In this paper we study the distribution of scattering resonances for a multi-dimensional semi-classical Schrödinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.

We prove the nonlinear asymptotic stability of the gravitational hydrostatic equilibrium for the general equation of state of pressure-density relation in the framework of vacuum free boundary problem of spherically symmetric compressible Navier-Stokes-Poisson equations in three dimensions.The results apply to white dwarfs and polytropes with γ≥2, for which even the linearized asymptotic stability results are not available in literature, to the best of the authors' knowledge. Detailed decay rates of perturbations are given.

We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently de- veloped formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well- posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.

This paper is concerned with the Cauchy problem of the one-dimensional compressible Navier--Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A result on the existence and uniqueness of a globally smooth nonvacuum solution is obtained provided that the -norm of the initial perturbation) for some positive constant independent of . Here is the adiabatic gas constant. This is a Nishida--Smoller type global solvability result with large data.

The Boltzmann equation without Grads angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial