On the cometary flow equations with force fields

The Cauchy problem of a nonlinear kinetic equation modeling the time evolution of a cometary flow interacting with a force field is discussed, two kinds of existence results for weak solutions are established for initial data having finite mass and finite kinetic energy. The first one concerns a given force field which is assumed to be divergence free with respect to the velocity variable, it is shown that there exists a nonnegative weak solution to the Cauchy problem when the initial datum and the force field have reasonable integrability. As a special case, we also consider a Lorentz field and give another type of existence result. The second one deals with self-consistent electrostatic field, we show that when the initial datum has an L2 integrability the system has a global nonnegative solution which extends a previous result obtained by one of the authors.