Tjon-Wu Model Of The Boltzmann Equation

Boltzmann equation is the practically 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. However in view of the complication of the collision operator the study of the equation is very difficult. Tjon—Wu equation([l]—[4]) or Tjon—Wu model of the Boltzmann equation with easier form was derived from the spatially homogeneous Boltzmann equation by use of the Abel transformation. Several authors investigated the equation(see, e.g.:[5]—[11]).This paper studies a generalized Tjon—Wu equation proposed by A.Lasola[11].First,the Cauchy problem for the equation is considered. 1. We obtain the existence and uniqueness for the solution of the Cauchy problem by means of semigroup theory. 2. We show positivity and estimates of moments of senior order for the solution of the Cauchy problem in some special conditions by using the properties of fixed point ,the basic moment inequality and the method of successive approximations.Second, we discuss the stationary problem for the equation. 1. We establish the existence of the stationary solution through verifying the condition of the Schauder fixed point theorem. 2. We show that the stationary solution is positive by means of positivity of the operator P and the properties of fixed point. 3. We obtain the uniqueness and the asymptotic stability of the stationary solution (A.Lasola [11] has proved that the stationary solution for the equation is exponentially stable in weak topology ). The positivity of the stationary solution plays an important role in the proof of stability. It allows us to apply the Kantorovich—Rubinstein maximum principle to prove that the stationary solution is globally asymptotically stable in strong topology. 4. We show by dulity and embedding method that the stationary solution is C~∞in some special conditions.