In this paper, we study the global exisitence of L∞ weak entropy solution to the initial boundary value problem for compressible Euler equations with relaxtion and the large time asymptotic behavior of the solution. Motivated by the sub-characterisitic conditions, we proposed some structural conditions on the relaxation term comparing with the pressure function. These conditions are proved to b...

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz coefficients and even with coefficients which grow at most linearly. For super-linearly growing coefficients convergence in the strong and numerically weak sense remained an open question. In this article we prove for many stochastic differential equations with su...

We construct a numerical algorithm for the approximate solution of a nonlinear elastic limiting strain model based on the Fourier spectral method. The existence and uniqueness of the numerical solution is proved. Assuming that the weak solution to the boundary-value problem possesses suitable Sobolev regularity, the sequence of numerical solutions is shown to converge to the weak solution of th...

Standard weak solutions to the Poisson problem on a bounded domain have squareintegrable derivatives, which limits the admissible regularity of inhomogeneous data. The concept of solution may be further weakened in order to define solutions when data is rough, such as for inhomogeneous Dirichlet data that is only square-integrable over the boundary. Such very weak solutions satisfy a nonstandar...

We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coercivity conditions on the coefficients of the stochastic differential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an H– valued continuous process...

We introduce a class of reaction diffusion systems of which weak solution exists global-in-time with relatively compact orbit in L. Reaction term in this class is quasi-positive, dissipative, and up to with quadratic growth rate. If the space dimension is less than or equal to two, the solution is classical and uniformly bounded. Provided with the entropy structure, on the other hand, this weak...

This paper is devoted to the study of finite volume methods for the discretization of scalar conservation laws with a multiplicative stochastic force defined on a bounded domain D of R with Dirichlet boundary conditions and a given initial data in L(D). We introduce a notion of stochastic entropy process solution which generalizes the concept of weak entropy solution introduced by F.Otto for su...

In this paper, we numerically study the semiclassical limit of the SchrödingerPoisson equations as a selection principle for the weak solution of the VlasovPoisson in one space dimension. Our numerical results show that this limit gives the weak solution that agrees with the zero diffusion limit of the Fokker-Planck equation. We also numerically justify the multivalued solution given by a momen...

We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu’s theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotics of an unstead...

We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set Γo a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.