# H\"{o}lder continuous weak solution of 2d Boussinesq equation with diffusive temperature

## TL;DR

This paper proves the existence of Hölder continuous weak solutions to the 2D Boussinesq equations with diffusive temperature, which match prescribed kinetic energy profiles, demonstrating non-uniqueness and regularity properties.

## Contribution

It constructs Hölder continuous weak solutions to the 2D Boussinesq equations with diffusive temperature that realize any smooth kinetic energy profile.

## Key findings

- Existence of Hölder continuous weak solutions with prescribed energy.
- Solutions satisfy the Boussinesq equations in the distributional sense.
- Solutions exhibit specific regularity properties in space and time.

## Abstract

We show the existence of H\"{o}lder continuous periodic weak solutions of the 2d Boussinesq equation with diffusive temperature which satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,1]\rightarrow R_+$ and $\varepsilon\in (0, \frac{1}{10})$, there exist $v\in C^{\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2), \theta\in C_t^{1,\frac{1}{20}-\frac{\varepsilon}{2}}C_x^{2,\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2)$ which solve boussinesq equation in the sense of distribution and satisfy e(t)=\int_{{\rm T}^2}|v(t,x)|^2dx, \quad \forall t\in [0,1].