用いたプロファイルの決定および与えられた admissible なプロファイルをもつ解の構成を P. Felmer 氏, S. Martinez 氏らと共に行い成功した.
また 2 次の相互作用をもつ非線形シュレディンガー方程式系について研究代表者は分担者小澤および林氏と共に研究に取り組み, 初期値問題の局所および大域可解性, さらには定在波解の存在を様々な設定の下で行った. また研究分担者小薗は Lax-Milgram 定理の一般化およびその楕円型方程式系への応用を, また連携研究者足達, 佐藤は準線形楕円型方程式, 非線形シュレディンガー方程式系の解の漸近挙動の研究等を行い, 塩路は非線形楕円型方程式の球対称解の研究を行い, 既存の結果をほぼすべて含む, 球対称解の一意性定理を導いた.

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(S))

Project Year :

2012
-
2016

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(1) 自由境界問題：Navier-Stokes方程式の自由境界問題を線形化して得られるStokes方程式の自由境界条件下での解作用素の R-有界性を半空間のモデル問題についてsurface tensionがついていない場合とついている場合について分けて示した。次に有界領域および一様な非有界領域においてそのレゾルベント評価を行った. さらにここでの手法を半空間の場合の結果を用いて, 有界領域および一様な非有界領域における解作用素の R-有界性を surface tensionがついていない場合に示した。
(2)流れの安定性：2次元の場合の物体を横切る圧縮性粘性流体の安定性を示すための鍵となる、外部領域でのStokes方程式の解の減衰度を2次元外部領域の場合に示した。また2次元以上の有界領域における圧縮性粘性流体流れを考え、その線形化問題の解の指数安定性が質量項の摂動平均がゼロとなる場合に示し、これを用いて流の安定性を示した。2次元の円柱周りの定常Oseen方程式について、有限要素法による数値解析を行った。線形反復解法としてGMRES法を用いることで、Reynolds数が100程度まで、双子渦を再現することに成功した。またあるReynolds数以下では渦が生じないことも再現できた。
(3)メゾレベルからの粘性流体の運動方程式の導出: 最終目標は決定論的なモデル化が不可能な多重スケール流体運動のモデル方程式を導出することとした。本年度はNavier-Stokes方程式から導かれるレイリープリセットーケラー方程式に、実験からの類推で確率項を付けたものの数値解析を行いある程度実験値を再現していることを確認後、Funaki-Inoueの確率Naiver-Stokes方程式から、流体極限の方法を用いてレイリープリセットーケラー方程式型の確率常微分方程式を導いた。

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(挑戦的萌芽研究)

Project Year :

2013
-
2015

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非線型シュレディンガー方程式の連立系に対して、ラグランジアンを具体的に書き下す事に成功した。このラグランジアンの導入により、シュレディンガー方程式の連立系における質量共鳴の解析力学・ラグランジュ幾何からの把握が系統的に出来るようになり、二次の相互作用における波動函数の複素共軛の存在が便宜的・形式的に必要だったのではなく、その役割が本質的である事が説明可能となった。また、質量共鳴に伴う単色波振動因子の持つ位相変調により得られる楕円型方程式系を変分問題として定式化し、空間次元に関して2次元以上5次元以下ではコンパクト性及び再配列理論を用いる事により、また、コンパクト性の失われる6次元では、スケーリングの議論を用いる事により、その基底状態の存在（本質的な）及び一意性を示す事が出来た。更に、質量共鳴条件下で初期データの大きさを任意に小さく取っても解の爆発現象を生み出す相互作用を、ホップ・コール型の変換に基づいて具体的に書き下す事が出来た。更に、質量共鳴条件に現れる質量比と爆発時刻のオーダーとの密接な関係を、初期データの大きさの観点から見出した。また、初期データが空間遠方で指数函数的に減衰している場合に、質量共鳴条件下で時刻 t ≠ 0 となった途端、解が解析的になるという現象、即ち、解析的平滑効果を証明した。同時に、連立系の波動函数の収束半径に相当するパラメータの比が質量共鳴条件に現れる質量比と一致することも見出し、質量共鳴と解の解析性との関係に関し、新たな知見を得た。

• 科学研究費助成事業(早稲田大学)  科学研究費助成事業(挑戦的萌芽研究)

Project Year :

2012
-
2014

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マリンライザーと呼ばれる海底から石油を移送するために, 基地から海底へ垂直に下したパイプの，深さ x に於ける水平方向方向の変位を u(x) とした時の u(x) の満たす方程式（ riser equation）は次の，ダンピング項 a u_t を持つ4階の準線形波動方程式で与えられる．
u_tt + a u_t + 2b u_xxxx - 2[(c x+d) u_x]_x + (b/3) [(u_x)%3]_xxx - 2[(c x+d) (u_x)%3]_x - b [(u_xx)%2 u_x]_x = 0, 0<x<L, 0<t<T,
u(0,t) = u(L,t) = u_xx(0,t) = u_xx(L,t) = 0, 0<t<T, u(x,0) = u_0(x), u_t(x,0) = u_1(x), 0<x<L.
この方程式の形式解に対しては，次のエネルギー保存則が成り立つことが示される．
E(u(t),v(t)) = E(u_0,u_1) + [(v)%2 の (0,L)x(0,t)上の積分 ]， v(t) = u_t(t),
E(u,v) = [ (b/2)(u_x u_xx)%2 + (c x+d) (u_x)%2 + (1/4)(c x+d) (u_x)%4 + b (u_xx)%2 ] の (0,L) 上の積分．（ここで，u_x, u_xx, u_t は x, t に関する偏微分，(u)%n は u の n 乗を表す．）しかしながら，方程式には u に関する空間2階微分 u_xx の 2乗 という非線形項が含まれるため，エネルギー保存則だけでは，弱解の構成が極めて困難であった．ここでは，この方程式に強減衰項 εu_xxxxt を加えた緩和問題を考え，解の存在のための第一歩である，アプリオリ評価を導出した．

• Project Year :

2009
-
2013

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We studied nonlinear elliptic equations arising in various fields of mathematical physics by means of variational analysis, ordinary differential equations, and viscosity techniques. We studied orbital stability of standing waves, explicit blow-up solutions, and exponential decay of ground states for systems of nonlinear Schr"odinger type equations.

• Project Year :

2011
-
2012

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Blow-up phenomenon of positive solutions to the Cauchy problem for nonlinear heat equation of Fujita type is studied. The blow-up phenomenon was first observed and proved by Hiroshi Fujita about half a century ago. There is a large literature on the subject, especially on the formation of blow-up solutions in configuration space up to the blow-up time. We have proved that the blow-up time is characterized by means of the spherical average of the Cauchy data. We removed the assumptions of uniformity and isotropy on the Cauchy data, which were necessary in the former works by Lee and Ni (Trans. Ams, 1992) and Gui and Wang (JDE 1995). We described how the ODE structure controls the blow-up phenomenon.

