Data In Hs Research Articles

Well-posedness of the Hele–Shaw–Cahn–Hilliard system

We study the well-posedness of the Hele–Shaw–Cahn–Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in Hs, s>d2+1, the existence and uniqueness of solution in C([0,T];Hs)∩L2(0,T;Hs+2) that is global in time in the two dimensional case (d=2) and local in time in the three dimensional case (d=3) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood–Paley theory in order to establish certain key commutator estimates.

GLOBAL WEAK SOLUTIONS FOR A PERIODIC HUNTER–SAXTON EQUATION WITH WEAK DISSIPATION

In this paper, we consider a periodic Hunter–Saxton equation with weak dissipation. We obtain the existence of global weak solutions to the equation for any initial data in H1(𝕊).

Ill-posedness of modified kawahara equation and kaup-kupershmidt equation

In this article, we consider the Cauchy problems for the modified Kawahara equation∂tu+μ∂x(u3)+α∂x3u+β∂x3u+γ∂xu=0and the Kaup-Kupershmidt equation∂tu+μu∂x2u+α∂x5u+β∂x3u+γ∂xu=0.Using the general well-posedness principle introduced by I. Bejenaru and T. Tao, we prove that the modified Kawahara equation is ill-posed for the initial data in Hs(R) with s<−14 and that the Kaup-Kupershmidt equation is ill-posed for the initial data in Hs(R) with s < 0.

Well-Posedness for the Fifth Order KdV Equation

We consider the Cauchy problem of the fifth order KdV equation with low regularity data. We cannot apply the iteration argument to this problem when initial data is given in the Sobolev space Hs for any s ∈ R. So we give initial data in Hs,a = Hs ∩ Ha with a ≤ min{s, 0}. Then we recover more derivatives of the nonlinear term to be able to use the iteration method. Therefore we obtain the local well-posedness in Hs,a in the case s ≥ max{–1/4, –2a – 2}, –3/2 –1/4. The main tool is a variant of the Fourier restriction norm method, which is based on Kishimoto's work (2009).

Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H1(T3)

A refined trilinear Strichartz estimate for solutions to the Schr\"odinger equation on the flat rational torus T^3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic Nonlinear Schr\"odinger Equation in H^s(T^3) for all s \geq 1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.

Well-posedness of a higher order modified Camassa–Holm equation in spaces of low regularity

In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the Hs(R) with \({s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}\) As a consequence of the conservation of the energy \({{||u||_{H^{1}(R)},}}\) we have the global well-posedness for the initial data in H1(R).

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

This paper is devoted to studying the initial-value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces Xs, b(R2) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I-method as well as L2 conservation law, we show that this fifth-order shallow water wave equation is globally well-posed for the initial data in the Sobolev spaces Hs(R) with \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$s{>}-\frac{63}{58}$\end{document}. Copyright © 2010 John Wiley & Sons, Ltd.

Random data Cauchy problem for supercritical Schrödinger equations

In this paper we consider the Schrödinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space Hs if s is large enough and strongly ill-posed is s is below some critical threshold sc. Here we use the randomisation method of the inital conditions, introduced by N. Burq and N. Tzvetkov, and we are able to show that the equation admits strong solutions for data in Hs for some s<sc.

Global well-posedness and scattering for the defocusing [formula omitted]-subcritical Hartree equation in [formula omitted

We prove the global well-posedness and scattering for the defocusing H12-subcritical (that is, 2<γ<3) Hartree equation with low regularity data in Rd, d⩾3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space Hs(Rd) with s>4(γ−2)/(3γ−4), which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for any s>max(1/2,4(γ−2)/(3γ−4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution Iu, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m(ξ)⋅〈ξ〉p. As a byproduct of our proof, we obtain that the Hs norm of the solution obeys the uniform-in-time bounds.

Existence theorem and blow‐up criterion of the strong solutions to the magneto‐micropolar fluid equations

AbstractIn this paper we study the magneto‐micropolar fluid equations in ℝ3, prove the existence of the strong solution with initial data inHs(ℝ3) for$s&gt;{3\over2}$, and set up its blow‐up criterion. The tool we mainly use is Littlewood–Paley decomposition, by which we obtain a Beale–Kato–Majda‐type blow‐up criterion for smooth solution (u, ω,b) that relies on the vorticity of velocity ∇ × uonly. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Global well-posedness of the 2D critical dissipative quasi-geostrophic equation in the Triebel–Lizorkin spaces

We prove global well-posedness of the 2D critical dissipative quasi-geostrophic equation for small initial data in the Triebel–Lizorkin spaces F p , q s , s > 2 p , p , q ∈ ( 1 , ∞ ) . In particular, our result implies the global existence for small initial data in the Sobolev space H s , p , s > 2 p , p ∈ ( 1 , ∞ ) since F p , 2 s = H s , p . The key ingredients of our proof are the pointwise exponential decay estimate of the semigroup e − t κ ( − Δ ) α and maximum principle.

