Infinite Energy Solutions And Eternal Solutions Of Vlasov-Poisson System And Boltzmann Equation

Many important developments in mathematics originate with the N-body problem. It describes the motion of N mass points which move accord to Newton's equations of motion under the influence of their mutual attraction governed by Newton's law of gravity. According to a statistical average, many kinetic equations are obtained. We mainly discuss Boltzmann equation and Vlasov-Poisson system.Now, the infinite energy solutions is popular. Perthame and Castella considered the infinite energy solutions of Vlasov-Poisson equation and Vlasov-Poisson-Fokker-Planck equation with initial data f₀∈L¹âˆ©L^∞, separately. Using the similar method, we also discuss infinite energy solutions of Vlasov-Poisson equation and Vlasov-Poisson-Fokker-Planck in L^p. The key step is to estimate macroscopic densityρ(t,x) , field E(t,x) and low moment on the variable speed. All the estimates are new, and the results expand the work of Perthame and Castella. Moreover, using probability distance (Wasserstein distance) method, we show the uniqueness of solution to Vlasov-Poisson equation under the conditon that the macroscopic density is boumded in L^∞.In addition, for soft potentials (Including Maxwell's molecules model), we prove the global existence of infinite energy distributional solutions to Boltzmann equation with initial data close to the local Maxwellian. According to Kaniel and Shinbrot iteration scheme, we find two functions l₀ and u₀ satisfing the beginning condition by some estimates and the solution of differential equation. Then we prove the existence and uniqueness of solution to the Boltzmann equation. Since the initial data near the local Maxwell distribution (M₀(x,ξ) = exp(?), the solution is also located in the Maxwell distribution exp(?). Obviously, this solution is infinite mass and energy. The work improve the results obtained by Mischler and Perthame, who proved the global existence of distributional solutions with infinite energy to the Boltzmann equation only for the Maxwell's molecules .Finally, using a new iterative method, a variant of Kaniel-Shinbrot iterative method, we prove the existence and uniqueness of eternal solution to Boltzmann equation for the hard interactions and rigid spheres model. The ingenuity of the new iteration method is that when the beginning condition is satisfied, the iteration can continue, and the iterative sequence is monotonous. we have proved that the Cauchy problem of the Boltzmann equation, with cut-off soft potential and initial data close to a travelling Maxwellian, has a unique positive eternal solution. C. Villani conjectured that except for Maxwell's distributions, the nonlinear Boltzmann equation has no other type of positive eternal solutions with finite kinetic energy. The conjecture was first verified by Bobylev and Cercignani for the spatially homogeneous case. But our results gave a negative answer to Villani conjecture for the spatially inhomogeneous case.