The 2D incompressible Boussinesq equations with general critical dissipation

Abstract

This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $\alpha +\beta =1$ when ${\Lambda }^{\alpha }\equiv (-\Delta {)}^{\frac{\alpha }{2}}$ and ${\Lambda }^{\beta }$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $\alpha +\beta =1$ and $0.9132\approx {\alpha }_0<\alpha <1$. The cases when $\alpha =1$ and when $\alpha =0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by exploiting the global regularity of a generalized critical surface quasi-gesotrophic equation as well as the regularity of a combined quantity of the vorticity and the temperature.

30 pages

References in corpus (1)

• Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation

Cited by in corpus (5)

• Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation
• Global regularity for the 2D Oldroyd-B model in the corotational case
• A global regularity result for the 2D Boussinesq equations with critical dissipation
• Global smooth solution to the 2D Boussinesq equations with fractional dissipation
• Global well-posedness to the subcritical Oldroyd-B type models in 2D