Holder continuity for a drift-diffusion equation with pressure

TL;DR

This paper proves that weak solutions to a linear drift-diffusion equation with nonlocal pressure maintain their H"older continuity over time, even when the drift velocity is at a critical regularity level, using a maximum principle approach.

Contribution

It establishes persistence of H"older continuity for solutions with critical regularity drift, without requiring smallness conditions on the drift velocity.

Findings

H"older continuity persists over time for solutions with critical drift.

The proof uses Campanato's characterization and a maximum principle argument.

The estimate depends only on scale-invariant quantities, not smallness of the drift.

Abstract

We address the persistence of H\"older continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressure \[ u_t + b \cdot \grad u - \lap u = \grad p,\qquad \grad\cdot u =0 \] on $[0,\infty) \times \R^{n}$, with $n \geq 2$. The drift velocity $b$ is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of H\"older spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of $u$. We provide an estimate that does not depend on any local smallness condition on the vector field $b$, but only on scale invariant quantities.