In this paper we discuss the dissipative property of near-equilibrium classical solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann System in the whole space @math when the positive charged ion flow provides a spatially uniform background. The most key point of studying this coupled degenerately dissipative system here is to establish the dissipation of the electromagnetic field which turns out to be of the regularity-loss type. Precisely, for the linearized non-homogeneous system, some @math energy functionals and @math time-frequency functionals which are equivalent with the naturally existing ones are designed to capture the optimal dissipation rate of the system, which in turn yields the optimal @math - @math type time-decay estimates of the corresponding linearized solution operator. These results show a special feature of the one-species Vlasov-Maxwell-Boltzmann system different from the case of two-species, that is, the dissipation of the magnetic field in one-species is strictly weaker than the one in two-species. As a by-product, the global existence of solutions to the nonlinear Cauchy problem is also proved by constructing some similar energy functionals but the time-decay rates of the obtained solution still remain open.
In this paper, we are concerned with the one-species Vlasov–Poisson–Boltzmann 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.