Gradient Flow Solutions For Porous Medium Equations with Nonlocal L\'{e}vy-type Pressure

TL;DR

This paper develops a variational approach to construct weak solutions for a porous medium equation involving a nonlocal Lévy operator, extending classical fractional Laplacian models and addressing unique technical challenges.

Contribution

It introduces a novel variational scheme to handle porous medium equations with general Lévy-type pressures, broadening the scope beyond fractional Laplacians.

Findings

Constructed weak solutions for nonlocal Lévy operators

Extended the mathematical framework for nonlocal porous medium equations

Addressed technical challenges due to lack of interpolation techniques

Abstract

We study a porous medium-type equation whose pressure is given by a nonlocal L\'{e}vy operator associated to a symmetric jump L\'{e}vy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical fractional Laplace operator. For the class of L\'evy-operators, we construct weak solutions using a variational minimizing movement scheme. The lack of interpolation techniques is ensued by technical challenges that render our setting more challenging than the one known for fractional operators.