Regularity results for viscous 3D Boussinesq temperature fronts

TL;DR

This work studies the 3D Boussinesq system without heat diffusion, focusing on temperature fronts and their interface regularity. It develops a parabolic-elliptic framework, leveraging a heat-kernel decomposition $u = e^{t\Delta}u_0 - (\partial_t-\Delta)_0^{-1}\mathbb{P}(u\cdot\nabla u) + (\partial_t-\Delta)_0^{-1}\mathbb{P}(\theta e_3)$ and the interface law $Z_t(\alpha,t)=u(Z(\alpha,t),t)$ to propagate regularity of fronts that are piecewise Hölder. The paper proves local existence for arbitrary data and global existence for small data in critical spaces, and establishes propagation of regularity for $C^{1+\gamma}$, $W^{2,\infty}$, and $C^{2+\gamma}$ fronts by combining maximal-regularity estimates, paradifferential calculus, and contour-dynamics techniques. This yields a unified approach to front persistence in 3D Boussinesq flows, including non-constant patches, with potential implications for understanding interface evolution in geophysical and environmental fluid dynamics.

Abstract

This paper is about the dynamics of non-diffusive temperature fronts evolving by the incompressible viscous Boussinesq system in $\mathbb{R}^3$. We provide local in time existence results for initial data of arbitrary size. Furthermore, we show global in time propagation of regularity for small initial data in critical spaces. The developed techniques allow to consider general fronts where the temperature is piecewise Hölder (not necessarily constant), which preserve their structure together with the regularity of the evolving interface.

Theorems & Definitions (12)

• Remark 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Theorem 3.1
• Remark 3.2
• Theorem 4.1
• Remark 4.2