Variation on a theme by Kiselev and Nazarov: H{ö}lder estimates for non-local transport-diffusion, along a non-divergence-free BMO field

Abstract

We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator, and a general advection field in BMO, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by L. Silvestre (2012), our advection field does not need to be bounded. A similar result can be obtained in the super-critical case if the advection field is Hölder continuous. Our proof is inspired by A. Kiselev and F. Nazarov (2010) and is based on the dual evolution technique. The idea is to propagate an atom property (i.e. localization and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.

Paralinearization of the Muskat Equation and Application to the Cauchy Problem

“…This means that the lifespan necessarily depends on the profile of the initial data. This is a well-known problem in the analysis of several dispersive equations, or other parabolic equations (see in particular [37,13,26,48,50,43]).…”

On the dynamics of the roots of polynomials under differentiation

“…We can see that this last equation shares many characteristics with the Hele-Shaw equation, the Muskat equation or the dissipative surface quasi-geostrophic equation, to name a few. For the dissipative SQG equation, the global regularity has been proved by Kiselev, Nazarov and Volberg [27], Caffarelli-Vasseur [8] and Constantin-Vicol [19] (see also [26,37,41,32]). The nonlinearity in the Muskat equation is more complicated.…”