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.

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound u (t,) u (t,)= O (1)(1+ t) | ln | on the distance between an exact BV solution u and a viscous approximation u, letting the viscosity coefficient 0. In the proof, starting from u we construct an approximation of the viscous solution u by taking a mollification u* and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed . Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. 2004 Wiley Periodicals, Inc.

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^ 2$ or $\bT^ 2$. Higher regularity of mild solutions with arbitrary initial data in H^{2-} is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic H^{2-} data. Some decay in time estimates are also provided.

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the <i>C</i>⁰-norm of the initial data is sufficiently small, while the <i>C</i>¹-norm of the initial data can be large. The analysis is based on <i>a priori</i> estimates, which are obtained by a generalised Lax transformation.

For systems of hyperbolic conservation laws, a new Glimm functional was recently constructed when the linearly degenerate manifold in each characteristic field is either the whole space or it consists of a finitely many smooth and transversal manifolds of codimension one. This new functional leads to the neat consistency and convergence rate estimation of the Glimm scheme. In this paper, by the motivation of the result in Ref. 2, we show that the corresponding new Glimm functional can be constructed for general systems only under the strict hyperbolicity assumption.