Boltzmann Equation And Its Kinetic Models

Kinetic equations arising from rarefied gas dynamics are important research objectsin mathematical physics. This topic involves many fields of natural science and has extensive applications, for example: nonequilibrium statistical physics, astrophysics,plasma physics, semiconductor, aeronautical engineering, nuclear reactor and so on. The present thesis is devoted to studying the classical Boltzmann equation, the (Gauss-)BGK model and an equation of Boltzmann type arising from the theory of dissipative gases, we mainly focus on some matters remaining to be settled in this field, and establish a series of qualitative mathematical results.In chapter 1, we briefly introduce several equations of mathematical physics which are the main objects in this thesis, then the current situation and problems to be solved relating to these equations are analysed, finally we give the main results obtained in this thesis, which will be discussed and proved in the following chapters.Chapter 2 is devoted to studying a Boltzmann type equation describing the evolutionof a dissipative gas consisting of granular media, this is a new direction in kinetic theory. Concerning the physical background of this problem, we assume that when two molecules in the gas collide, the conservation of momentum holds but a definite part of kinetic energy is lost. Furthermore, we also assume that the gas is put in a thermal bath. Under those assumptions, the microscopic state of the gas is governedby a partial differential equation of Fokker-Planck-Boltzmann type. We discuss smoothness and long time behavior of its spatially homogeneous solutions in the case of quasi-Maxwell molecules approximation. First, under the very weak assumption that the initial distribution is a probability density having null bulk velocity and finite temperature, we prove the smoothness of its solutions and establish the asymptotic estimates of its Sobolev's norms near t = 0. Then, combining this regularity result and interpolation method we show that the solution converges exponentially towards a unique equilibrium as t→∞, furthermore, an estimate of the exponent governing the exponential convergence is given. These results sharpen some theorems obtained by M. Bisi, J. A. Carrillo and G. Toscani[24].In chapter 3, we study the spatially homogeneous probability solutions of the classical Boltzmann equation for hard potentials with Grad's weakly angular cut off assumption. Firstly, we can define a continuous linear functional on a continuous function space by the Boltzmann collision operator, as a consequence, we can extendthe domain of the collision operator to the space consisting of some probability measures. Having this result in mind, we give an equivalent formulation of the spatiallyhomogeneous Boltzmann equation in the space of probability measures. Secondly,by using the method introduced in [44], we establish some a priori estimates for probability solutions, and prove the existence and uniqueness of conversation probabilitysolutions for initial distributions respectively in P₂(R³) and P₄(R³). Finally, Using extended Povzner's inequality given by Mischler and Wennberg [37] and inspiredby Desvillettes method [42], we prove the production of higher moments for probability solutions. Those results extend some of the classical theorems obtained in [33, 36, 37, 39, 40, 41, 42, 44] to the probability solutions; meanwhile, we also improve some results on probability solutions obtained in [38, 56, 57, 58].In chapter 4, for the general Boltzmann equation for soft potentials with Grad's weakly angular cut off, we discuss a class of its distributional solutions having infiniteenergy and decaying exponentially at infinity along its characteristics. Firstly, we establish an a priori estimate of the collision operator for a class of measurable functionsdecaying exponentially at infinity (maybe having infinite energy). Combining this result and Kaniel-Shinbrot method [59], we obtain the existence, uniqueness and stability of solutions with small amplitude. Then, we prove, for general initial data, the stability of solutions in this function class (but without existence result) and establishtheir long time behavior. These results generalize theorems obtained by Mischler and Perthame [70] for Maxwell molecules to soft potentials. It should be mentioned here that in [74] an important representation of the collision operator for continuous functions was given and similar results for mild continuous solutions were established, here we use Kaniel-Shinbrot method.In chapter 5, we study the BGK model in R^N, which is extremely important in kinetic theory. Firstly, we prove that the global distributional solution, which was constructed by Perthame in his major work [12], propagates higher moments. Then, supposing that the collision frequency satisfies v =ρ^μ(t,x)(μ≥1), we prove the uniqueness of solutions to the BGK model in a weighted L^∞space and give a local existence theorem, which partially improves some known results obtained in [87, 88]. Chapter 6 is devoted to studying the L^p solutions to the BGK model in R^N. Firstly, we extend the method used to prove the L^∞estimates for hydrodynamic quantities in [87] and obtain some desired L^p estimates. On the basis of these estimates, we can establish weighted L^p estimates for local Maxwellians which is very efficient for us to proceed. Secondly, with these estimates, we obtain a uniform L^p estimate for solutions to the BGK model, then we employ velocity averaging lemma and moment lemma to prove the strong convergence of approximating solutions and as a consequence we obtain an L^p solution; further, we also get a series of properties for this kind of solutions, for example conservation of mass, momentum and energy. Finally, we prove that the solutions constructed above propagate some of the L^p moments. We note that the existence result obtain in this chapter is different from Perthame's theorem [12].In chapter 7, we discuss a corrected version of the BGK model, i. e., the ellipsoidal statistical model, also called Gauss-BGK model. It is an extremely complicated kineticmodel, as far as we know, there is no mathematical results about its solvability. Here we discuss a special case: spatially homogeneous solutions. Firstly, we give a representation of its spatially homogeneous solution, and analyse the structure of this representation. Consequently, we obtain a Maxwellian lower bound for the solution. Secondly, we prove the solution converges exponentially (with an explicit exponent) towardsits equilibrium as time goes to infinity. An entropy theorem is also established.