Since the work by Guo (Invent Math 153(3):593630, 2003), it has remained an open problem to establish the global existence of perturbative classical solutions around a global Maxwellian to the VlasovMaxwellBoltzmann system with the whole range of soft potentials. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper solves this problem in the whole space provided that initial perturbation has sufficient regularity and velocity-integrability.

In this paper, we will introduce a precise classification of characteristic functions in the Fourier space according to the moment constraint in the physical space of any order. Based on this, we construct measure-valued solutions to the homogeneous Boltzmann equation with the exact moment condition as the initial data.

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.

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The global existence of weak solutions to the compressible NavierStokes equations with vacuum attracts many research interests nowadays. For the isentropic gas, the viscosity coefficient depends on density function from physical point of view. When the density function connects to vacuum continuously, the vacuum degeneracy gives some analytic difficulties in proving global existence. In this paper, we consider this case with gravitational force and fixed boundary condition. By giving a series of <i>a priori</i> estimates on the solution coping with the degeneracy of vacuum, gravitational force and boundary effect, we give global existence and uniqueness results similar to the case without force and boundary. Copyright 2006 John Wiley &amp; Sons, Ltd.