The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions

Abstract

We study the relation between the porous medium equation $u_t=\Delta (u^m),m>1$, and the eikonal equation ${\nu }_t=|D\nu {|}^2$. Under quite general assumptions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as $m\downarrow 1$ to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.