Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation

In this article we apply the technique proposed in Deng-Hou-Yu [7] to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric regularity of the level sets of θ, we obtain global regularity results with improved growth estimate on │∇^⊥θ│. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum │∇^⊥θ│. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of θ in our theorems are satisfied. Therefore these theorems provide a good explanation of the double exponential growth of │∇^⊥θ│ observed in this and past numerical simulations.