Rigidity aspects of singular patches in stratified flows

TL;DR

This paper advances the understanding of the local well-posedness of 2D stratified flow models with singular patches, improving previous assumptions and analyzing both inviscid and viscous cases with delicate diffusion effects.

Contribution

It significantly improves existing results by replacing compatibility assumptions and establishes uniform local well-posedness for the viscous case with vanishing conductivity.

Findings

Enhanced well-posedness results for singular patch flows.

Uniform local well-posedness in viscous regimes with vanishing conductivity.

Refined analysis of transport-diffusion equations with singular potentials.

Abstract

We explore the local well-posedness theory for the 2d inviscid Boussinesq system when the vorticity is given by a singular patch. We give a significant improvement of \cite{Hassainia-Hmidi} by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We establish a uniform LWP theory with respect to the vanishing conductivity. This issue is much more delicate than the inviscid case and one should carefully deal with various difficulties related to the diffusion effects which tend to alter some local structures. The weak a priori estimates are not trivial and refined analysis on transport-diffusion equation subject to a logarithmic singular potential is required. Another difficulty stems from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing effects of transport-diffusion equation advected by a velocity field which scales slightly below the Lipschitz class.