Global Wellposedness Of The Vlasov-Maxwell-Boltzmann System In The Perturbative Framework And Related Problems

In this thesis,we will study some mathematical theories of some fundamental equa-tions arising in the kinetic theory of diluted gases.The results obtained in this the-sis include the global well-posedness of the Cauchy problems of the Vlasov-Maxwell-Boltzmann system and the Boltzmann equation with frictional force in the perturbative framework,and the fluid dynamical limit of the Boltzmann equation to the level of the compressible Euler equation.This thesis is divided into the following five chapters:? The first chapter focuses on the background and some recent progress on the math-ematical theories of some complex kinetic equations,which include the Boltzmann equation,the Vlasov-Maxwell-Boltzmann system and the Boltzmann equation with frictional force as typical examples.Some recent progress on the mathematical the-ories related to these equations are reviewed first,then we give the main problems to be studied in this thesis and the results obtained;? The second chapter is concerned with the construction of the global-in-time classical solution to the Cauchy problem of the noncutoff Vlasov-Maxwell-Boltzmann system near Maxwellians with strong angular singularity.Our analysis is based on a refined time-velocity weighted nonlinear energy method and the interpolation techniques between negative Sobolev norms and positive Sobolev norms without linear decay analysis.Compared with the result obtained in[28],the regularity assumption and the smallness condition we imposed on the initial perturbation are weaker.Meanwhile,the time decay estimates of the derivatives with respect to the spatial variables of solutions to the Boltzmann equation is obtained;? The third chapter is devoted to the global existence of small-amplitude solutions near a global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for non-cutoff soft potentials with weak angular singularity,which is based on a refined time-velocity weighted energy method and interpolation techniques.For the case of strong angular singularity studied in[28],the corresponding lin-earized Boltzmann collision operator has a nice dissipative effect like the linearized Landau collision operator,but for the case of weak angular singularity,the Fourier transform method in[28]does not work.Our main strategy is to make great effort to design suitable weight functions ?l(k)(?,?),or in other words,to design suitable working space for the case of weak angular singularity.Thus the results obtained in[28]and the last chapter for the case of strong angular singularity and the result ob-tained in this chapter together complete the research on the global solvability near a given global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for whole space;? In chapter four,we develop a general method for proving the optimal time decay estimates of the higher-order derivatives with respect to the spatial variables of solutions to the Boltzmann-type and Landau-type systems in the whole space,for both hard potentials and soft potentials.With the help of this method,we establish the global existence and temporal decay rates of solution near a given global Maxwellian to the Cauchy problem of the cutoff Boltzmann equation with frictional force for very soft potentials,i.e.-3<?<-2;? In the fifth chapter,we give a global version of the compressible Euler equations limit of the Boltzmann equation for the case when the Cauchy problem of the compressible Euler equations admits a unique global smooth solution.A by-product of such a result is the existence of a global strong solution of the Boltzmann equation for initial data close enough to a local Maxwellian uniquely determined by the global solution of the compressible Euler equations.Our analysis is motivated by Caflisch's approach[15]and the main ingredient of our analysis is a new esti1ate on the L2 norm of the low velocity function.It is worth to pointing out that such a result together with the the result obtained in[141]on the global existence of smooth solutions to the one-dimensional compressible Euler equations imply that even for algebraic decay(with respect to the microscopic velocity)initial perturbation,one can indeed construct global strong solution to the Cauchy problem of the Boltzmann equation with slab symmetry.Note that the result obtained in[101]asks that the initial perturbation decays(with respect to the microscopic velocity)exponentially.