L ^p stability of solutions of Boltzmann equation with external force in soft potentials

In this paper, we present a uniform L ^p stability theory for the solutions to Boltzmann equation with small or large external forces in vacuum in soft potentials. We assume that the initial data are sufficiently small and a fast decay in space and velocity. Our stability analysis is divided into two parts. For small external forces, we first consider the time derivative of the pth power of L ^p norm of the distance between two mild solutions f and g, then define nonlinear functionals apply the Grönwall's inequality to obtain the L ^p stability. For large external forces, we make use of the constructive assumptions on external force, Minkowski's inequality Grönwall's inequality to derive the L ^p stability estimate. © 2010 American Institute of Physics.