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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 6

### Uniqueness of weak solutions to the Boussinesq equations without thermal diffusion

Pages: 1595 – 1624

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n6.a5

#### Authors

#### Abstract

This paper focuses on the general $d$-dimensional $(d \geq 2)$ Boussinesq equations with the fractional dissipation $(-\Delta)^{\alpha} u$ and without thermal diffusion. Our primary goal here is the uniqueness of weak solutions to this partially dissipated system in the weakest possible setting. The issue of the uniqueness of weak solutions is very important and can be quite difficult as in the case of the Leray–Hopf weak solutions to the 3D Navier–Stokes equations. We present two main results. The first is the global existence and uniqueness of weak solutions which assesses the global existence of $L^2$-weak solutions for any $\alpha \gt 0$ and the uniqueness of the weak solutions when $\alpha \geq \frac{1}{2} + \frac{d}{4}$ for $d \geq2$. Especially the 2D Boussinesq equations without thermal diffusion have unique and global $L^2$ weak solutions. The second result establishes the zero thermal diffusion limit with an explicit convergence rate for the aforementioned weak solutions. This convergence result appears to be the very first one on weak solutions of partially dissipated Boussinesq systems.

#### Keywords

Besov space, Boussinesq equation, uniqueness, weak solution

#### 2010 Mathematics Subject Classification

35Q35, 76D03

Received 12 July 2018

Accepted 1 July 2019

Published 26 December 2019