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.

Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hlder regularity (dimension three) of the shear flows. Remarks are made on the quadratic and linear laws of front speed expectation in the small and large root mean square regimes.

We prove that the strong detonation travelling waves for a viscous combustion model are nonlinearly stable, that is, for a given perturbation of a travelling wave, the solution converges to a shifted travelling wave asymptotically as<i>t</i> . The rate of convergence is obtained. This work is a continuation of a previous work by done T.P. Liu and the first author in [7].

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative

The ellipsoidal BGK model was introduced in\cite {Ho} to fit the correct Prandtl number in the Navier-Stokes approximation of the classical BGK model. In the paper we establish the global existence of mild solutions to the Cauchy problem on the model for a class of initial data allowed to have large oscillations. The proof is motivated by a recent study of the same topic on the Boltzmann equation in\cite {DHWY}.