The Viscosity Analysis On The Boltzmann Equation

Boltzmann equation is one of the most important models for the non-equilibrium statistical physics.From mathematical viewpoint,Boltzmann equation is an integro-differential equation.The right hand term of Boltzmann equation,which is called also collision operator, has five multi-integral.In the meantime,the collision operator only acts on the velocity variable v,and is local in t,x.This gives the equation certain degeneracy in some senses. Therefore,the study for Boltzmann equation is presently the one of the most challenges in mathematics,not only in numerical computation,but in the theory of partial differential equation.For the homogenous dilute gases,the spatially homogeneous Boltzmann equation with hard potentials has been fully analyzed.Of cause,few results for soft potentials have been obtained so far.However,the studies for inhomogeneous Boltzmann equation have been confronted with many difficulties.The only satisfy result was the renormalized theory introduced by DiPerna and Lions twenty years ago.But the uniqueness and the differentiability for the renormalized solution are the most open problems.Since then,mathematicians make a lot of works on the Boltzmann equation,especially on collision operator,and many properties have been found.Nevertheless,to win the open problem may need new mathematical tools and new theories.For the stability of the solution to the Boltzmann equation on the initial value,many results have been established[4,58,100].In this paper we shall study the viscosity analysis on the Boltzmann equation.However,to our knowledge this problem has not been studied so far.Here the viscosity equation is derived by adding a Laplacian in v,which is also called Fokker-Plank term,into the Boltzmann equation.On physically,the Plank-Fokker term is caused by the diffusion of molecules.It must be stressed that the viscosity equation can be obtained by a Laplacian term in x.However,this kind of approximation has no clear explain on physically.We obtain two main results:the one is that some accurate estimates on the viscosity approximation are presented for homogeneous Boltzmann equation with angular cut off,including Maxwellian and hard potentials.Noting that all these estimates depend on time,which indicates the viscosity term has diffusion effect.The other is that the asymptotic behaviour of the viscosity solution is studied in the frame of renormalization for inhomogeneous Boltzmann equation.We obtain that the viscosity approximation solution convergesin L~1 to the renormalized solution with defect measure of the Boltzmann equation, as the viscosity coefficients goes to 0. In Chapter 1 the preliminary on Boltzmann equation is introduced,including the basic definitions,Boltzmann's H-theorem,some basic a priori estimates and the properties of collision operator.Some known results on the spatially homogeneous Boltzmann equations are provided in Chapter 2,which include that the existence of the solution and uniqueness,the estimates on moments,the propagation of smoothness and singularity,the trend to the equilibrium,and the renormalization theory for inhomogeneous Boltzmann equations and the velocity averaging theory for the transport equations.It must be emphasized that we here are concerned with the Cauchy problem of Boltzmann equation,especially in the case of hard potentials with angular cutoff.We study the viscosity approximation on the Maxwellian molecules in Chapter 3.The existence of the mild solution to the viscosity equation and the uniqueness are obtained,and an estimate on the viscosity approximation is presented accurately.In Chapter 4,we study the viscosity approximation on the homogeneous Boltzmann equation with angular cutoff and hard potentials.We prove the existence of the viscosity weak solution by Schauder fixed point,and show the uniqueness by Gronwall's inequality. At last,an estimates on the approximation in the sense of L_k~1 and L_k~2 are derived.The viscosity analysis on the inhomogeneous Boltzmann equation is put on Chapter 5. We study the uniform estimate on the viscosity renormalized solution by weak compactness analysis,and then investigate the asymptotic behavior of the viscosity renormalized solution.Our results mainly rely on the interpolation inequality,Mouhot-Villani's decomposition on Q~+ and the velocity averaging lemma.