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.
All Science Journal Classification (ASJC) codes
- Eventual regularity
- Global regularity
- Lower bounds for fractional laplacian
- Supercritical SQG