A compactness theorem and its application to a system of partial differential equations

Abstract

An existence theorem is established for the initial-boundary-value problem for the system $\frac{\partial}{\partial t} \alpha (u) - \Delta u \ni \sigma (u)|\nabla\phi|^2,$ div$(\sigma (u) \nabla \phi) =0$ in the case where $\alpha$ is a maximal monotone graph in $\Bbb R$ and $\sigma$ is a continuous, nonnegative function on $\Bbb R$ such that $ \sigma(s)=0$ if and only if $s\ge a$ for some $a>0.$ A solution is constructed as the limit of a sequence of classical weak solutions of the regularized problems. Due to the possible degeneracy of the system, oscillations in the approximating solution sequence can persist. We prove that the failure of strong convergence is concentrated in a set which does not really matter in our passage to the limit.