On the global regularity for the supercritical SQG equation

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 Λ^γ = (-Δ)^γ/2 is normalized to 1. We show that, given a smooth initial datum with ∥θ₀∥_L2^γ/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 λ^γ 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.