Methods and Applications of Analysis

Volume 30 (2023)

Number 4

Global well-posedness and decay estimate for the 2-D Boussinesq system in critical spaces

Pages: 149 – 174

DOI: https://dx.doi.org/10.4310/MAA.2023.v30.n4.a2

Authors

Dongjuan Niu (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Lu Wang (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Abstract

The aim of this paper is to investigate the Cauchy problem of two-dimensional Boussinesq equation with variable viscosity. We establish the global well-posedness of strong solution in critical spaces $(L^2 \cap \dot{B}^ {−1}_{\infty, 1}(\mathbb R^2)) \times \dot{B}^{\varepsilon}_{\frac{2}{\varepsilon}_,{1}}(\mathbb R^2)$ and prove the global well-posedness and time-decay estimates of strong solutions in either critical spaces $(\dot{B}^{{-1} + \frac{2}{p}}_{p,1} \cap L^2 (\mathbb R^2))$ or the almostscaling invariant spaces $(\dot{B}^{0}_{{2,1}} \cap \dot{B}^{{-\varepsilon}}_{{2,1}} (\mathbb R^2) \times (\dot{B}^{1}_{{2,1}} \cap \dot{B}^{{1 +\varepsilon}}_{{2,1}} (\mathbb R^2)$ under the suitable assumption of initial temperature. Compared with the result of Abidi (J. Math. Pure Appl. 91 (2009) 80-99), we weaken the initial assumption.

Keywords

Boussinesq system, global well-posedness, time-decay estimates

2010 Mathematics Subject Classification

35Q30, 76D03

Received 13 August 2023

Accepted 19 July 2024

Published 7 August 2024