Global existence in L1 for the square-well kinetic equation

An attractive square-well is incorporated into the Enskog equation, in order to model the kinetic theory of a moderately dense gas with intermolecular potential. The existence of solutions to the Cauchy problem in L¹. global in time and for arbitrary initial data. is proved.

A simple derivation of the square-well kinetic equation is given. Lewis's method is used~ which starts from the Liouville equation of statistical mechanics. Then various symmetries of the collisional integrals are established. An H-theorem for entropy, mass, and momentum conservation is obtained, as well as an energy estimate, and key gain-loss estimates.

Approximate equations for the square-well kinetic equation are constructed that preserve symmetries of the collisional integral. Existence of nonnegative solutions of the approximate equations and weak compactness are obtained. The velocity averaging lemma of Golse is then a principal tool in demonstrating the convergence of the approximate solutions to a solution of the renormalized square well kinetic equation. The existence of weak solution of the initial value problem for the square well kinetic equation is thus proved.