The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

Global Existence and Scattering in L2 for the Mass-critical Hartree Equation With H0,1 Initial Data

In this paper we consider the Cauchy Problem for the mass-critical Hartree equation I(e)tu △u=μ(|x|2*|u|2)u,(t,x)∈R×Rn,n≥3,(1) u(0,x)=φ(x), x∈Rn,(2)     

On the Well-Posedness for Stochastic Schr¨odinger Equations with Quadratic Potential

The authors investigate the influence of a harmonic potential and random perturbations on the nonlinear Schrödinger equations. The local and global well-posedness are proved with values in the space Σ(? ⁿ ) = {f ∈ H ₁(? ⁿ ), |·|f ∈ L ²(? ⁿ )}. When the nonlinearity is focusing and L ²-supercritical, the authors give sufficient conditions for the solutions to blow up in finite time for both confining and repulsive potential. Especially for the repulsive case, the solution to the deterministic equation with the initial data satisfying the stochastic blow-up condition will also blow up in finite time. Thus, compared with the deterministic equation for the repulsive case, the blow-up condition is stronger on average, and depends on the regularity of the noise. If ? = 0, our results coincide with the ones for the deterministic equation.     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative

In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.     

半线性Schr(o)dinger方程初值问题的H1-适定性

在RN(N≥2)中讨论一类具时间依赖系数的半线性Schr(o)dinger方程初值问题的H1-适定性.通过Strichartz估计和解的先验估计,得到了系数零点阶数λ与临界适定指数σ之间的关系,在一定条件下证明了该问题的H1-局部适定性和整体适定性.     

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.     

Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,

     

Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation

We establish the local well-posedness for the viscous Degasperis-Procesi equation. We show that the blow-up phenomena occurs in finite time. Moreover, applying the energy identity, we obtain a global existence result in the energy space.     

Global well-posedness for the Euler-Boussinesq system with axisymmetric data

In this paper we prove the global well-posedness for the three-dimensional Euler-Boussinesq system with axisymmetric initial data without swirl. This system couples the Euler equation with a transport-diffusion equation governing the temperature.     

Well-posedness,Decay Estimates and Blow-up Theorem for the Forced NLS

In this article we prove that the following NLS iu_t = u_{zz}-g|u|^{P-1}u,g > O, x, t > 0 with either Dirichlet or Robin boundary condition at x = 0 is well-posed. L^{p + 1} decay estimates, blow-up theorem and numerical results are also given.     

Global well-posedness for the fifth-order mKdV equation

We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg–de Vries equation in Sobolev spaces H~s(R) for s-(3/(22)).The main approach is the"I-method"together with the multilinear multiplier analysis.     

Global well-posedness of partially periodic KP-I equation in the energy space and application

In this article, we address the Cauchy problem for the KP-I equation

${?}_{t}u+{?}_x^{3}u?{?}_x^{?1}{?}_y^{2}u+u{?}_{x}u=0$

for functions periodic in y. We prove global well-posedness of this problem for any data in the energy space $\mathbf{E}=\{u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_{x}u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_x^{?1}{?}_{y}u\in L^2(\mathbb{R}\times \mathbb{T})\}$. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.     

The defocusing energy-supercritical NLS in higher dimensions

We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.