The 2D Boussinesq-Navier-Stokes Equations with Logarithmically Supercritical Dissipation

Abstract

We study the global well-posedness of the initial-value problem for the 2D Boussinesq-Navier-Stokes equations with dissipation given by an operator L that can be defined through both an integral kernel and a Fourier multiplier. When the operator L is represented by $ \frac{|\xi|}{a(|\xi|)}$ with a satisfying $ \lim _{|\xi| \rightarrow \infty} \frac{a(|\xi|)}{|\xi|^{\sigma}}=0$ for any σ>0, we obtain the global well-posedness. A special consequence is the global well-posedness of 2D Boussinesq-Navier-Stokes equations when the dissipation is logarithmically supercritical.

Keywords

Supercritical Boussinesq-Navier-Stokes equations / global regularity

Acknowledgements

Regmi was partially supported by Presidential Summer Incentive Award 2023-24, University of North Georgia. Wu was partially supported by the National Science Foundation of USA grants DMS 2104682 and 2309748. The authors thank Professor Dongho Chae for discussions.

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