A Simulated-linearization Method For Boltzmann Equation And The Regularity Of The Solution Of Kac Equation

The kinetic theory of gases is devoted to the study of behaviors of gas in the phase space of position and velocity of gas particle, and now appear in a variety of science and application such as,astrophysics,aerospace engineering ,nuclear engineering,semiconductor technology,social science and bioligy technology and so on.Boltzmann equation,as the oldest but the most fundamental equation in the kinetic theory, is very inportant and valued. although there has been 150 years,since it was founded,there are many problems still open,for example the global existence and uniqueness etc. And it is a long load for us.In this paper, we develop a new linearized method for Boltzmann equation and give some applicaitons. Besides this,we establish the regularity of the solution of the Kac-Boltzmann equation.1. The most difficult of the study of Boltzmann equation is in the nonlinear operator in the right. We put forward a new method which can linearize the collision operator. Using this method, we can get an expected result but with a strong condition and in some idiographic equation ,for example the Kac-Boltzmann equation,the result is satisfying2. For the regularity of the Kac-Boltzmann eqation with energy,the most important step is estabishing the Povzner-type inequation .Here, the Povzner-type inequation and the L~p estimates for the collision operator are established. With these estimates, the regularity is proved.