Uniform L~p Stability For The Spatially Inhomogeneous Boltzmann-enskog Equation With External Force

Boltzmann equation is a kind of nonlinear differential integral equation which satisfies the probability density, and it has been a hot issue in the study of its mathematical theory.As its extended form Enskog equation has also been concerned by people. However, in order to satisfy some properties of the Enskog equation, People are more inclined to study the Boltzmann-Enskog equation of the modified equation. At present, a lot of researches have been done on the existence and uniqueness of the solution of the Boltzmann-Enskog equation, and there have been some research results about its stability. The global existence, uniqueness and stability of the solution of Boltzmann-Enskog equation are studied by Toscani and Bellomo. Then, estimation of uniform L1 stability of homogeneous BoltzmannEnskog equation of the external force field studied by Jiang et al. This paper researches Lp stability of mild solutions for a Boltzmann-Enskog equation with external force field in the hard sphere model when the initial value of f0 is sufficiently small and fast decay on polynomial or exponential time.The main proof idea of this paper comes from Seung-Yeal Ha and Se Eun Noh in 2011,but it is proved that the stability of the classical solution of the spatially inhomogeneous Boltzamnn-Enskog equation without the external force field. In this paper, an external force field is added to the equation. In the course of the proof, we use the H ¨older inequality and the idea of Lu. The first is to estimate Lpnorms with a weight(1 + |v|2)?2 of mild solutions so that we can get the time-integrability of weighted Lp, 1 < p < 3 norms. The second is to estimate the weighted Lpnorms of the difference of mild solutions of the Boltzmann-Enskog equation with initial data f0and?f0respectively. Finally, we can obtain the stability proof by using an important inequality.