On the global regularity for the supercritical SQG equation

Abstract

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t θ+ \mathcal{R}^\perp θ\cdot \nabla θ+ Λ^γθ= 0, \qquad θ(\cdot,0) =θ_0 \] on ${\mathbb{T}}^2=[0,1{]}^2$, with $\gamma \in (0,1)$. The coefficient in front of the dissipative term ${\Lambda }^{\gamma }=(-\Delta {)}^{\gamma /2}$ is normalized to $1$. We show that given a smooth initial datum with $\Vert {\theta }_0{\Vert }_{L^2}^{\gamma /2}\Vert {\theta }_0{\Vert }_{{\dot{H}}^2}^{1-\gamma /2}\le R$, where {\em $R$ is arbitrarily large}, there exists ${\gamma }_1={\gamma }_1(R)\in (0,1)$ such that for $\gamma \ge {\gamma }_1$, the solution of the supercritical SQG equation with dissipation ${\Lambda }^{\gamma }$ does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, that relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.

12 pages