• Project Year :

2009
-
2012

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Various types of nonlinear PDEs (nonlinear elliptic equations, nonlinear diffusion equations, nonlinear wave equations, nonlinear Schrodinger equations) arising in physics and engineering were synthetically studied from the viewpoint of the theory of nonlinear evolution equations by using the techniques from the theory of nonlinear functional analysis, the theory of functions of a real variable, the theory of ordinary differential equations and the calculus of variations.

• Project Year :

2008
-
2011

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We study nonlinear problems via variational approaches. Especially (1) we study singular perturbation problems for nonlinear Schrodinger equations and systems. We introduce a new purely variational method which enables us to construct concentrating solutions in a very general setting. (2) We study nonlinear elliptic equations and systems in various settings. We give a new variational construction of radially symmetric ground states. We also study stability and instability of solutions. (3) We also study highly oscillatory solutions in 1-dimensional singular perturbation problems. We give characterization and existence result.

• Project Year :

2004
-
2008

• Project Year :

2004
-
2007

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(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever
(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.
iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long.

• A characterization of weighted Sobolev spaces and its applications

Project Year :

2004
-
2006

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1.We proved a weighted version of the Sobolev-Lieb-Thirring inequality. As an application we showed a weighted LP-Sobolev-Lieb-Thirring inequality which is a new result even in the non-weighted case.2.We proved that the wavelet basis is a greedy basis in weighted Triebel-Lizorkin spaces. As an application we determined the approximation spaces of the weighted Triebel-Lizorkin spaces by means of the non-linear approximation by wavelets.3.We proved a wavelet characterization of weighted Herz spaces. Tang and Yang showed a vector valued inequality in weighted Herz space although there are some mistakes in their condition on weights. We gave a correct condition on weights and proved the wavelet characterization.4.We studied the decay estimate of the solution of a non-linear elliptic partial differential equations with a potential. We characterize the class of weights in the decay estimate of the solution by the potential. We proved the global existence of a solution with small amplitude for several dispersive equations such as modified Boussinesq equations, improved Boussinesq equations and semi relativistic Hartree equations.5.We studied the existence of a singular solution and its property of a semilinear degenerate elliptic equation with p-harmonic operator in the principal term. In particular we investigated a linearized degenerate elliptic operator and proved the non-negativeness of the smallest eigenvalue and its relation to the Hardy type inequality

• Project Year :

2002
-
2004

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We studied identifying the discontinuity of the medium such as inclusions, cavities, cracks and the physical property of the medium. For identifying the discontinuity for the medium, we improved and adopted the probe method and enclosure method. Especially, we studied the behavior of the reflected solution and the unique continuation property which are essential for the probe method, and we accomplished the probe method. As for the enclosure method, we enlarged its application by replacing the complex geometric optic solution which is difficult to construct and localize by introducing the osciallating-decaying solution. We also showed that the reconstruction methods for the inverse boundary value problem such as the probe method, singular source method, no response test are unified into the no response test, and the probe method and singular source method are the same methods. For the inverse scattering problem, we solved the difficulty of the linear sampling method by proposing two new reconstruction methods. Moreover, we succeeded in establishing the probe method for the one space dimensional parabolic equation and giving the theoretical frame work for Shirota's computational method for identifying the discontinuity of the coefficient for the wave equation.As for identifying the physical property of the medium, we studied two inverse problems for identifying the residual stress and the damage of steel-concrete connected beam. We gave the dispersion formula of the speed of the Rayleigh wave and applied it for the former inverse problem. For the latter problem, we established identifying the damage from the frequency response function which is a practical measured data. We also studied identifying the coefficient for the nonlinear wave equation and succeeded in observing that we can identify the linear and the quadratic part of the coefficient by linearizing the Dirichlet to Neumann map

• Project Year :

2001
-
2004

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In this research project, various space-time behavior of solutions to nonlinear dispersive equations, such as nonlinear Schrodinger equations (NLS) and KdV type equations, nonlinear hyperbolic equations, such as nonlinear wave and Klein-Gordon equations, and coupled systems of those equations, such as nonlinear field equations. The main results are the following :
(1)Asymptotic completeness in the energy space H^1(R^3) for NLS with repulsive case has been proved.
(2)A unified treatment for small data scattering for nonlinear field equations has been given in terms of critical and subcritical setting.
(3)Existence and uniqueness of self-similar solutions for nonlinear wave equations have been proved in the framework of weak Lebesgue spaces.

• Project Year :

2001
-
2002

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In this research we studied about the characterization of weighted function spaces by means of the ψ-transform of Frazier and Jawerth and its appilications to partial differential operators. We proved a generalization of the Lieb-Thirring inequality which gives an estimate of the moments of the negative eigenvalues of the Schrodinger operator with negative potentials. Our result is about higher order degenerate elliptic operators. We gave a generalization of Egorov-Kondrat'ev's result.We also proved a generalization of the Sobolev-Lieb-Thirring inequality. Our result is a weighted version of the original one, which is a generalization of Ghidaglia-Marion-Temam's theorem or Edmunds-Ilyin's theorem. Our results are expected to be applied to the estimate of the Hausdorff dimension of the attractor of nonlinear partial differential operators

• Project Year :

1999
-
2001

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Arai studied harmonic analysis on negatively curved manifolds. Let M be a complete, simply connected Riemannian manifold whose sectional curvatures K_M satisfy -∞ < -k^2_2 【less than or equal】 K_M 【less than or equal】 -k^2_1 < 0, where k_1 and k_2 are positive constants. Arai obtained several results on elliptic harmonic functions on M. In particular he established fundamental part of harmonic analysis on M by proving theorems related to Hardy spaces, BMO, VMO, Carleson measures, Green's potential. As applications, he also studied Bloch function theory on manifolds and the regularity problem of degenerate harmonic measures. Ozawa studied by using real variable method nonlinear Schrodinger equations, nonlinear wave equations and nonlinear Krein-Gordon equations. Yajima obtained some results on the fundamental solutions of Schrodinger equations. Kanjin proved Paley's inequality for Jacobi expansions and studied the Hausdorff operator acting on real Hardy spaces.

• Project Year :

1997
-
2000

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Mathematical reality and mechanism is studied on the nonlinear classical fields such as those described by nonlinear wave equations, nonlinear Klein-Gordon equations, and nonlinear Schrodinger equations. Special attention is given to global theories which are beyond the scope of available general theories. Small data scattering is proved for the above equations in the Sobolev spaces of any nonnegative order by specifying the associated critical nonlinearities. Large time behavior and regularity of solutions is also studied.

• Project Year :

1996
-
1997

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In this research project, we established the calssification of the singularities of solution surfaces of quasi-homogeneous first order partial differential equations, viscosity solutions of Hamilton-Jacobi equations with one-space variable and multivalued solutions of conservation laws which are a part of the main purpose. Moreover, we extend the theory ofviscosity solutions to the case when the second order non-degenerate equation with non-local effects (one-space variable). This research is important to describe the crystal growth with fascet surfaces. We also give some iimportant new examples of Riemannian manifolds with integrable geodesic flows.
On the other hand, we have shown the existence of stable solutions for Ginzburg-Landau equation in a rotain domain. We have given a characterization of the symplectic and Lagarangian stablity of isotropic submanifolds with corank one by using a kind of transversality theorem. As a result on Algebraic and Geometric Topology we have developed an elementary tools for calculating the cohomology of heyper eliptic mapping class groups over finite fields.
These results are contained in the areas of the border of Geometry and Analysis. We expect to apply these results for studying Partial Differentail Equations in near future.

