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 (Comm. PDE, 2005) 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 $\theta$, we obtain global regularity results with improved growth estimate on $| \nabla^{\bot} \theta |$. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum $| \nabla^{\bot} \theta |$. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of $\theta$ in our theorems are satisfied. Therefore these theorems provide a good explanation of the double exponential growth of $| \nabla^{\bot} \theta |$ observed in this and past numerical simulations.