H\"{o}lder continuous weak solutions of the 3D Boussinesq equation with thermal diffusion

TL;DR

This paper proves the existence of Hölder continuous weak solutions to the 3D Boussinesq equation with thermal diffusion, approximating Onsager's critical regularity and matching prescribed kinetic energy.

Contribution

It constructs Hölder continuous solutions with specific regularity that satisfy the Boussinesq equations and prescribed energy, advancing understanding of weak solutions in fluid dynamics.

Findings

Existence of Hölder continuous weak solutions with regularity below Onsager's critical threshold.

Solutions satisfy prescribed kinetic energy distribution over time.

Solutions are constructed on the 3D torus with thermal diffusion included.

Abstract

In this paper, we show the existence of H\"{o}lder continuous periodic weak solutions of the 3D Boussinesq equation with thermal diffusion, which apprroximate the Onsager's critical spatial regularity and satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,T]\rightarrow \mathbb{R}_+$ and $\beta\in (0, \frac{1}{3})$, there exist $v\in C^{\beta}([0,T]\times {\mathbb{T} }^3)$ and $ \theta\in C_t^{1,\frac{\beta}{2}}C_x^{2,\beta}([0,T]\times {\mathbb{T} }^3)$ which solve (\ref{e:boussinesq equation}) in the sense of distribution and satisfy \begin{align} e(t)=\int_{{{\mathbb{T} }^3}}|v(t,x)|^2dx, \quad \forall t\in [0,T].\nonumber \end{align}