• 科学研究費助成事業(北海道大学)  科学研究費助成事業(奨励研究(A))

Project Year :

1996

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• 科学研究費助成事業(北海道大学)  科学研究費助成事業(奨励研究(A))

Project Year :

1995

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分散性媒質中を複数個の波が伝播するとき,これらの波の間で振動数と波数または速度に関する共鳴条件が満たされるとエネルギー授受を伴う強い共鳴相互作用が発生する。このような相互作用の一つとして波長のスケールが極端に異なる二種類の波の間で起こる長波短波相互作用がありプラズマ波,表面張力重力波,密度成層流体,二層流体等多くの系で観測される。この共鳴現象を記述する簡単なモデルとして知られている非線型シュレディンガー方程式と波動方程式との連立系を研究した。特に以下の成果を得ることができた。
(1)共鳴方程式の線型化作用素に付随する振動積分の有界性が成立する函数空間を導入した。
(2)共鳴方程式の非線型項に現われる一階微分の因子が引起こす「微分の損失」が発生する状況を(1)との関連で分類した。
(3)共鳴方程式に対して縮小写像の原理が適用可能な枠組を設定した。その結果共鳴方程式を函数解析的に取扱うことに成功した。同時に非線型項の結合定数が可解性には影響しないことを示すことができた。従って従来より行われてきた逆散乱法による取扱いが如何に問題を限定していたかという事情を広い視点から説明することができた。また数値実験で実証されていた共鳴方程式の示すカオス現象の理論的解明についても研究を行った。その結果波のもつ滑らかさはソボレフの意味では少なくとも1/2の指数で時間的に安定であることが証明できた。一次元空間でソボレフ指数の1/2は臨界指数に相当するので現象としてはカオス的側面をある程度把握したことになったと考えられる。

• Project Year :

1993
-
1995

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Motion of crystal surface in crystal growth is a typical example of phase-boundaries (interface). Such a phenomena attracts interdeciplinary interest as nonequilibriun nonlinear phenomena. Interface controlled model is an important class of evolution equations of phase boundaries. This is the case when heat and mass diffusion is negligible so that the evolution is determined by geometry of surface. Phenomena that facets appear on interface arises, for example, in the growth of Helium crystal growth in low temperature. In this situation, the governing equation has a nonlocal term and it is difficult to describe. So far the evolution law is described by restricting a class of evolving interfaces. The head investigator gave a formulation to this problem which is comparible with partial differential equations. It is based on the theory of nonlinear semigroups and nowadays it is called Fukui-Giga formulation. By this formulation curve evolution by crystalline energy can be understood as a limit of evolution by smooth anisotropic energy.
In motion of interfacial energy having anisotropy, it is important whether or not there is a self-similar shrinking solution. If interfacial energy is isotropic and there is no external force, the equation becomes the famous curve shortening equation. It is known that the only self-similar solution is a circle. However, the proof is rather complicated. Head investigator gave an elementary proof. For motion by anisotropic curvature be proved the existence of self-similar solution in an elementary way. However, uniqueness is shown only for evolution law that does not depend the orientation of curves.
The above research is a study of important example of nonlinear parabolic equations.Investigator studied large time asymptotic behaviors of solutions of nonlinear Schrodinger equation describing dispersive phenomena and discovered a nonlinear effect that is not tractable as a linear phenomena.

• 科学研究費助成事業(北海道大学)  科学研究費助成事業(奨励研究(A))

Project Year :

1994

View Summary

Stark効果をもつSchrodinger作用素に対する数学的散乱理論は1977年のAvron-Herbst及びVeselic-Weidmaunの独立した共同研究によりその短距離理論が完成したがこの理論の適用限界となるポテンシャルのクラスを特定することについては未解決点を残していた。この問題に関するVeselic-Weidmann予想について研究代表者は1991年に肯定的解答を与えStark効果の下での短距離力と長距離力との分類について明確な指針を与えた。次いでJensenとの共同研究においてこの量子力学的分類が古典力学的分類と対応しないことを証明しStark効果の下では形式的対応原理が破錠することを示した。Stark散乱の長距離理論は上記二つの仕事をもって始まった。White,Yajima,Jensen,Grafらによる各々独自の長距離理論が構築されそれに付随した変形波動作用素も次々に提案されたが対応原理の破錠を合理的に解消したものはGrafによるものであった。研究代表者はJensenとの共同研究においてGrafの変形波動作用素の存在の為の必要十分条件をポテンシャルの減衰度で特徴づけ対応原理の破錠に対する合理的説明を完全な形で行うことに成功した。

• 科学研究費助成事業(北海道大学)  科学研究費助成事業(奨励研究(A))

Project Year :

1993

View Summary

非線型のシュレディンガー方程式やクライン・ゴルドン方程式に代表される古典場の偏微分方程式は場の量子論の正当性を根底で支える理論的な拠り所であるばかりでなく非線型偏微分方程式論全般からみても重要な数学的対象である。本研究に於てはこれらの代表的な方程式に対する散乱問題を扱った。不思議なことに物理に登場する重要な方程式の多くは解の漸近解析に関し既存の数学的一般論では丁度扱うことの出来ない境界に位置する。具体的に云えば1+1次元の三次非線型性や1+2次元の二次非線型性に従う場は時間無限大に於て漸近自由場にはならない。本研究では線型理論との類推から位相の歪みを記述するドラ-ド型の修正漸近自由場とは如何に定義されるべきものであるかという問題の考察から始まりその修正漸近自由場に収束する解を構成せよという問題を最終的に解くという定式化に基づいて非線型遠距離散乱の理論の基礎を確立した。位相函数はフーリエ変換を媒介とし擬微分作用素として導入されるべきものである一方元の方程式に戻って考察すればある種のハミルトン・ヤコビ方程式を満足することが必要となる。この厳密解として位相函数を定めるのも一方法であるが本研究では漸近状態によって具体的に表し得る近似解を導入し時間無限大での近似度を非常に取扱い易い条件に置き換えた。修正波動作用素はこの修正漸近自由場に対する摂動方程式と見做される特異積分方程式を縮小写像の方法で解くことによって定義される。非線型散乱問題に於てこのようなプログラムを提出し実際に解いてみせたのは始めての試みであった。更にこの方法は応用が広いことも確認されその他の方程式にも適用できることが解明されつつある。

• 科学研究費助成事業(早稲田大学)  基盤研究(A)

Project Year :

1904.01
-

• 科学研究費助成事業(早稲田大学)

