We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation (Equation presented) on T2 = [0,1]2 with γ ∈ (0,1). The coefficient of the dissipative term Λγ = (-Δ)γ/2 is normalized to 1. We show that, given a smooth initial datum with ∥θ0∥L2γ/2 ∥θ0∥H2γ/2 ≤ R, where R is arbitrarily large, there exists γ1 = γ1(R) ∈ (0,1) such that, for γ ≥ γ1, the solution of the supercritical SQG equation with dissipation λγ 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, which relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.
- Eventual regularity
- Global regularity
- Lower bounds for fractional laplacian
- Supercritical SQG
ASJC Scopus subject areas