Holder continuity for a drift-diffusion equation with pressure

Abstract

We address the persistence of Hölder 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 $[0,\infty )\times {\mathbb{R}}^n$, with $n\ge 2$. 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 Hölder 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.

References in corpus (2)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics

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