View Summary

キャビテーション，衝撃波の伝播，多成分流体，大気の循環などの流体解析では，ミクロとマクロの境界で発生するマルチスケール現象や非平衡系の数学的解析が重要である．本研究では，複雑流体のモデリング，マルチスケール構造の解明と数学解析手法の確立を目的とする．2016年度は，モデリング，数学解析と応用に分けて研究を推進した．流体のモデリングについては，有限自由度の離散的な非平衡熱力学系についての変分的な定式化を行った．また，キャビテーション気泡と気泡クラウドに関する実験，レイリー・ベナール対流の解析を中心に行った．数学解析としては，ナビエ・ストークス方程式，オイラー方程式，非線型シュレディンガー方程式をはじめとする非線型発展方程式の初期値問題の時間大域解の存在を保障する先験評価に重要な役割を果たす対数型ソボレフ埋蔵に就いて研究し，放物型方程式に対してはその散逸構造に因り通常のソボレフ埋蔵の枠組で閉じている事を明らかにした．また，非圧縮粘性流体の自由境界問題を有界領域の場合に考察し、時間局所解の一意存在と時間大域解の一意存在及び解の漸近挙動を示した．さらに，複数の保存量を持つ微視的な系から非線形流体力学揺動理論を経て導かれると予想される，多成分 KPZ 方程式について考察し，殆ど全ての初期値に対し方程式は大域的適切性を持つことを示した．また，質量保存アレン-カーン方程式にノイズを加えて得られる確率偏微分方程式について，極限で確率的摂動を持つ質量保存平均曲率運動が導かれることを示した．数値解析として，準離散化方程式の対称性と保存則を研究し，ネーター定理を導き，高階の場の理論のために，マルチシンプレクティック構造の研究を行った．非線形力学の応用として，ミクロスケールでの高分子鎖の捩れ運動から生じる幾何学的位相を見出し，回転型分子モーターの回転軸運動の粗視化モデルに適用した．概ね順調に進んでいるが，研究代表者と分担者４名で研究内容が多岐に渡るため，全体のまとまりを考えて，本来の研究目的に沿って組織的に研究を遂行する必要があると考えている．複雑流体のモデリングについては，連続的な非平衡系としての変分的定式化の確立，レリー・ベナール対流の解析，気泡クラウドのマクロモデルの構築，確率的な気泡ダイナミクスの変分的定式化と解析を中心に進める．数学解析では，部分積分に依る時間発散型の高次繰り操みエネルギーとブレジス・ガルエの論法を駆使し，半相対論的方程式の高次相互作用が数学的に実現されるかどうか，検討する．また，２相流問題の考察を行い，まずは非圧縮・非圧縮の２相問題についての有界領域で外側の境界条件が自由境界条件の場合を考える．つづいて，一方が有界領域、外側がその補集合である全空間での２相問題を考え，この時間局所解，時間大域解、解の漸近挙動を示す．さらに，多成分KPZ方程式について，カップリング定数が3重線形性を満たさない場合の不変測度の研究，大規模相互作用系からのKPZ方程式の導出を目指した研究，ノイズ項が空間変数にも依存する確率アレン-カーン方程式や確率的平均曲率運動に関する研究などを進める．数値解析については，マルチシンプレクティック構造の研究を続け，流体力学への応用，特に拘束を含む系への応用を考え，離散的ディラック構造の研究も進行する予定である．非線形力学の応用として，高分子鎖の捩れ運動から生じる幾何学的位相の効果を，ベン毛微生物の遊泳機構の解明に応用する．できるだけ，分担者からの使用ができるだけ均一になるように早めの確認と連絡をしたい．また，大学院生による研究調査や協力のための謝金にも利用したい

• 　View Summary

The challenging problem on global well-posedness of the Navier-Stokes equations had been so fully investigated that several remarkable results are obtained. Furthermore, our DNS of the uniformly isotropic turbulence is still by far the larger computational performance so that we could deal with the turbulent fluid with the high Reynolds number without any error of the experiment and indeterminacy. Our study has been based on the DNS of such a world highest standard and we could succeed to overcome difficulty of turbulence with the high Reynolds number. In this way, our research projects have developed the modern mathematical analysis, the applied mathematics, computational science and hydrodynamics and hopefully will lead the relevant subjects to the world-wide level

• 科学研究費助成事業（早稲田大学）

• 科学研究費助成事業(早稲田大学)

View Summary

非局所項を伴う非線形楕円型方程式に対して特異摂動問題を中心に研究を行った. 一般的な非線形項を伴う場合を S. Cingolani 氏と共に研究を行い, ポテンシャル関数の極小点に凝集する解を構成すると共に, ポテンシャル関数の極小点集合の位相的性質 (cup length) により凝集解の個数を下から評価することに成功した. このような凝集解の存在結果は局所問題である非線形シュレディンガー方程式の場合に S. Cingolani 氏, L. Jeanjean 氏と共に極限方程式に対して一意性, 非退化性を仮定せずに求めているが, 今回得られた結果はその非局所問題に拡張したものである. ここでは単に結果を拡張するのみではなく, Moroz と Van Schaftingen により提示された locally sublinear case での特異摂動解の存在問題に対して肯定的な解答を与えることに成功している.
非線形シュレディンガー方程式に対する L2-制限問題 (normalized 解の存在問題) に関しては Langrange formulation を用いた新たなアプローチの開発に成功した. このアプローチは古くからある制限付き変分問題に対する方法に Pohozaev の等式関連したスケーリングの手法を加味したものであるが, 従来行われてきた sub additivity 不等式を用いた方法と比べると非常に柔軟な方法であり, 非常に一般的な設定で非線形項が奇関数の場合に無限個の解が生成できる等応用の幅も大きい. この手法は非線形 Choquard 方程式等の非局所項を含む非線形問題に対する L2-制限問題へのアプローチも可能にすると思われ, さらなる発展が期待される.
なお, 研究分担者, 小澤は semi-relativisitic シュレディガー-ポアソン-スラッター方程式の ground state に関する結果, 連携研究者 生駒は非局所問題である fractional ラプラシアンをもつスカラーフィールド方程式に関する結果を得ている.

### Presentations

• On the Poincaré and Related Inequalities

T. Ozawa  [Invited]

International Workshop on Multiphase Flows: Analysis, Modelling and Numerics

Presentation date： 2020.12

Event date：
2020.12

• Existence and Uniqueness of Classical Paths under Quadratic Potentials

T. Ozawa  [Invited]

The 37th Kyushu Symposium on Partial Differential Equations  (Fukuoka, Japan)

Presentation date： 2020.01

• ポワンカレの不等式・温故知新

T. Ozawa  [Invited]

微分方程式セミナー  (Osaka, Japan)

Presentation date： 2020.01

View Summary

一般領域上のポワンカレの不等式とその証明に就いて再考すると共に、幾つかの一般化を与える。

• T. Ozawa  [Invited]

PDE Workshop  (Peking, China)

Presentation date： 2019.11

View Summary

The existence of classical paths is shown to be realized as a minimizer of the action of the Lagrangean. The optimal bound on the time intervals for uniqueness of classical paths is also given. This talk is based on my recent jointwork with Kazuki Narita.

