As a continuation of [28], the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on the velocity field and the perfect conducting boundary condition is given for the magnetic field. Under an assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a L8 estimate on the error by multi-scale analysis.

In this paper, we first show that the regular solutions of compressible Euler equations in<i>R</i>³with damping will not be global if the initial density function has compact support. This implies that<i>c</i>²cannot be smooth across the boundary<i></i>separating the gas and the vacuum after a finite time, where<i>c</i>is the speed of sound. Then we study the local existence of solutions for isentropic gas flow in<i>R</i>when<i>c</i>, 0&lt;<i></i>1, is smooth across<i></i>, using the energy method and the characteristics method.

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf-sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier-Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to <i>p</i>-system which consists of two different families of shocks interacting at some positive time, we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.