The interface between regions where u<0 andu > 0in the porous medium equation

Abstract

We consider the porous medium equation with sign changes. In particular this equation describes the mixing of fresh and salt groundwater due to mechanical dispersion. The unknown function u, which denotes the velocity of the fluids, may take positive as well as negative values. Our main result is the following : under certain monotonicity hypotheses on the initial function, there exists a time T> 0 after which the regions where u < 0 and u > 0 are separated by an interface x = ζ(t) such that ζ is continuously differentiable on [T,∞]. The method of proof is based on a priori estimates for solutions of regularized problems and for their level lines

AMS(MOS):

• 35K60
• 35K65
• 35R35

Keywords:

• Nonlinear diffusion equation
• regularity of the free boundary