CONCENTRATION PHENOMENON FOR THE L2 CRITICAL AND SUPER CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS IN ENERGY SPACES

We show that the blow up solution with data in energy spaces H1 of the nonlinear Schrödinger equation i ut + Δu + |u|αu = 0 has a concentration behavior in the critical Sobolev space Ḣsα (0 &lt; sα = n/2 - 2/α ≤ 1) if the time variable tends to the blow up time. Moreover, the blow up points in ℝn are also quantified in the L2 critical case α = 4/n.

On Global Finite Energy Solutions of the Camassa-Holm Equation

We consider the Camassa-Holm equation with data in the energy norm H1(R1). Global solutions are constructed by the small viscosity method for the frequency localized equations. The solutions are classical, unique and energy conservative. For finite band data, we show that global solutions for CH exist, satisfy the equation pointwise in time and satisfy the energy conservation law. We show that blow-up for higher Sobolev norms generally occurs in finite time and it might be of power type even for data in H3/2−.

The Cauchy Problem for the Fifth Order Shallow Water Equation

The local well-posedness of the Cauchy problem for the fifth order shallow water equation $$ \partial _{t} u + \alpha \partial ^{5}_{x} u + \beta \partial ^{3}_{x} u + \gamma \partial _{x} u + \mu u\partial _{x} u = 0,\;x,t \in \mathbb{R}, $$ is established for low regularity data in Sobolev spaces $$ H^{s} {\left( {s \geqslant - \frac{3} {8}} \right)} $$ by the Fourier restriction norm method. Moreover, the global well-posedness for L 2 data follows from the local well-posedness and the conserved quantity. For data in H s (s > 0), the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L 2 norm of initial data.

WELL-POSEDNESS FOR THE CAUCHY PROBLEM TO THE HIROTA EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES

The local well-posedness of the Cauchy problem for the Hirota equation is established for low regularity data in Sobolev spaces Hs(s≥-¼). Moreover, the global well-posedness for L2 data follows from the local well-posedness and the conserved quantity. For data in Hs(s>0), the global well-posedness is also proved. The main idea is to use the generalized trilinear estimates, associated with the Fourier restriction norm method.

Ground state mass concentration in the $L^2$-critical nonlinear Schrödinger equation below $H^1$

We consider finite time blowup solutions of the L2-critical cubic focusing nonlinear Schrodinger equation on R 2. Such functions, when in H1, are known to concentrate a fixed L2-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in L2 are known to concentrate at least a small amount of mass. In this paper we consider the intermediate case of blowup solutions from initial data in Hs, with 1 > s > sQ, where sQ = 1 5 + 1 5 √ 11. Our main result is that such solutions, when radially symmetric, concentrate at least the mass of the ground state at the origin at blowup time.

Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schr�dinger Equation

We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.

Well-posedness of Cauchy problems for Korteweg-de Vries-Benjamin-Ono equation and Hirota equation

Abstract The well-posedness of the Cauchy problems to the Korteweg-de Vries-Benjamin-Ono equation and Hirota equation is considered. For the Korteweg-de Vries-Benjamin-Ono equation, local result is established for data in Hs (R)(s⩾−1/8). Moreover, the global well-posedness for data in L 2(R) can be obtained. For Hirota equation, local result is established for initial data in Hs (s⩾1/4). In addition, the local solution is proved to be global in Hs (s⩾1)if the initial data are in Hs (s⩾1) by energy inequality and the generalization of the trilinear estimates associated with the Fourier restriction norm method.

The initial–boundary-value problem for real viscous heat-conducting flow with shear viscosity

An initial–boundary-value problem for the nonlinear equations of real compressible viscous heat-conducting flow with general large initial data is investigated. The main point is to study the real flow for which the pressure and internal energy have nonlinear dependence on temperature, unlike the linear dependence for ideal flow, and the viscosity coefficients and heat conductivity are also functions of density and/or temperature. The shear viscosity is also presented. The existence, uniqueness and regularity of global solutions are established with large initial data in H1. It is shown that there is no shock wave, vacuum, mass concentration, or heat concentration (hot spots) developed in a finite time, although the solutions have large oscillations.

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H 1. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.