The dynamics of dilute electrons can be modelled by the fundamental VlasovPoissonBoltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [17, 19] by obtaining new entropy estimates for this physical model.

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 consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates the stochastic system as a sequence of deterministic ones. We prove for the γ-distribution that the hypocoercivity framework ensures the stability of this series and hence the stochastic stability of the underlying random kinetic equation. The presented approach also yields a convergent numerical approximation.

G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in" Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of

The existence of a discrete travelling wave is proved for the relaxing scheme. The main idea is to change the original scheme such that the resulting scheme is monotonic to which Jennings' result can be applied. The equivalence of the resulting scheme and the original one is shown when = 1 q. The u-component of the discrete travelling wave thus obtained is a discrete shock for a monotone conservative difference scheme, which approximates the corresponding conservation law.