The 2D Boussinesq-Navier-Stokes Equations with Logarithmically Supercritical Dissipation (Journal of Mathematical Study, 2024)
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 $\frac{|ξ|}{a(|ξ|)}$ with $a$ satisfying $ \lim_{|ξ|→ ∞} \frac{a(|ξ|)}{|ξ|^σ} = 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.