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).

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.

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].

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.

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.

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 > 0 and rho > 0 is small enough. The minimization problem is L-2 critical and in order to characterize the values alpha, beta > 0 such that I-alpha,I- beta(rho) > -infinity for every rho > 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 > 0 such that
1/S parallel to phi parallel to(L8/3(R3))/parallel to phi parallel to(1/2)(H1/2(R3)) <= (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.

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.

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.

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.

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.

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.

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.

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

Sharp remainder terms are explicitly given on the standard Hardy inequalities in L-p(R-n) with 1 < p < 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.

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

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.

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.

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.

In this note we study Hartree type equations with vertical bar del vertical bar(alpha) (1 < alpha <= 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 < gamma(1) <= 1 and small data scattering in H-S for s > 2 2-alpha/2 when 2 < gamma(1) <= d and 0 < gamma(2) <= 2. For this we use U-P-V-P space argument and Strichartz estimates.

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.

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.

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.

© 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.

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.

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

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

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.

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.

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.

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

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

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 > 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 -> 0 can be established.

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.

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.

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 >= 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.

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.

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.

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.

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.

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.

© 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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.
licensee Springer.

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 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.

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 >= 1, m >= 0, 1 < alpha < 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 < gamma < n, and 0 <= 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.

We study a system of nonlinear Schrodinger equations with quadratic interaction in space dimension n <= 6. The Cauchy problem is studied in L-2, in H-1, and in the weighted L-2 space < x > L--1(2) = F(H-1) under mass resonance condition, where < x > = (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.

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.

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.

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.

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

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

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.

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.

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

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.

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.

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.

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].

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 >= 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 < gamma < n and 0 <= delta <= 1. (C) 2010 Elsevier Ltd. All rights reserved.

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 < q <= 6. Global well-posedness of smooth solution is also proven for a regularization model of this Oldroyd-B system in two space dimension.

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.

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.

We prove the global existence of analytic solutions to the Cauchy problem for the cubic Schrodinger equation in space dimension n >= 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.

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.

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.

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.

In this paper, we derive some Sobolev inequalities for radially symmetric functions in. (H)over dot(s) with 1/2 < s < 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' 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.

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.

Some properties of distributions f satisfying x . del f is an element of L(p)(R(n)), 1 <= p < 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)).

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 < gamma < n, n >= 3. We establish a weighted L(2) Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.

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.

In this paper, we consider elliptic estimates for a system with smooth variable coefficients on a domain Omega subset of R(n), n >= 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 > 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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

We study the global Cauchy problem for nonlinear Schrodinger equations with cubic interactions of derivative type in space dimension n >= 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.

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.

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 --> 0 in R-n, n >= 1.

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.

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 -> 0. Solutions are considered in H-s space for all s > 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 > 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 -> infinity. We also prove non-existence of non-trivial asymptotically free solutions for 1 < p <= 2 under vanishing condition near zero frequency on asymptotic states.

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.

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.

In this paper, we consider the Cauchy problem of Schrodinger-IMBq equations in R-n, n >= 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 < s < min (2, p) and some p > n/2. We also provide a blowup criterion for n = 3 in Triebel-Lizorkin space containing BMO space naturally.

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 < gamma < n. We prove the existence and uniqueness of global solutions for 0 < gamma < (2n)/(n + 1), n >= 2 or gamma > 2, n >= 3, and the nonexistence of asymptotically-free solutions for 0 < gamma <= 1, n >= 3. We also specify asymptotic behavior of solutions as the mass tends to zero and in. nity.

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 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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 > 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.

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.

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 < 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 <= p(1) <= p(2) < 1 + 4/(n - 2s) for NLS and NLKG, and 1 + 4/(n - 1) <= p1 <= p(2) <= 1 + 4/(n - 2s) for NLW.

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 < p < infinity, since the former works for p greater than or equal to 1 + 4/n, while the latter for 1 + 2/n < 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 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 > 0 and 0 < v < 2 if s = n/2, and has an arbitrary growth rate if s > 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 < s* ≤ s if s > 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.

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 > 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 < σ ≤ 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.

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 > 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 < sigma < 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.

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 < n/2 using the generalized Strichartz estimates given by J. Ginibre and G. Velo (1995). (C) Elsevier, Paris.

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.

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 < gamma < 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 < sigma < 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 < nu < min(1, gamma-sigma/12), <(phi)over cap> 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) < infinity}.

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.

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.

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.

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) < infinity}.

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.

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.

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 --> 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.

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 < p2 < 5. It is shown that if phi is-an-element-of H1(R) and parallel-tophiparallel-to2(2) < 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.

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.

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.

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 > 4/(n + 2) to the optimal range p - 1 > 2/n in space dimension n = 3 and to an intermediate range in space dimension n greater-than-or-equal-to 4.

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.

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.

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) > 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 < gamma-1, gamma-2 < 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 < gamma-1 < gamma-2 = 2 is also treated according to the sign of lambda-2.

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.

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.

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 > 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.

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 < gamma less-than-or-equal-to 1/2, as \x\ --> 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 > 1, as x --> - infinity.

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.

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。

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．

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.

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.

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.

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.

(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.

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

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

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.

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

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.

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.

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.

非線形のSchrodinger方程式、Klein-Gordon方程式で長い有効距離をもつ相互作用系を散乱問題および初期値問題として扱った。これら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拠り所であるばかりでなく非線型偏微分方程式の理論体系から見ても重要な数学的対象である。非線型効果の長時間的影響には方程式のもつ様々な個性が関与する為既存の一般論では的確に取扱えないという困難を常に内包している。特に時空1+1次元の三次非線型性或いは1+2次元の二次非線型性は物理的にしばしば現れるモデルであるが時間無限大においてその波動函数は漸近自由解には収束しない。これは漸近自由解を立脚点とする従来の非線型散乱理論においては深刻な問題であり1970年代に始まり現在に至る迄少しずつ認識されてきた経緯がある。一般的に云ってこの問題は低次元空間における低次非線型性をもつ偏微分方程式に数多く起こり物理的には波動函数の位相因子の歪みとなって現れる。さて上記問題を合理的に解決せよというのがReadの問題であり長年に亘り未解決であった。研究代表者はまずFlato-Simon-TaflinのMaxwell-Dirac系を扱った研究を手掛りとし上記問題を空間一次元のSchrodinger方程式について解決した。次いでその理論の高次元化に取組みGinibreとの共同研究にてこれを成功させた。同様の試みはKlein-Gordon方程式についても成功するものと思われるが現在の所完成していない。但し空間二次元の二次非線型性の相互作用系についてはその数学的機構はほぼ明らかとなった。具体的には波動函数のもつ光円錐状での特異性発生の構造をポワンカレ群の生成作用素のなす不変ソボレフ空間にて解析的に記述することに成功し位相因子の歪みの統制が可能であることを示した。

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。

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.

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。

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。

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

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

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.

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。

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).

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).

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.

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.

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.

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.

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

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

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).

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).

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.

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.

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.

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.

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.

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。

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.

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.