A method of energy estimates in L∞ and its application to porous medium equations

Abstract

The existence of time local $C^{\infty }$-solutions is shown for Cauchy problem of the porous medium equations. Our arguments rely on the "$L^{\infty }$-energy method" developed in our previous paper [16] and a new method based on the theory of evolution equations in the $L^2$-framework which enables us to handle with perturbations which can be decomposed into monotone parts and small parts in Sobolev spaces of higher order.