The Cauchy problem for the BGK equation with an external force

In this paper, we study the Cauchy problem for the BGK equation with an external force. Firstly, we establish an L∞ existence result for this equation, and obtain some weighted L∞ estimates. Then, by means of the regularizing effects to the initial datum, we construct the approximate solutions and obtain some uniform estimates of the approximate solutions. Finally by using compactness method and passing to the limit, we prove the existence theorems of the L1 and Lp solutions and establish the propagation properties of the Lp moments.