Cecil Pompiliu Grunfeld
A Nonlinear Evolution Equation in an Ordered Space, Arising from Kinetic Theory
(460K, pdf)

ABSTRACT.  We investigate the Cauchy problem for a nonlinear evolution equation, 
formulated in an abstract Lebesgue space, as a generalization of various 
Boltzmann kinetic models. Our main result provides sufficient conditions for the existence, uniqueness, and positivity of global in time solutions. The proof is based on ideas behind a well-known monotonicity method, originally developed within the existence theory of the classical Boltzmann equation in $L^1$. Our application examples concern Smoluchowski's coagulation equation, a Povzner-like equation with dissipative collisions, and a Boltzmann model with chemical reactions.