The Generating Kac Equation

Recently, more and more scientist's interest focus on the study of the generating Kac equation([1]—[2]). The special equation have been extensively discussed since it was put forward by the famous physical scientist Boltzmann. The abundant outcome have been obtained for the equation. For example, L.Arkeryd([3]—[5]) have proved the uniqueness and existence of the solution, what's more, owing to the advance of the Maxwell model, it is possible for the exact estimate of the Boltzmann equation. In addition, A.V.Bobylev[6] particularly investigated the Maxwell model and found out most beautiful result. However, only a little result have been got for the inelastic interaction becauce of the dissipation of the energy.Since M.Kac put forward the Kac equation in 1950s, the study of inelastic Boltzmann equations are beginning to become active and the great achivement are acquired. Becauce the Kac equation possesses the relatively simple structure, it is possible for us to find out the exact rate of convergence to the Maxwell equilibrium. Tulvirenti and Toscani[7] have widely discussed the equation and made use of Fourier transform to put Ernst-Brito conjecture([8]—[9]). Moreover, A.V.Bobylev, C.Cerignani and G.Toscani([10]—[12]) got a lot of relative result on the basis of Ernst-Brito conjecture.The generating Kac equation, which describes more real particles, will be put forward in this paper and we will investigate the uniqueness, existence and asymptotic properties of the generating Kac Model in detail. Then the generic form of the generating Kac model will be given in the first part, we will seriously disscuss the connection of several probability metrics in the second part, the uinqueness and existence will be proved in the last part, in addition, we will also simply dicuss the similar solution in the last part.