Continuity estimates for porous medium type equations with measure data

Continuity estimates for porous medium type equations with measure data

We consider parabolic equations of porous medium type of the form ut − divA(x, t, u,Du) = μ in ET , in some space time cylinderET . The most prominent example covered by our assumptions is the classical porous medium equation ut −∆u = μ in ET . We establish a sufficient condition for the continuity of u in terms of a natural Riesz potential of the right-hand side measure μ. As an application we...

Pointwise Estimates for Porous Medium Type Equations with Low Order Terms and Measure Data

We study a Cauchy-Dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space-time cylinder for degenerate porous medium type equations with low order terms and a non-negative, finite Radon measure on the right-hand side. The central objective is to acquire linear pointwise estimates for weak solutions in terms of Riesz potentials. Our main result, Theorem 1.1, g...

The Porous Medium Equation with Measure Data

We study the existence of solutions to the porous medium equation with a nonnegative, finite Radon measure on the right hand side. We show that such problems have solutions in a wide class of supersolutions. These supersolutions are defined as lower semicontinuous functions obeying a parabolic comparison principle with respect to continuous solutions. We also consider the question of how the in...

Hamilton’s gradient estimates and Liouville theorems for porous medium equations

Logarithmic estimates for continuity equations

The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coe cient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6, 9, 12], where the usual requirement of bo...