UNIFORM TEMPORAL STABILITY OF SOLUTIONS TO DOUBLY NONLINEAR DEGENERATE PARABOLIC EQUATIONS
Jérôme Droniou, Robert Eymard, Kyle S. Talbot · HAL (Le Centre pour la Communication Scientifique Directe) · 2015
We show that solutions to a class of nonlinear degenerate parabolic initial-boundary value problems exhibit uniform temporal stability when the coefficients and data are perturbed. The class of equations encompasses the Richards model of groundwater flow, the Stefan problem and the parabolic p-Laplace equation (or, more generally, parabolic Leray-Lions operators). Beginning with a proof of temporally-uniform, spatially-weak stability, we strengthen the latter by relating the unknown to an underlying convex structure that emerges naturally from energy estimates on the solution. We do not assume uniqueness or additional regularity of the solution. The double degeneracy — shown to be equivalent to a maximal monotone operator framework — is handled with techniques inspired by a classical monotonicity argument and a simple variant of the compensated compactness phenomenon.