Existence of weak solutions for nonlinear drift-diffusion equations with measure data

TL;DR

This paper proves the existence of weak solutions for nonlinear drift-diffusion equations with measure data, extending results to cases with divergence-free drifts and applications to Navier-Stokes coupling.

Contribution

It introduces new existence results for weak solutions with measure data, relaxing conditions on the drift term and providing energy estimates for coupled systems.

Findings

Existence of nonnegative weak solutions under sub-scaling drift conditions.

Relaxed conditions for divergence-free drifts allowing supercritical scaling.

Construction of weak solutions for coupled nonlinear diffusion and Navier-Stokes equations.

Abstract

We consider nonlinear drift-diffusion equations (both porous medium equations and fast diffusion equations) with a measure-valued external force. We establish existence of nonnegative weak solutions satisfying gradient estimates, provided that the drift term belongs to a sub-scaling class relevant to $L^1$-space. If the drift is divergence-free, such a class is, however, relaxed so that drift suffices to be included in a certain supercritical scaling class, and the nonlinear diffusion can be less restrictive as well. By handling both the measure data and the drift, we obtain a new type of energy estimates. As an application, we construct weak solutions for a specific type of nonlinear diffusion equation with measure data coupled to the incompressible Navier-Stokes equations.