In this paper, we study the existence of stationary solutions to the VlasovPoissonBoltzmann system when the background density function tends to a positive constant with a very mild decay rate as| x|. In fact, the stationary VlasovPoissonBoltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln (e+| x|)) for some > 0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the VlasovPoissonFokkerPlanck system, J. Math. Anal. Appl. 202 (1996) 10581075] where the decay rate (1+| x|) 1 2 is assumed.

This paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries, which is described by a degenerate and strongly coupled parabolic system. We prove the global (in time) existence of weak solution by constructing an approximate solution with some standard smoothing. The proof is based on the physical nature of gas convection, in which the heat (energy) flux in convection is determined by the mass (vapor) flux in convection.

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions; the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium, to be precise. Together with the results of Parts I and II about the well-posedness of the Cauchy problem around the Maxwellian, we conclude this series with a satisfactory mathematical theory for the Boltzmann equation without angular cutoff.

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space dependent density, zero velocity and constant temperature. In this paper, we will study the nonlinear stability of these stationary solutions by using the energy method. The analysis combines the analytic techniques used for the conservation laws using the fluid-type system derived from the Boltzmann equation (cf. [14]) and the dissipative effects on the fluid and non-fluid components of the Boltzmann equation through the celebrated H-theorem. To our knowledge, this is the first result on the global classical solutions to the Boltzmann equation with external force and non-trivial large time behavior in the whole space.

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G-equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.