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 $\mathcal{L}$ that can be defined through both an integral kernel and a Fourier multiplier.  When the operator $\mathcal{L}$ is represented by $\frac{|\xi|}{a(|\xi|)}$ with $a$ satisfying $ \lim_{|\xi|\to \infty} \frac{a(|\xi|)}{|\xi|^\sigma} = 0$ for any $\sigma>0$, 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.