Communications in Mathematical Sciences

Volume 16 (2018)

Number 5

Global well-posedness for $n$-dimensional Boussinesq system with viscosity depending on temperature

Pages: 1427 – 1449

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n5.a12

Authors

Xiaoping Zhai (School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong, China)

Zhi-Min Chen (School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong, China)

Abstract

In this paper, we study the global well-posedness issue for the Boussinesq system with the temperature-dependent viscosity in $\mathbb{R}^n (n \geq 2)$. With a temperature damping term, we first get a global solution in $\mathbb{R}^2$, provided the initial temperature is exponentially small compared with the initial velocity field. Then, using a weighted Chemin–Lerner-type norm, we can also give a global large solution in $\mathbb{R}^n$ if the initial data satisfies a nonlinear smallness condition. In particular, our results imply the global large solutions without any smallness conditions imposed on the initial velocity.

Keywords

global well-posedness, Boussinesq system, Littlewood–Paley theory

2010 Mathematics Subject Classification

35Axx, 35Q30, 76D03

Full Text (PDF format)

Received 15 November 2017

Received revised 8 June 2018

Accepted 8 June 2018

Published 19 December 2018