The Compactness Of Solution Sequence To Stationary Kinetic Equaitons, The Existence And Large Time Behavior Of The Solution To Boltzmann Equations

In this thesis, we consider following three problems:the compactness of the sequence of solutions to the inhomogeneous stationary equations; large time be-havior of the solution to the Landau Equation with specular reflective boundary condition; global solutions to the Vlasov-poisson-Boltzmann System with hard po-tentials. Next, we will introduce the three problems step by step.(Ⅰ):We consider the locally relative compactness of the sequence of solutions to the inhomogeneous kinetic equations in some given spaces:(v·▽x+F·▽v)fλ=gλ, the locally relatively compactness of the sequence of solutions to the inhomogeneous kinetic equation with rough force in space of LP(Rxn×Rvn) and v·▽xfλ=gλ, the locally relatively compactness of the sequence of solutions to the inhomogeneous kinetic equation in space of Bp,qs(Rxn×Rvn).We obtain the locally relative compactness of the sequence of solutions to sta-tionary kinetic equation with external force in Lp space and in Bp,qs space without external force by mainly using the Fourier multiplier methods.(Ⅱ):We consider one-dimensional Landau equation as follows ft+v1fx=Q(f,f), f(0,x,v)=f0(x,v),(1.1) and the specular reflective boundary condition f(t,0,v)=f(t,0,Pv).(1.2) Here f(t, x,v)≥0is the spatial distribution function for the particles at time t≥0, position x∈R+with velocity v=(v1, v2, v3)∈R3. And Pv=(-v1, v2,v3).We consider the Landau collision operator as where (?)i=(?)ui etc.The nonnegative matrix φ is given by The original Landau collision operator with Coulombic interactions corresponds to the case γ=-3.we are concerned with the case γ≥-2.A half space problem for the one-dimensional Landau equation with specular re-flective boundary condition is investigated.By using energy methods,we show that the solution to the Landau equation converges to a global Maxwellian.Moreover,a time-decay rate is also obtained.(Ⅲ):Consider the Vlasov-Poisson-Boltzmann system for two species of particles (?)tF+v·▽xF++E·▽vF+=Q(F+,F+)+Q(F+,F-),(?)tF-v·▽xF--E·▽vF-=Q(F-,F-)+Q(F-,F-),(1,1) with initial data F±(0,x,v)=F0,±(x,v). Here F±(t,x,v)represent the number density functions for ions(+)and electrons(-)respectively at time t≥0,with velocity v∈R3and spatial coordinate x∈Ω where Ω is the torus T3or the whole space R3.And the normalized Boltzmann collision operator is given by,cf.[8], Here ω∈S2and v’=v-[(v-u)·ω]ω,u’=u+[(v-u)·ω]ω.(1.3)In this paper,we assume that B(θ)satisfies the Grad angular cutoff assumptions,0<B(θ)≤C|cosθ|and γ∈[0,1], which are called hard potentials.The self-consistent electrostatic field E(t,x)=-▽xφ(t,x)and the electric po-tential φ(t,x)satisfiesIf we study the position x∈T3,∫T3φ(t,x)=0holds. we construct global solution near Maxwellian to the Vlasov-Poisson-Boltzmann system with hard potentials in both the torus and the whole space. Our results in the torus generalize the results in the torus with hard sphere case in [30] to hard potentials. Our results in the whole space extend the existence results in [62] and time decay rate in [73] of the Vlasov-Poisson-Boltzmann system with hard sphere case to hard potentials. We use the energy methods in [71,72,77] and semigroup methods to to get the time decay of the solutions to this system, and use some techniques in [34,77] to close the energy estimates and obtain global existence.