Global Solutions To A Class Of Cometary Flow Equations With Force Fields

The study of nonlinear kinetic system is complicated but very useful. The Cauchyproblems for three dimensional cometary flow equations with external force fields areinvestigated, three theorems for global existence of weak solutions are proved for initialdata with L~1andL~p integrability and having finite first order velocity moments. The firstone presents two types of existence results, on the one hand, it is shown that there existsa nonnegative weak solution to the Cauchy problem when a given smooth field which isassumed to be divergence free with respect to the velocity, on the other hand, we study theLorentz field as a special case, besides the existence, the second order velocity momentsof the weak solutions are proved finite. The second one deals with a given field having L~q integrability for q>1such that1/p+1/q=1(p,q>1), then the Cauchy problem has a nonnegativesolution. The third one considers the existence of a nonnegative solutions when the externalfield is divided into two parts: one has the L~p integrability, the other verifies linear growth.We should point out that method of the first main result is normal, the second and thirdone solved by the ingenious way which we integrated utilization of moment lemma andasymptotic method.