Global regularity for 2D Boussinesq temperature patches with no diffusion

TL;DR

This paper proves that the boundary curvature of temperature patches in the 2D Boussinesq system remains bounded over time without diffusion, using novel cancellations and Sobolev space techniques.

Contribution

It introduces new cancellation methods to control boundary regularity and extends regularity results to $C^{2+gamma}$ patches in the no-diffusion Boussinesq system.

Findings

Bounded curvature for temperature patches with $W^{2,gamma}$ boundary

New proof of $C^{1+gamma}$ regularity using Sobolev spaces

Persistence of $C^{2+gamma}$ regularity for patches

Abstract

This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space $\mathbb{R}^2$. We prove that for initial patches with $W^{2,\infty}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound $\nabla^2u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+\gamma}$ regularity result in [23], $0<\gamma<1$, using the scale of Sobolev spaces for the velocity. Furthermore, taking advantage of the new cancellations, we go beyond to show the persistence of regularity for $C^{2+\gamma}$ patches.