In this article, we study a class of divergence Schrödinger equations with intercritical inhomogeneous nonlinearity [Formula: see text] where 2 − n < b < 2, c ≥ b − 2, and 2(2 − b) − bp < np − 2 c < (2 − b)( p + 2). We prove the blow-up of radial solutions for the negative energy by using a virial-type estimate. In addition, we derive a generalized Gagliardo–Nirenberg inequality and use it to establish the general blow-up criteria for radial solutions with non-negative energy.

We study the vanishing dispersion limit of strong solutions to the Cauchy problem for the Schrödinger-improved Boussinesq system in a two dimensional domain. We show an explicit representation of limiting profile in terms of the initial data. Moreover, the first approximation is also represented as a pair of solutions of a linear system with coefficients and forcing term given by the limiting profile.

<p>In this paper, we prove some new -estimates of the velocity by the technique of -energy method.</p>

Abstract

We study the Cauchy problem for the Schrödinger-improved Boussinesq system in a two-dimensional domain. Under natural assumptions on the data without smallness, we prove the existence and uniqueness of global strong solutions. Moreover, we consider the vanishing “improvement” limit of global solutions as the coefficient of the linear term of the highest order in the equation of ion sound waves tends to zero. Under the same smallness assumption on the data as in the Zakharov case, solutions in the vanishing “improvement” limit are shown to satisfy the Zakharov system.

<p lang="fr">&lt;abstract&gt;&lt;p&gt;First, we prove uniform-in-$ \epsilon $ regularity estimates of local strong solutions to the Chern-Simons-Schrödinger equations in $ \mathbb{R}^2 $. Here $ \epsilon $ is the dispersion coefficient. Then we prove the global well-posedness of strong solutions to the limit problem $ (\epsilon = 0) $.&lt;/p&gt;&lt;/abstract&gt;</p>

<p lang="fr">&lt;abstract&gt;&lt;p&gt;In this paper, we will use the Banach fixed point theorem to prove the uniform-in-$ \epsilon $ existence of the compressible full magneto-micropolar system in a bounded smooth domain, where $ \epsilon $ is the dielectric constant. Consequently, the limit as $ \epsilon\rightarrow0 $ can be established. This approximation is usually referred to as the magnetohydrodynamics approximation and is equivalent to the neglect of the displacement current.&lt;/p&gt;&lt;/abstract&gt;</p>

<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>.

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 &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.

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 &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.

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 &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.

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 &lt; s &lt; 3/2. In case 1/2 &lt; s &lt;= 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 &gt;= 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 &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.

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 &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.

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 &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.

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.

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).

In this paper we show the existence of a non-trivial solution to a fractional Schrödinger equation, which turns out to be a global minimizer of a variational problem. We give a sharp condition for the existence of global minimizers. To do this we develop a new compactification scheme of angularly regular solutions.

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&apos;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 &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.

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.

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 &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.

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 &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.

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 &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)).

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.

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 &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.

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
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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 &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.

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 --&gt; 0 in R-n, n &gt;= 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 -&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.

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 &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.

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.

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 &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.

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 &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.

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 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.

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.

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.

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.

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 &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}.

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
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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}.

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 --&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.

This paper is concerned with the initial value problem for nonlinear Schrodinger equations of the form
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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.

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 &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.

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) &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.

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 &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.

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.

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.