Gradient Flow Solutions For Porous Medium Equations with Nonlocal Lévy-type Pressure
Guy Foghem, David Padilla-Garza, Markus Schmidtchen
TL;DR
This work extends gradient-flow methods for porous medium-type equations to a broad class of nonlocal pressures governed by symmetric Lévy operators with radial kernels. By developing a minimizing movement (JKO) scheme in the Wasserstein space and leveraging energy-dissipation structures along with symbol-based surrogate estimates, the authors establish existence and convergence of weak solutions beyond the classical fractional Laplacian. The analysis yields a robust framework including discrete-to-continuum convergence, weak formulations, and entropy-based regularity, while highlighting limitations such as the lack of decay estimates without interpolation. The results unify and extend prior fractional-pressure results to general Lévy kernels, and provide a toolbox of spaces and conditions (e.g., symbol bounds, unimodality) to verify applicability to concrete nonlocal operators. This contributes a rigorous path to analysing nonlocal porous media dynamics in probabilistic and PDE contexts, with potential implications for mean-field and gradient-flow models in heterogeneous media.
Abstract
We study a porous medium-type equation whose pressure is given by a nonlocal Lévy operator associated to a symmetric jump Lévy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical fractional Laplace operator. For the class of Lévy-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.