This is a continuation of the paper [15] on nonlinear boundary layers of the Boltzmann equation where the existence is established and shown to be strongly dependent on the Mach number <i>M</i> ^<i></i> of the Maxwellian state at far field. In this paper, when <i>M</i> ^<i></i>&lt;1, we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers.

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using LittlewoodPaley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting.

In this paper, we are concerned with the one-species VlasovPoissonBoltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional.

This paper is concerned with the Cauchy problem on the VlasovPoissonBoltzmann system for hard potentials in the whole space. When the initial data is a small perturbation of a global Maxwellian, a satisfactory global existence theory of classical solutions to this problem, together with the corresponding temporal decay estimates on the global solutions, is established. Our analysis is based on time-decay properties of solutions and a new timevelocity weight function which is designed to control the large-velocity growth in the nonlinear term for the case of non-hard-sphere interactions.

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and NavierStokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation