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

Boltzmann equation is a kind of important differential equations, whose mathematical theory has been one of the most challengeable field, especially the research of properties of solutions to the Boltzmann equations. In this paper, we discuss stability of wild solution to the Boltzmann equations with external forces and cut-off and inverse power law potentials under the assumption that initial data are sufficiently small and decay fast enough alge-braically or exponentially, which including part of soft potentials and Maxwellian model (-3/4<?? 0) and hard potentials and hard sphere model (0<?? 1). Prior to this, Duan-Yang-Zhu give the global existence of wild solutions when the solutions of charac-teristic equation satisfy certain conditions in a 2005 article. However, we add a limit of ?0??E(t)||Lx?dt?< Co to gain the stability theory of wild solutions.The main proof train of thought of stability comes from the article of Ha-Lee-Yun in 2009, but it proved the stability of classical solutions to the spatially inhomogeneous Boltzmann equations with small external forces, in this paper, the external forces are larger than its, and the index of damping exponent can be optimized to ?2>? for any sufficiently small e> 0 under hard potentials and hard sphere model. The first is to estimate Lp norms with a weight (1+|u|2)n/2 of wild solutions so that we can get the time-integrability of weighted Lp< 3 norms. The second is to estimate the derivative with respect to t of weighted Lp norms of the difference of wild solutions of the Boltzmann equation with initial data ?0 and?0 respectively. The last is to get the proof of stability by Gronwall inequality, the stability is proved by generalized Gronwall inequality in the case of hard potentials and hard sphere model.