Convergence of the point vortex method for 2-D vortex sheet

We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

     

On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

We establish a rigorous well-posedness results for the Marchenko system associated to the scattering theory of the one dimensional Gross-Pitaevskii equation (GP). Under some assumptions on the scattering data, these well-posedness results provide regular solutions for (GP). We also construct particular solutions, called Nsoliton solutions as an approximate superposition of traveling waves. A study for the asymptotic behaviors of such solutions when t → ± ∞ is also made.     

具阻尼的Gross-Pitaevskii(GP)方程在二维空间中的稳定性

本文讨论出现在吸引玻色-爱因斯坦凝聚中的一类带调和势的阻尼非线性Schr dinger方程.对照玻色-爱因斯坦凝聚的物理性质,证明了阻尼参数存在一个门槛值,即当阻尼参数大于该门槛值时,初值问题的解整体存在;当阻尼参数小于该门槛值时,其初值问题的解将在有限时间内坍塌.     

改进欧拉格式求解随机微分方程的收敛性

采用改进的欧拉格式求解随机微分方程,当方程的偏移系数和扩散系数均满足全局Lipschitz条件和线性增长条件时,证明改进格式的强收敛的阶是1/2.     

Naimark-Sacker Bifurcations in the Euler Method for a Delay Differential Equation

We consider the application of the Euler method to a delay differential equation which has a Hopf bifurcation, and prove that Naimark-Sacker bifurcations, i.e., Hopf bifurcations of maps, occur in the discretization by using the center manifold theorem and the Naimark-Sacker theorem. We also present obstacles to the generalization of the result.     

Zero Dissipation Limit to Rarefaction Waves for the 1-D Compressible Navier-Stokes Equations

The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investi...     

Profiles of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Cauchy problem for Gross-Pitaevskii equation.In terms of Merle and Raphёel＇s arguments as well as Carles＇ transformation,the limiting profiles of blow-up solutions are obtained.In addition,the nonexistence of a strong limit at the blow-up time and the existence of L2 profile outside the blow-up point for the blow-up solutions are obtained.     

Upper bound of weak-limitation for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of Gross-Pitaevskii equation. We obtain the upper bound of weak-limitation for the blow-up solutions by using the method of Cazenave (2003) [3] as well as the concentration compact principle.     

利用齐次平衡法求GP方程的Jacobi椭圆函数解

描述玻色-爱因斯坦凝聚（BEC）的有效而方便的方程是著名的Gross-Pitaevskii（GP）方程。本文在将GP方程变换为非线性薛定谔方程（NLS）的基础上，利用齐次平衡法求出了Gross-Pitaevskii（GP）方程的一系列Jacobi椭圆函数解。     

The Inviscid Limit for the Steady Incompressible Navier-Stokes Equations in the Three Dimension*

In this paper, the authors consider the zero-viscosity limit of the three dimensional incompressible steady Navier-Stokes equations in a half space R⁺×R². The result shows that the solution of three dimensional incompressible steady Navier-Stokes equations converges to the solution of three dimensional incompressible steady Euler equations in Sobolev space as the viscosity coefficient going to zero. The method is based on a new weighted energy estimates and Nash-Moser itera...     

Semiclassical Limit of Nonlinear Schrodinger Equation (II)

In this paper, we use the Wigner measure approach to study the semiclassical limit of nonlinear Schrödinger equation in small time. We prove that: the limits of the quantum density: ρ^∈ =: |ψ^∈|² and the quantum momentum: J^∈ =: ∈Im(\overline{ψ^∈}∇ψ^∈) satisfy the compressible Euler equations before the formation of singularities in the limit system.     

Hydrodynamic limits: some improvements of the relative entropy method

The present paper is devoted to the study of the incompressible Euler limit of the Boltzmann equation via the relative entropy method. It extends the convergence result for well-prepared initial data obtained by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80]. It explains especially how to take into account the acoustic waves and relaxation layer, and thus to obtain convergence results under weak assumptions on the initial data.     

The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases

This paper concerns the threshold of global existence and finite time blow up of solutions to the time-dependent focusing Gross-Pitaevskii equation describing the Bose-Einstein condensation of trapped dipolar quantum gases. Via a construction of new cross-constrained invariant sets, it is shown that either the corresponding solution globally exists or blows up in finite time according to some appropriate assumptions about the initial datum.     

非线性散焦立方薛定谔方程的半古典极限

全空间上非线性散焦立方薛谔方程的半古典极限被证明，由量子力学中的薛定谔方程的解所定义的电子密度和电流密度当Planck常数趋于零时收敛于古典力学中的非线性可压缩欧拉方程的解，同时相应的Wigner函数的测度意义下收敛于非线性Viasov方程的解，这些结果的证明基于“改变”的Kinetic能量函数的估计和Winger测度方法。     

Limiting profile of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Gross-Pitaevskii equation. Using the concentration compact principle and the variational characterization of the corresponding ground state, we obtain the limiting profile of blow-up solutions with critical mass in the corresponding weighted energy space. Moreover, we extend this result to small super-critical mass case by the variational methods and scaling technique. This work was supported by National Natural Science Foundation of China (Grant No. 10771151) and Scientific Research Fund of Sichuan Provincial Education Department (Grant No. 2006A068)     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Vortex Motion Law of an Evolutionary Ginzburg-Landau Equation in 2 Dimensions

We study the asymptotic behavior of solutions to an evolutionary Ginzburg-Landau equation. We also study the dynamical law of Ginzburg-Landau vortices of this equation under the Neuman boundary conditions. Away from the vortices, we use some measure theoretic arguments used by F.H.Lin in [1] to show the strong convergence of solutions. This is a continuation of our earlier work [2].