Trilinear Lp estimates with applications to the Cauchy problem for the Hartree-type equation

In this paper, $L^p$ estimates for a trilinear operator associated with the Hartree type nonlinearity are proved. Moreover, as application of these estimates, it is proved that after a linear transformation, the Cauchy problem for the Hartree-type equation becomes locally well posed in the Bessel potential and homogeneous Besov spaces under certain regularity assumptions on the initial data. This notion of well-posedness and the functional framework to solve the equation were firstly proposed by Y. Zhou.     

Sharp threshold of blow-up and scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2<M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, then the solution $u(t)$ is globally well-posed and scatters; if $M{[u_0]}^{\frac{s?s_c}{s_c}}E[u_0]<M{[Q]}^{\frac{s?s_c}{s_c}}E[Q]$ and $M{[u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2>M{[Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.     

Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

We consider the defocusing, -critical Hartree equation for the radial data in all dimensions (n5). We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.     

Well-posedness of the Cauchy problem for Ostrovsky, Stepanyams and Tsimring equation with low regularity data

In this paper we prove that the initial value problem of the OST equation ut+uxxx+η(Hux+Huxxx)+uux=0 (x∈R, t?0), where η>0 and H denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space Hs(R) when , and globally well-posed in Hs(R) when s?0.     

Energy-critical Hartree equation with harmonic potential for radial data

In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions (n≥5) and show the global well-posedness and scattering theory in the space Σ=H¹∩FH¹. We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao’s approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory.     

On the Cauchy problem for the noncompact Landau-Lifshitz-Gilbert equation

Local well-posedness of the Cauchy problem for the noncompact Landau-Lifshitz-Gilbert equation is investigated via the pseudo-stereographic projection. Existence of global solutions is established for small initial data. In the case of one space dimension global existence theorems are proved for large initial data.     

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

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On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

Finite dimensional global attractor for a dissipative anisotropic fourth order Schrödinger equation

We study the long-time behavior of the solutions to a nonlinear damped anisotropic fourth order Schrödinger type equation in ${\mathbb{R}}^2$. We prove that this behavior is described by the existence of regular finite-dimensional global attractor in the energy space.     

On cauchy problem of the Benney-Lin equation with low regularity initial data

In this work we prove that the initial value problem of the Benney-Lin equation ut ＋ uxxx ＋ β（uxx ＋ u xxxx） ＋ ηuxxxxx ＋ uux = 0 （x ∈ R, t ≥0 0）, where β 〉 0 and η∈R, is locally well-posed in Sobolev spaces HS（R） for s ≥ -7/5. The method we use to prove this result is the bilinear estimate method initiated by Bourgain.     

Cauchy problem for the multi-dimensional Boussinesq type equation

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt−Δu+Δ²u=Δσ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ∈C(R), σ(s) is of polynomial growth order, say p (>1), either , s∈R, where β>0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N=1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown.     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

The Cauchy problem for a generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces $H^s$, $s>3/2$ for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in $H^r$-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

Standing waves for the NLS on the double-bridge graph and a rational–irrational dichotomy

We study standing waves of NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$. We study the solutions with possibly nontrivial components on the half-lines and a cnoidal component on the circle. The problem is equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter ω. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of negative frequencies ${\omega }_n$. Moreover we show that the frequency sequence admits cluster points and any negative real number can be a limit point of frequencies choosing a suitable irrational geometry $L_1/L_2$. These results depend on basic properties of diophantine approximation of real numbers.     

Vanishing viscosity limit of short wave–long wave interactions in planar magnetohydrodynamics

We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schrödinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an aurora-type phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters, to a solution of the limit decoupled system involving the compressible Euler equations and a nonlinear Schrödinger equation. The vanishing viscosity limit serves to justify the SW–LW interactions in the limit equations as, in this setting, the SW–LW interactions cannot be defined in a straightforward way, due to the possible occurrence of vacuum.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.