Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space

Aiting Le; Chenyin Qian

doi:10.3934/dcdsb.2022057

Article Contents

2022, Volume 27, Issue 12: 7625-7656. Doi: 10.3934/dcdsb.2022057

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Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space

Aiting Le and
Chenyin Qian^,

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

^* Corresponding author: Chenyin Qian
^* Corresponding author: Chenyin Qian

Received: November 2021

Revised: February 2022

Early access: March 2022

Published: December 2022

The second author is supported by Natural Science Foundation of Zhejiang Province, No. LY20A010017

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Abstract

In this paper, we investigate the fractional dissipation 2D Boussinesq equations with initial data in the critical space $ u_0\in H^{2-2\alpha}(\mathbb{R}^2) $ and $ \theta_0\in H^{2-2\beta}(\mathbb{R}^2) $. The local well-posedness for the equations is firstly established by using some a priori estimates for the solution in $ L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\alpha}(\mathbb{R}^2))\times L^{p}(0, T;{H}^{2-\frac{p-1}{p} 2\beta}(\mathbb{R}^2)) $ with some suitable $ p $. And then the generalized blow-up criterion and smoothing effect are obtained in turn, which improves some of the previous results for (critical, subcritcial or supcritical) Boussnesq equations. The results of the present paper are based on the Littlewood-Paley theory and the nonlinear lower bounds estimates for the fractional Laplacian, and can be treated as a generalization of results for 2D quasi-geostrophic equation.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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