Global well-posedness for the 2D fractional Boussinesq Equations in the subcritical case

Daoguo Zhou, Zilai Li, Haifeng Shang, Jiahong Wu, Baoquan Yuan and Jiefeng Zhao

Vol. 298 (2019), No. 1, 233–255

DOI: 10.2140/pjm.2019.298.233

Abstract

We study the global regularity of solutions to the 2D Boussinesq equations with fractional dissipation, given by ${\left(-\Delta \right)}^{\alpha ∕2}u$ in the velocity equation and by ${\left(-\Delta \right)}^{\beta ∕2}\theta$ in the temperature equation. We establish the global regularity for $\frac{2}{3}<\alpha <1$, $\alpha +\beta >1$ and $\alpha >\frac{1}{1+\beta }$. This result is for the subcritical regime $\alpha +\beta >1$ and the point here is to obtain the global regularity for the largest possible range of $\alpha$.

Keywords

Boussinesq equations, fractional dissipation, global regularity

Mathematical Subject Classification 2010

Primary: 35B65, 35Q35

Secondary: 76B03, 76A10

Milestones

Received: 7 March 2017

Revised: 12 January 2018

Accepted: 25 May 2018

Published: 2 February 2019