On the BGK equation with some force fields in LpLp space

The Cauchy problem for the BGK equation with two kinds of force fields is studied in this paper. The first one considers a given force field F(t,x,v)F(t,x,v) which is assumed to be divergence free with respect to the velocity variable and has an LpLp integrability. It is shown that there exists a nonnegative distributional solution which extends a previous result obtained by the authors. The second one deals with a Lorentz field E(t,x)+v×B(t,x)E(t,x)+v×B(t,x) with E(t,x)E(t,x) and B(t,x)B(t,x) belonging to LpLp space, which gives a similar existence result to the Cauchy problem.