On the global regularity for the supercritical SQG equation

TL;DR

This paper proves that for the supercritical SQG equation with sufficiently large fractional dissipation, solutions remain regular globally in time, using a novel approach based on nonlinear lower bounds and maximum principle.

Contribution

It introduces a new concise proof of eventual regularity for the supercritical SQG equation, applicable for large enough dissipation parameter.

Findings

Solutions do not blow up in finite time for large enough beta.

The proof relies solely on nonlinear lower bounds and maximum principle.

Regularity results hold even for arbitrarily large initial data within certain bounds.

Abstract

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t \theta + \mathcal{R}^\perp \theta \cdot \nabla \theta + \Lambda^\gamma \theta = 0, \qquad \theta(\cdot,0) =\theta_0 \] 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 $\|\theta_0\|_{L^2}^{\gamma/2} \|\theta_0\|_{\dot{H}^2}^{1-\gamma/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $\gamma_1 = \gamma_1(R) \in (0,1)$ such that for $\gamma \geq \gamma_1$, 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.