• T. Ozawa  [Invited]

International Conference “Actual Problems of Analysis, Differential Equations and Algebra”  (Nur-Sultan, Kazakhstan)  National academy of sciences of the Republic of Kazakhstan

Presentation date： 2019.10

View Summary

A class of self-similar solutions to the derivative nonlinear Schr"odinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions. This talk is based on my recent jointwork with Kazumasa Fujiwara (Tohoku) and Vladimir Georgiev (Pisa).

• T. Ozawa  [Invited]

第9回岐阜数理科学研究会  (Gifu, Japan)

Presentation date： 2019.09

View Summary

微分型相互作用をもつ非線型シュレディンガー方程式の自己相似解の構成法を説明する。

• T. Ozawa  [Invited]

12th ISAAC Congress  (Aveiro, Portugal)

Presentation date： 2019.07

View Summary

A class of self-similar solutions to the derivative nonlinear Schr"odinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions. This talk is based on my recent jointwork with Kazumasa Fujiwara (Tohoku) and Vladimir Georgiev (Pisa).

• Self-similar solutions to the derivative nonlinear Schrödinger equation

T. Ozawa  [Invited]

Conference on "Nonlinear Dispersive Waves, Solitons and related topics" - IML, Djursholm, Sweden, 10-14 June 2019  (Djursholm, Sweden)

Presentation date： 2019.06

View Summary

A class of self-similar solutions to the derivative nonlinear Schr"odinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions. This talk is based on my recent jointwork with Kazumasa Fujiwara (Tohoku) and Vladimir Georgiev (Pisa).

• T. Ozawa  [Invited]

PDE Workshop Waseda - GSSI, L'Aquila-Pisa  (Pisa, Italy)

Presentation date： 2019.02

View Summary

An elementary and explicit approach to the Cauchy problem for the transverse wave equation is given in the framework of evolution equations. References S. Alinhac, "Hyperbolic Partial Differential Equations," Springer, 2009. L. Hörmander, "Lectures on Nonlinear Hyperbolic Differential Equations," Springer, 1997.

• T. Ozawa  [Invited]

PDE Workshop in Miyazaki  (Miyazaki, Japan)

Presentation date： 2019.01

• T. Ozawa  [Invited]

Mathematical Fluid Mechanics and Related Topics - in honor of Professor Hideo Kozono's sixtieth birthday -  (Tokyo, Japan)

Presentation date： 2018.09

View Summary

We rewrite the standard Hardy inequality in the form of an equality with radial derivative in $\mathbb{R}^n$ for $n \geq 3$. Then we present another Hardy inequalities with radial and spherical derivatives in $\mathbb{R}^n$ for $n \geq 2$. We also discuss those optimality and nonexistence of nontrivial extremizers.

• T. Ozawa  [Invited]

Nonlinear Dispersive Equations at Florianopolis  (Florianopolis, Brazil)

Presentation date： 2018.07

• T. Ozawa  [Invited]

PDE Seminar  (Shanghai, China)

Presentation date： 2018.06

View Summary

This talk is based on my recent jointwork with Kazumasa Fujiwara, Scuola Normale Superiore di Pisa. We give a sharp upper bound of lifespan of blowup solutions to the Cauchy problem for a generalized derivative nonlinear Schrödinger equations with periodic boundary condition.

• T. Ozawa  [Invited]

Celebrating Approximate 60s -- An International Conference on Nonlinear PDEs and Its Applications at NYU Shanghai  (Shanghai, China)

Presentation date： 2018.06

View Summary

We rewrite the standard Hardy inequality in the form of an equality with radial derivative in $\mathbb{R}^n$ for $n \geq 3$. Then we present another Hardy inequalities with radial and spherical derivatives in $\mathbb{R}^n$ for $n \geq 2$. We also discuss those optimality and nonexistence of nontrivial extremizers.

• T. Ozawa  [Invited]

作用素論セミナー  (Kyoto, Japan)

Presentation date： 2018.05

• Lifespan estimates of solutions to NLS without gauge invariance

T. Ozawa  [Invited]

PDE Seminar  (Pekin, China)

Presentation date： 2018.05

• Improved Hardy inequalities

T. Ozawa  [Invited]

PDE Workshop  (Sichuan, China)

Presentation date： 2018.04

• On improved Hardy inequalities

T. Ozawa  [Invited]

Workshop in Hangzhou 2018  (Hangzhou, China)

Presentation date： 2018.04

View Summary

Hardy inequalities are studied in view of both radial and angular derivatives in the framework of equalities.

• T. Ozawa  [Invited]

Workshop on Harmonic analysis and Nonlinear Evolution Equations  (Pisa, Italy)

Presentation date： 2018.02

View Summary

Hardy inequalities are studied in view of both radial and angular derivatives in the framework of equalities.

• T. Ozawa  [Invited]

Hyperbolic Partial Differential Equations and Related Topics--in honor of the 60th birthday of Professor Tokio Matsuyama--  (Tokyo, Japan)

Presentation date： 2018.01

View Summary

An explicit lifespan estimate is presented for nonlinear Schr"odingerequations on one-dimensional torus. This talk is based on my recent joint-work with Kazumasa Fujiwara (JSPS fellow, PD).

• T. Ozawa  [Invited]

Noninear Dispersive Equations in Kumamoto, 2018  (Kumamoto, Japan)

Presentation date： 2018.01

• T. Ozawa  [Invited]

Recent topics on PDEs  (Tokyo, Japan)

Presentation date： 2017.11

• T. Ozawa  [Invited]

RIMS Workshop Nonlinear Wave and Dispersive Equations  (Kyoto, Japan)

Presentation date： 2017.08

View Summary

An explicit lifespan estimate is presented for nonlinear Schr"odingerequations on one-dimensional torus. This talk is based on my recent joint-work with Kazumasa Fujiwara (JSPS fellow, PD).

• 数理物理学に於ける co-presence

T. Ozawa  [Invited]

co-presence研究会  (Chiba, Japan)

Presentation date： 2017.03

• T. Ozawa  [Invited]

The 9th Nagoya Workshop on Differential Equations  (Aichi Nagoya, Japan)

Presentation date： 2017.03

• T. Ozawa  [Invited]

Zhejiang-Tohoku International Workshop for Nonlinear Partial Differential Equations 2017  (Miyagi Sendai, Japan)

Presentation date： 2017.03

• T. Ozawa  [Invited]

Critical Exponents and Nonlinear Evolution Equation 2017  (Tokyo, Japan)

Presentation date： 2017.02

View Summary

We revisit Kato’s theory on Landau-Kolmogorov (or KallmanRota) inequalities for dissipative operators in an algebraic framework in a scalar product space. This is a joint-work with Masayuki Hayashi.

