Global persistence of geometrical structures for the boussinesq equation with no diffusion

TL;DR

This paper proves that higher regularity of temperature patches in the Boussinesq system persists over time, with results varying based on initial data size and spatial dimension.

Contribution

It establishes the global-in-time persistence of higher regularity for temperature patches in the Boussinesq system, extending previous results limited to $C^1$ regularity.

Findings

Higher regularity persists globally in 2D for large initial data.

Higher regularity persists globally in higher dimensions for small initial data.

The results depend on the spatial dimension and initial data size.

Abstract

Here we investigate the so-called temperature patch problem for the incompressible Boussinesq system with partial viscosity, in the whole space $\mathbb{R}^N$ $(N \geq 2)$, where the initial temperature is the characteristic function of some simply connected domain with $C^{1, \varepsilon}$ H{\"o}lder regularity. Although recent results in [1, 15] ensure that an initially $C^1$ patch persists through the evolution, whether higher regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue globally in time, in the 2-D case for large initial data and in the higher dimension case for small initial data.