System of porous medium equations

The geometry of dissipative evolution equations: the porous medium equation

We show that the porous medium equation has a gradient ow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations

Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they c...

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

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...

Globally Convergent Multigrid Methods for Porous Medium Type Equations