• T. Ozawa  [Invited]

International Conference for the 70th Anniversary of Korean Mathematical Society  (Seoul, Korea)

Presentation date： 2016.10

View Summary

This talk is based on a recent joint work with Kazuya Yuasa, Department of Physics, Waseda University. We study the Schrödinger–Robertson uncertainty relations in an algebraic framework. Moreover, we show that some specific commutation relations imply new equalities, which are regarded as equality versions of well-known inequalities such as Hardy's inequality.

• T. Ozawa  [Invited]

Workshop on Differential Equations in Hiroshima  (Hiroshima, Japan)

Presentation date： 2016.10

View Summary

Fractional Leibniz estimates with appropriate correction terms are introduced for homogeneous fractional derivatives of order larger than or equal to one. Those fractional Leibniz estimates have the flexibility in arbitrary redistribution of fractional derivatives in the corresponding bilinear estimates. The method of proof depends on the Coifman-Meyer theory. This talk is based on a recent joint-work with Kazumasa Fujiwara and Vladimir Georgiev.

• T. Ozawa  [Invited]

New trends in Partial Differential Equations  (Pisa, Italy)

Presentation date： 2016.10

View Summary

This talk is based on a recent joint work with Kazuya Yuasa, Department of Physics, Waseda University. We study the Schrödinger–Robertson uncertainty relations in an algebraic framework. Moreover, we show that some specific commutation relations imply new equalities, which are regarded as equality versions of well-known inequalities such as Hardy’s inequality.

• T. Ozawa  [Invited]

Nonlinear Wave and Dispersive Equations, Kyoto 2016  (Kyoto, Japan)

Presentation date： 2016.09

• T. Ozawa  [Invited]

Recent Topics on Dispersive Equations  (Tokyo, Japan)

Presentation date： 2016.08

• T. Ozawa  [Invited]

International Conference on Navier-Stokes equations and related PDEs : In honor of the 60th birthday of Professor Hi Jun Choe  (Daejeon, Korea)

Presentation date： 2016.06

View Summary

We give an explicit upper bound of life span of solutions to non gauge invariant nonlinear Schrödinger equations on n-dimensional tours. This talk is based on a recent joint work with Kazumasa Fujiwara.

• T. Ozawa  [Invited]

Seminar of Differential Equations  (Osaka, Japan)

Presentation date： 2016.06

View Summary

量子力学の基礎を担う不確定性関係を、等式の形で記述する。 更に、具体的な作用素の交換関係に適用し、新しい等式を生み、それらがハーディの不等式をはじめとする良く知られた不等式の等式版と見做される事情を説明する。この内容は湯浅 一哉 教授（早稲田大学先進理工学部物理学科）との共同研究に基づくものである。

• T. Ozawa  [Invited]

PDE Seminar  (Hokkaido, Japan)

Presentation date： 2016.05

View Summary

We give an explicit upper bound of life span of solutions to non gauge invariant nonlinear Schrödinger equations on n-dimensional tours. This talk is based on a recent joint work with Kazumasa Fujiwara.

• T. Ozawa  [Invited]

Centre International de Rencontres Mathématiques (CIRM)「Recent Trends in Nonlinear Evolution Equations」  (Marseille, France)

Presentation date： 2016.04

• T. Ozawa  [Invited]

Nonlinear Partial Differential Equations and Mathematical Physics Workshop  (Sanya, China)

Presentation date： 2016.01

View Summary

We revisit Kato’s theory on Landau-Kolmogorov (or KallmanRota) inequalities for dissipative operators in an algebraic framework in a scalar product space. This is a joint-work with Masayuki Hayashi.

• T. Ozawa  [Invited]

The 13th Japanese-German International Workshop on Mathematical Fluid Dynamics  (Darmstadt, Germany)

Presentation date： 2016.01

• On the Hardy type inequalities

T. Ozawa  [Invited]

Second Workshop on Nonlinear Dispersive Equations  (Campinas, Brazil)

Presentation date： 2015.10

• Life span of solutions to nonlinear Schrödinger equations on torus

T. Ozawa  [Invited]

Workshop on Partial Differential Equations  (Zhejiang, China)

Presentation date： 2015.03

• Finite time extinction for nonlinear Schrödinger equation

T. Ozawa  [Invited]

Workshop on Nonlinear Partial Differential Equations  (Hangzhou, China)

Presentation date： 2015.03

• Quadratic interactions in dispersive systems

T. Ozawa  [Invited]

Division Conference  (Kyoto, Japan)  Division of Functional Equations The Mathematical Society of Japan

Presentation date： 2014.12

• 古典場の理論に現れる非線型波動方程式

T. Ozawa  [Invited]

中央大学理工学部数学教室 Encounter with Mathematics 62波動方程式－古典物理から相対論まで－  (Tokyo, Japan)

Presentation date： 2014.09

• Systems of quadratic dispersive equations

T. Ozawa  [Invited]

Roman Summer School and Workshop KAM Theory and Dispersive PDEs.  (Roma, Italy)

Presentation date： 2014.09

• Refinements of Hölder's inequality

T. Ozawa  [Invited]

Seminari di Equazioni alle Derivate Parziali  (Pisa, Italy)

Presentation date： 2014.02

• Bilinear estimates in the Sobolev spaces

T. Ozawa  [Invited]

Analyse Numérique et Equations aux Dérivées Partielles  (Paris, France)

Presentation date： 2014.01

• ヘルダーの不等式の精密化

T. Ozawa  [Invited]

九州関数方程式セミナー  (Fukuoka, Japan)

Presentation date： 2013.11

• 基礎的不等式再考 (Part I, II)

T. Ozawa  [Invited]

第3回室蘭非線形解析研究会  (Hokkaido, Japan)

Presentation date： 2013.11

• Hardy type inequalities on balls

T. Ozawa  [Invited]

第2回岐阜数理科学研究会  (Gufu, Japan)

Presentation date： 2013.09

• Hardy type inequalities on balls

T. Ozawa  [Invited]

Mexico-Japan Joint Meeting on PDE's at Morelia

Presentation date： 2013.09

• Hardy type inequalities on balls

T. Ozawa  [Invited]

The 9th ISAAC Congress  (Poland)

Presentation date： 2013.08

• Mass resonance in a system of nonlinear Schrödinger equations

T. Ozawa  [Invited]

Linear and Nonlinear PDE  (Pisa, Italy)

Presentation date： 2013.08

• T. Ozawa  [Invited]

The Asian Mathematical Conference 2013 (AMC 2013)  (Busan, Korea)

Presentation date： 2013.06

• T. Ozawa  [Invited]

The 8th Japanese-German International Workshop on Mathematical Fluid Dynamics  (Tokyo, Japan)

Presentation date： 2013.06

• On a system of Schrödinger equations

T. Ozawa  [Invited]

Harmonic Analysis and PDEs on Manifolds  (Tokyo, Japan)

Presentation date： 2013.04

• Bilinear estimates on the Klein-Gordon equation

T. Ozawa  [Invited]

The 19th Machikaneyama Seminar on PDEs  (Osaka, Japan)

Presentation date： 2013.03

• T. Ozawa  [Invited]

