Existence And Uniqueness Of The Classical Solution Of Inelastic Kac Equation In R~1

The Kac equation is deduced from Kinetic model of gas molecule, which is an in-elastic Boltzmann equation. In this model, the molecule is rarefied and at any momentthe collision occur between two molecule, the quality is conserved, but the momentumis not conserved and the energy is dissipative. We obtain the the global existence anduniqueness of the classical solution under the only assumption that the initial energyis finite. We also obtain the estimates of the regularity using the moment inequalities,and get the classical solution of the equation.In proving the existence, we cut the collision core and get the approximating so-lutions which are the solutions of the equations with core being cuto?. In the processof finding the approximating solutions, we construct a subspace of L1, and obtain theapproximating solutions in L21∩L2 which prepare for the existence of the Kac Equation.we also use weak convergency and compact embedding theorem in the functional anal-ysis, the latter is realized by the estimates of the moments. When approximating thesolution of the original equation, we need the compact imbedding to get a transition,as everyone knows, the imbedding relation wouldn't hold in unbounded areas, but theexistence of the moment make up for it. When we choose the subsequence, we use avariational technique, combining with the uniform bound about time and parameter m,M, we could use Lebesgue dominant convergent theorem to approximate the solutionof the original equation. In this paper, we get the estimates of the higher moments ofthe inelastic collision operators and get the regularity using interpolation and Sobolevembedding theorem. Since the raise of the moments and di?erentiability are obtainedalternately, and the imbedding relation can't hold when the dimension is large, so thedi?erentiability can't be raised and so do the regularity. In this article, we assumethe dimension is 1. In proving the uniqueness, we get some important estimates usingKato inequality which is more delicate and prove the theorem by iteration.