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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 L 1. 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 equatioll 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 irutial value problem for the squarewell
kinetic equation is thus proved.