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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 6

### Uniqueness of weak solutions to the Boussinesq equations with fractional dissipation

Pages: 1531 – 1548

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n6.a4

#### Authors

#### Abstract

This paper examines the existence and uniqueness of weak solutions to the ddimensional Boussinesq equations with fractional dissipation $(-\Delta)^{\alpha}u$ and fractional thermal diffusion $(-\Delta)^{\beta}\theta$. The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting $u\in L^{\infty}(0,T;B_{2,1}^{{d/2}-2\alpha+1}(\mathbb{R}^d))$ and $\theta\in L^{\infty}(0,T;B_{2,1}^{d/2}(\mathbb{R}^{d}))$ when $\alpha>{1/4}$, $\beta\geq 0$ and $\alpha+2\beta\geq 1$. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term, which allows us to close the estimates in the aforementioned Besov spaces.

#### Keywords

Boussinesq equations, Littlewood–Paley, weak solution, uniqueness

#### 2010 Mathematics Subject Classification

35Axx, 35Q35, 76Dxx

Ji is supported by the National Natural Science Foundation of China (Grants No. 12001065) and Creative Research Groups of the Natural Science Foundation of Sichuan (Grants No. 2023NSFSC1984), Wu is partially supported by NSF grant DMS 2104682 and the AT&T Foundation at Oklahoma State University.

Received 16 November 2019

Received revised 30 October 2022

Accepted 18 November 2022

Published 22 September 2023