Stability of Global Solution to Boltzmann-Enskog Equation with External Force

Abstract

In the presence of external forces depending only on the time and space variables, the Boltzmann-Enskog equation formally conserves only the mass of the system, and its entropy functional is also nonincreasing. Corresponding to this type of equation, we first give some hypotheses of its bicharacteristic equations and then get some results about the stability of its global solution with the help of two new Lyapunov functionals: one is to describe interactions between particles with different velocities and the other is to measure the $L^1$ distance between two mild solutions. The former Lyapunov functional yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the $L^1$ stability of global mild solutions to the Boltzmann-Enskog equation for a moderately or highly dense gas in the influence of external forces.