On The Cauchy Problem For The Inelastic Boltzmann With External Force And VPB Equation

Boltzmann build Boltzmann equation which based on Maxwell[1,2]and other workin1872[3]. Boltzmann equation can be used to describe the evolution of rarefied gas.Theexternal force or in the self-consistent gravitational repulsion effect can be described byBoltzmann equation with external force and Vlasov-Poisson-Boltzmann(VPB for short).According to whether the energy loss in the collision process can be divided into two casesof elastic and inelastic.There are a large number of charged particles in the universe, Charged particles existnot only between the short-range force (collision) the existence of Coulomb static electricityand gravitation. Coulomb electrostatic is greater than the gravitational force and thus the roleof gravity can be ignored, and the Coulomb electrostatic interactions and collisions decidedState of the charged particles and particle interactions in the induced electric field by thecollision and the Boltzmann equation can only be simulated touch Hit role for the role of theCoulomb electrostatic force will fail. VPB equation can describe the Coulomb electrostaticforce and the collision of these two kind of force, which is used to simulate the thin chargedparticle state.We consider the existence and stability under soft potential of inelastic Boltzmannequation with external force and the VPB equation.Elasticity coefficient η is constant andits range is [13,1] constants. First of all, We obtain the estimate of the collision operator ofexternal Boltzmann equation. Second, we construct a mapping of the Banach space andprove its compression.Boltzmann equation with external force has classic solution couldbe proved by Banach fixed point theorem.The L~1uniform stability of Boltzmann equationwith external force can be obtained by mild form of it and collision operator estimate. Weobtain the classic Boltzmann equation solution existence and L~1consistent with stability.Finally, the existence of classical solution and local L~1stability of VPB system are obtainedby the Boltzmann equation has with external force.Finally, by using the classic solutionof Boltzmann equation with external force and iterative method,We prove the existence ofclassical solutions of VPB system.The VPB equation’s local L~1stability can be obtained byconstruct nonliear functions.