Tohoku University Scienceweb GCOE The 5th GCOE International Symposium"Weaving Science Web beyond Particle Matter Hierarchy"  (Miyagi Sendai, Japan)

Presentation date： 2013.03

• Sharp Morawetz estimates

T. Ozawa  [Invited]

北海道大学数学談話会  (Hokkaido, Japan)

Presentation date： 2013.02

• T. Ozawa  [Invited]

UK-Japan Winter School, Nonlinear Analysis  (London, UK)

Presentation date： 2013.01

• クライン・ゴルドン方程式に対する双線型評価とフーリエ制限問題

T. Ozawa  [Invited]

Tohoku University Monday Analysis Seminar  (Miyagi Sendai, Japan)

Presentation date： 2012.11

• クライン・ゴルドン方程式の双線型評価

T. Ozawa  [Invited]

Kyushu Function Equation Seminar  (Fukuoka, Japan)

Presentation date： 2012.11

• Sharp bilinear estimate on the Klein-Gordon equation

T. Ozawa  [Invited]

Seminar on Differential Equations in Osaka, 2012-in honor of Professor Hiroaki Tanabe's 80th birthday -  (Osaka, Japan)

Presentation date： 2012.08

• A sharp bilinear estimate for the Klein-Gordon equation in two space-time dimensions

T. Ozawa  [Invited]

9th international Conference on Harmonic Analysis and Partial Differential Equations  (Madrid, Spain)

Presentation date： 2012.06

• T. Ozawa  [Invited]

CONFERENCE ON Evolution Equations, Related Topics and Applications  (Tokyo, Japan)

Presentation date： 2012.03

• Mass resonance in a system of nonlinear Schrödinger equations

T. Ozawa  [Invited]

PDE Seminar  (Paris,France)

Presentation date： 2012.03

• Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension

T. Ozawa  [Invited]

International Conference on Fluid and Gas Dynamics  (Zhejiang, China)

Presentation date： 2011.09

• T. Ozawa  [Invited]

One Forum, Two Cities : Aspect of Nonlinear PDEs  (Taipei, Taiwan)

Presentation date： 2011.08

• Life span of positive solutions for a semilinear heat equation with general non-decaying initial data

T. Ozawa  [Invited]

International Workshop on Differential Equations and Applications  (Tainan, Taiwan)

Presentation date： 2011.01

• T. Ozawa  [Invited]

International Workshop “Nonlinear PDE’s @ IMPA”  (Rio de Janeiro, Brazil)

Presentation date： 2010.08

• Remarks on some dispersive estimates

T. Ozawa  [Invited]

International Workshop “Fourier Analysis and Partial Differential Equations”  (Göttingen, Germany)

Presentation date： 2010.06

• Analytic smoothing effect for nonlinear Schrödinger equations

T. Ozawa  [Invited]

8th AIMS Conference  (Dresden, Germany)

Presentation date： 2010.05

• T. Ozawa  [Invited]

2009 International Workshop on Differential Equations and Their Applications  (Tainan, Taiwan)

Presentation date： 2009.12

• Global Cauchy problem for 2D Klein-Gordon equations

T. Ozawa  [Invited]

Work shop on Nonlinear dispersive and geometric evolution problems:singularities and asymptotics  (Vancouver, Canada)

Presentation date： 2009.08

• Analyticity of solutions to nonlinear Schrödinger equations

T. Ozawa  [Invited]

Nonlinear PDE in Zhang Jia Jie  (Zhang Jia Jie City, China)

Presentation date： 2009.08

### Specific Research

• 2020

View Summary

ラグランジュ力学における古典経路の存在は、解析力学の数学的基礎付けの最も重要な課題であり、ファインマン経路積分の定式化の基礎を成すことから、量子力学においても重要な課題である。変分解析の研究の進展によって「直接法」の方法論はラグランジュ力学の強固な基礎を形成するに至っているが、明らかにしているのは古典経路の存在のみであり、一意性は殆ど未解決である。そこで、本研究では一意性について研究した。鍵となる不等式としてポワンカレ型の新しい不等式を見出し、証明を与えた。その最良定数と最良定数を与える函数を特徴付けた。一意性を保障する時間幅（初期時刻と終期時刻の差）をその最良定数を用いて記述した。更に、その時間幅が最良である事を周期解を用いて具体的に示した。

• 2019

View Summary

微分型シュレディンガー方程式の初期値問題は近年盛んに研究されており、総電荷の閾値が4πである事も変分解析的に理解できるようになってきた。しかし、一般の爆発理論は全くの未解決問題として残ったままである。これは、通常の非線型シュレディンガー方程式に対して有効なビリアル法が破綻するからである。本研究では、具体的に自己相似解としての爆発解を構成して、その突破口を開いた。先ず、時空変数について自己相似的な波動函数を振幅函数と位相函数で表したとき、両者が満たすべき非線型常微分方程式系を特定することができた。更に、その解の存在を函数解析的に証明し、解の属す函数空間を見出すことができた。

• 2018   藤原和将

View Summary

ユークリッド空間上の非線型シュレディンガー方程式や複素ギンツブルグ・ランダウ方程式は、その非線型自己相互作用が引力型の場合、解は自己集中して爆発する。これは、レーザーの自己集束現象を数学的に説明するものとして興味深い理論である。その際、方程式のもつ時空尺度構造、ゲージ構造、変分構造が本質的に用いられる。特に、説明の鍵となる二次モーメントの消滅機構は空間伸長による対称性から定まるもので、通常、ビリアル論法と呼ばれている。　一方、円環（トーラス）上で考えた場合の分散型非線型発展方程式は、数値解析的研究は数多く行われているが、時空の対称性が限られるため、ビリアル論法は役に立たない。[1]において研究代表者は、トーラス上の上記分散型非線型発展方程式に対し、ゲージ構造の破綻が爆発解の存在に結び付く事例を幾つか見出した。　本研究では、[1]で見出した単調整の議論を更に深め、トーラス上の微分型非線型シュレディンガー方程式に対しても、単調性に由来する爆発現象を見出し数学的証明を与えた[2]。また、これらの結果を統一的に説明する理論体系に纏め上げ、[3]として発表した。

