Continuity results for degenerate diffusion equations with $L^{p}_t L^{q}_{x}$ drifts

TL;DR

This paper establishes local uniform continuity of solutions to degenerate diffusion equations with drift terms in certain Lebesgue spaces, identifying sharp conditions for continuity based on the drift's integrability and divergence properties.

Contribution

It provides sharp conditions for local continuity of solutions to degenerate diffusion equations with $L^{p}_t L^{q}_x$ drifts, including the critical case with divergence-free drifts.

Findings

Sharp subcritical condition for $p,q$ ensuring continuity.

Continuity depends on the modulus of $B$ in the critical case.

Divergence-free condition is crucial in the critical region.

Abstract

In this paper, we study local uniform continuity of nonnegative weak solutions to degenerate diffusion-drift equations in the form \[ u_{t} = \Delta u^{m} + \nabla\cdot \left( B (x,t) \, u\right), \quad \text{for } m \geq 1 \] assuming a vector field $B \in L^{p}_t L^{q}_{x}$. Regarding local H\"{o}lder continuity, we provide a sharp condition on $p$ and $q$, which is referred to as the subcritical region. In the critical region, the divergence-free condition is essential to providing uniform continuity which depends on the modulus continuity of $B$.