A gradient flow for the Porous Medium Equations with Dirichlet boundary conditions

TL;DR

This paper establishes a gradient flow framework for the porous medium equations with Dirichlet boundary conditions, constructing weak solutions via a modified Wasserstein distance and characterizing them as curves of maximal slope.

Contribution

It introduces a novel gradient flow structure for porous medium equations with Dirichlet conditions using a modified Wasserstein distance and constructs weak solutions through a minimizing movement scheme.

Findings

Constructed weak solutions via the minimizing movement scheme.

Characterized solutions as curves of maximal slope.

Extended gradient flow framework to porous medium equations with boundary conditions.

Abstract

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy functional with respect to $Wb_2$ distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130]. Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.