• 2013   柴田良弘, 吉村浩明, 吉田善章, 三浦英昭

View Summary

様々な特異現象に普遍的に現れるの本質を探る為、数学的設定が明確に記述可能な、流体やプラズマにおけるモデル方程式の初期値問題の解に現れる特異性の研究を行った。流体の速度場や渦度の爆発現象の研究では、爆発時刻が有限であれば時空の適当なノルムが無限大となる事が期待されており、実際多くの時空のノルムが無限大となる事が証明されているが、その相互関係については不明な部分も多く、最弱ノルムも同定されていないのが現状である。本研究ではナビエ・ストークス方程式とQテンソル系の連立系について、その最弱ノルムの同定を目指し、ナビエ・ストークス方程式で現在迄知られている最良の結果に相当する条件を、負の可微分性を示す斉次ベゾフ空間で与え、スケール不変性の観点から自然な形で説明出来る形に整備した。空間2次元の電磁流体力学の連立系については、水平方向の流体速度場の散逸構造と磁場の拡散係数との間に、爆発現象を抑制する積分構造を見出し、電磁場と速度場の相互作用に現れる特異性を部分積分を通じて消滅させる方法論を確立した。これは双曲型の非線型偏微分方程式である非線型波動方程式連立系の研究で見出された零構造に相当するものと考えられ、双曲型と放物型の非線型偏微分方程式の連立系における零構造の研究の出発点となるものと期待される。プラズマ系に現れる渦については、三波相互作用系としての非線型シュレディンガー方程式の連立系をモデル方程式として、その爆発解の発現機構について研究した。単独方程式では初期データが閾値より小さいと爆発は起こらない事が知られているが、連立系はそうとは限らず、任意に小さなデータを取ったとしても有限時刻で必ず爆発現象を起こす相互作用を見出し、この現象は連立系特有のものである事を明らかにした。実際の物理モデルでは単独系より連立系が圧倒的に多く用いられるので、プラズマにおけるはどんな小さな初期状態からも発生する可能性があると云う事情を充分説明するものと考えられる。

• 2010   山内雄介

View Summary

n次元ユークリッド空間に於けるp乗自己相互作用を持つ藤田型非線型熱方程式の初期値問題の正値解の爆発現象の解析について研究を行った。解が爆発する事自体については、約50年前の藤田宏の先駆的研究以来、良く知られており、Friedman, Kohn, Weissler, 儀我、柳田、溝口らの大きな寄与によって、爆発解の形状など空間的振無いについては精密な解析が進んでいる。一方、その爆発時刻が初期条件にどの様に依存するのかと云う問題については、LeeとNiの結果(Trans. AMS, 1992)とGuiとWangの結果(JDE 1995)のみが知られているところであるが、その仮定として、初期条件の空間遠方での挙動に極めて強い条件を課したものであり、充分解明されているとは言い難い。本研究では、初期函数の空間遠方での挙動について、単位球面上に射影した下極限函数に着目し、その球面平均が爆発時刻の上からの評価を具体的に与える事を明らかにした。より具体的には、球面平均の(1-p)乗で爆発時刻は上から評価される事が示された。先行結果に課されていた仮定の一様性・等方性を外し、爆発現象の常微分方程式的構造を記述する事に成功した。この研究成果は、国内外における幾つかの会議で発表され、高い評価を得た。本論文としてまとめられたものは、国際的に著名な学術専門誌 Journal of Mathematical Analysis and Applications に掲載が決定され、現在印刷中の段階であり、2011年中には出版の見込みである。

### Syllabus

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   full year

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   fall semester

• School of Advanced Science and Engineering

2022   spring semester

• School of Advanced Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   an intensive course(spring and fall)

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Fundamental Science and Engineering

2022   an intensive course(spring and fall)

• Graduate School of Fundamental Science and Engineering

2022   fall semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   spring semester

• Graduate School of Creative Science and Engineering

2022   an intensive course(spring and fall)

• Graduate School of Creative Science and Engineering

2022   fall semester

• Graduate School of Creative Science and Engineering

2022   spring semester

• Graduate School of Fundamental Science and Engineering

2022   full year

• Graduate School of Creative Science and Engineering

2022   an intensive course(spring and fall)

• Graduate School of Creative Science and Engineering

2022   fall semester

• Graduate School of Creative Science and Engineering

2022   spring semester

2022   fall semester

2022   spring semester

2022   fall semester

2022   spring semester

2022   full year

2022   full year

2022   an intensive course(spring and fall)

2022   fall semester

2022   spring semester

2022   fall semester

2022   spring semester

2022   full year

2022   fall semester

2022   spring semester

2022   full year

2022   full year

2022   full year

2022   full year

2022   full year

2022   an intensive course(spring and fall)

2022   full year

### Committee Memberships

• 2021
-
Now

日本学術会議  物理学委員会 物性物理学・一般物理学分科会 プラズマサイエンス小委員会 委員

• 2021
-
Now

日本学術会議  数理科学委員会 数学分科会 委員長

• 2021
-
Now

核融合研究所  運営会議 委員

• 2021
-
Now

東京理科大学  総合研究機構 アドバイザリー委員会 委員

• 2020
-
Now

日本学術会議  数理科学委員会 委員長

• 2019
-
Now

京都大学  数理解析研究所 運営委員会委員

• 2019
-
Now

京都大学  数理解析研究所 専門委員会委員

• 2016
-
Now

Science Council of Japan  Council Member

• 2016
-
Now

日本学術会議  会員

• 2006
-
Now

日本学術会議  数理科学委員会 数学分科会 委員

• 2006
-
Now

日本学術会議  数理科学委員会 委員

• 2006
-
Now

Japan National Committee for IMU  Committee Member

• 2006
-
Now

国際数学連合(IMU) 国内委員会  委員

• 2017
-
2020

日本学術会議  科学者委員会 研究計画・研究資金検討分科会 委員

• 2017
-
2020

日本学術会議  数理科学委員会 数学分科会 副委員長

• 2017
-
2020

日本学術会議  数理科学委員会 副委員長

• 2012
-
2020

日本学術会議  数理科学委員会 IMU分科会 委員長

• 2012
-
2020

Japan National Committee for IMU  Chair

• 2012
-
2020

国際数学連合(IMU) 国内委員会  委員長

• 2015
-
2019

東京理科大学  総合研究機構 アドバイザリー委員会委員

• 2013
-
2017

京都大学  数理解析研究所 運営委員会委員

• 2009
-
2017

京都大学  数理解析研究所 専門委員会委員

• 2016

日本数学会  解析学賞選考委員長代理

• 2016

日本学術会議  学術の大型研究計画検討分科会・数理科学分野の大型研究計画評価小分科会 幹事

• 2014
-
2016

日本数学会  解析学賞選考委員

• 2006
-
2016

Science Council of Japan  Associate Member

• 2006
-
2016

日本学術会議  連携会員

• 2015

日本数学会  解析学賞選考委員長

• 2014

日本物理学会  第70回年次大会実行委員

• 2008
-
2011

日本数学会  教育研究資金問題検討委員会運営委員

• 2008
-
2010

日本数学会  福原賞選考委員会 選考委員長

• 2006
-
2009

Mathematical Society of Japan  Board of Trustees, Committee of International Relations

• 2006
-
2009

日本数学会  理事 国際交流担当, MSJ Memoirs担当

• 2005
-
2008

Mathematical Society of Japan  Division of Functional Equations Chair

• 2005
-
2008

日本数学会  函数方程式論分科会 連絡責任評議員

• 2005
-
2008

日本数学会  評議員

• 2003
-
2008

Mathematical Society of Japan  Division of Functional Equations Committee Member

• 2003
-
2008

日本数学会  函数方程式論分科会委員

• 1997

日本数学会  評議員

### Social Activities

• 副島隆彦を囲む会  （（財）全電通労働会館 全電通ホール）

2015.05

• キタノ工務店