| The Boltzmann equation is an important model in statistics physics and kinetic theory, which describes the time evolution of the dilute gas. The mathematical theory of Boltzmann equation has been one of the most challenging fields in the differential equations, especially for the research on the properties of its solutions.In this paper, we study the stability of mild solutions to the Boltzmann equation with two types of external forces. One type of external force satisfies some “small” conditions. The other is according to the assumptions on the bicharacteristic generated by external force which can be arbitrarily large(needn’t to satisfy some “small” conditions). Under the two types of external forces, the existence of the solutions to the Boltzmann equation near vacuum has been studied. In this paper, following from the exponential decay estimate for velocity variable, we obtain the uniform stability of the mild solution to the Boltzmann equation with an external force for hard potentials and soft potentials by using the Gronwall’s inequality and Lu’s[44] trick.In Section 1, we give the research background and the research status of the Boltzmann equation, and describe the problem we need to solve. In Section 2, we present the main results and the assumptions about two types of external force. In Section 3, we give some fundamental lemmas and prove some conclusions for later use. Finally, the uniform stability estimate is obtained in Section 4. |
