Existence theorems for the initial value problem of the cometary flow equation with an external force

The initial value problem of the cometary flow equation with a given external force is investigated. By assuming that the initial microscopic density has finite mass and finite momentum and belongs to LpLp for some p>1p>1, three existence results of weak solutions with mass conservation and local estimates for the kinetic energy are established for different external forces, each of which is assumed to be divergence free with respect to particle velocities. The first result deals with a bounded smooth force and a Lorentz force with bounded smooth electric and magnetic intensities, and the second one concerns a force belonging to LqLq with 1p+1q=1. In the third theorem, we discuss a force that can be divided into two parts: one is in LqLq and the other is linearly growing at infinity; in this case we need to assume further that the initial density has finite first order spatial moment.