In the present paper, we consider the initial value problem for the bipolar Vlasov-Poisson-Boltzmann (bVPB) system and its corresponding modified Vlasov-Poisson-Boltzmann (mVPB). We give the spectrum analysis on the linearized bVPB and mVPB systems around their equilibrium state and show the optimal convergence rate of global solutions. It was showed that the electric field decays exponentially and the distribution function tends to the absolute Maxwellian at the optimal convergence rate (1+ t)^{-3/4} for the bVPB system, yet both the electric field and the distribution function converge to equilibrium state at the optimal rate (1+ t)^{-3/4} for the mVPB system.

In this paper, we study the strong convergence of a sequence of uniform<i>L^p</i>_loc(<b>R</b><b>R</b>⁺) bounded approximate solutions {<i>u</i>(<i>x</i>,<i>t</i>)} to the following scalar conservation laws[formula]with initial data[formula]Without the convexity assumption and growth condition at infinity for<i>f</i>(<i>x</i>,<i>t</i>,<i>u</i>), we prove strong convergence of a subsequence of {<i>u</i>(<i>x</i>,<i>t</i>)}. Under a more general growth condition than those in the previous work, we prove the existence of weak solution for the equation. The result obtained here generalizes those in earlier work. Some applications of the results are also given at the end of this paper.

Large-time behavior of solutions to the inflow problem of full compressible NavierStokes equations is investigated on the half line \mathbf{R}_+=(0,+\infty). The wave structure, which contains four wavesthe transonic (or degenerate) boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave, and the 3-rarefaction wave to the inflow problemis described, and the asymptotic stability of the superposition of the above four wave patterns to the inflow problem of full compressible NavierStokes equations is proven under some smallness conditions. The proof is given by the elementary energy analysis based on the underlying wave structure. The main points in the proof are the treatments of the degeneracies in the transonic boundary layer solution and the wave interactions in the superposition wave.

METHODS AND APPLICATIONS OF ANALYSIS. c 2009 International Press Vol. 16, No. 4, pp. 507520, December 2009 006

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