Rugularity And Uniqueness Of The Boltzmann Type Equations

Boltzmann's equation is certainly the most famous and successful mathematical model to describe the time and space evolution ,at a statistical level,of the position and velocity distribution function of particles in a dilute gas. The class of equations, beautiful and elegant, are powerful instrument both in the analysis of thermodynamical properties of fluids and in the solution of fluid-dynamical problems,which are often relevant in modern technology.The question of existence and uniqueness of solutions to the Boltzmann equation was first addressed by Carleman[1] in 1957. The existence theory in L¹ can be found in Arkeryd's paper [2]from 1972, where two existence proofs are given under certain hypothesis on the initial data. One of the proofs is based on a weak stability result and on the Povzner inequality [2, 3]. In the other case ,the proof depends on a monotonicity argument [4]. Then several authors investigated the question of uniqueness for the homogeneous Boltzmann equation [2, 5, 6, 7], However the perfect result was given by S. Mischer and B. Wennberg recently [8]. In 1988, R. J. DiPerna and P. L. Lions considered the spatially non-homogeneous Boltzmann equation perturbed by the Fokker-Planck operator and proved the global existence of weak solution(renormalized solution) [9, 10, 11]. In 2004, I.M.Gamba,V.Panferov and C.Villani studied homogeneous Fokker-Planck-Boltzmann equation, and proved that the solution of the equation is exist and unique [12].This paper study:1.the spatially homogeneous Fokker-Planck-Boltzmann equation. For solutions satisfying conservation of mass and momentum, and linear increasing of energy, new weighted L¹ and L² estimates on any time interval [8, T] C (0,∞) are given. With these results and the known L^p and Sobolev estimates of the collision operator, we establish the smoothness and uniqueness of the solutions of the equation. 2.the uniqueness of the spatially homogeneous Boltzmann equation. We show that the conservative solutions of the equation are unique if the collision kernel satisfies Grad's assumption and the initial datum f₀ belongs to L_s¹(R³)+∩L¹ logL¹ for some s > 2.