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.
Recently, following a combination of @cite_13 and @cite_4 , the optimal large-time behavior of the two-species Vlasov-Maxwell-Boltzmann system was analyzed in @cite_6 . The main finding in @cite_6 is that although the non-homogeneous Maxwell system conserves the energy of the electromagnetic field, the coupling of the Boltzmann equation with the Maxwell system can generate some weak dissipation of the electromagnetic field which is actually of the regularity-loss type. It should be pointed out that even though the form of two-species Vlasov-Maxwell-Boltzmann system looks more complicated than that of the case of one-species, the study of global existence and time-decay rate is much more delicate in the case of one-species because the coupling term in the source of the Maxwell system corresponds to the momentum component of the macroscopic part of the solution which is degenerate with respect to the linearized operator @math . Essentially, it is this kind of the macroscopic coupling feature that leads to some different dissipation properties between two-species and one-species for the Vlasov-Maxwell-Boltzmann system.
@cite_13 @cite_6 @cite_4
2046869394 1991114201 1989465898 1776952983
In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space amssym @math . The existence of global-in-time nearby Maxwellian solutions is known from Strain in 2006. However, the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t−3 2 + 3 (2r)) in the L (L)-norm for any 2 ≤ r ≤ ∞ if initial perturbation is smooth enough and decays in space velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to 0 are also provided. © 2011 Wiley Periodicals, Inc. 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. Abstract This paper studies the approximation of the Boltzmann equation by the Landau equation in a regime when grazing collisions prevail. While all previous results in the subject were limited to the spatially homogeneous case, here we manage to cover the general, space-dependent situation, assuming only basic physical estimates of finite mass, energy, entropy and entropy production. The proofs are based on the recent results and methods introduced previously in [R. Alexandre, C. Villani, Comm. Pure Appl. Math. 55 (1) (2002) 30–70] by both authors, and the entropy production smoothing effects established in [R. , Arch. Rational Mech. Anal. 152 (4) (2000) 327–355]. We are able to treat realistic singularities of Coulomb type, and approximations of the Debye cut. However, our method only works for finite-time intervals, while the Landau equation is supposed to describe long-time corrections to the Vlasov–Poisson equation. If the mean-field interaction is neglected, then our results apply to physically relevant situations after a time rescaling. In this article, we study the coupling of the Einstein field equations of general relativity to a family of models of nonlinear electromagnetic fields. The family comprises all covariant electromagnetic models that satisfy the following criteria: they are derivable from a sufficiently regular Lagrangian, they reduce to the linear Maxwell model in the weak-field limit, and their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the linear Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields, and thus avoid the use of electromagnetic potentials.