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  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.
Submitted on 18 Jan 2006 (v1), last revised 11 Apr 2006 (this version, v2). This work was in part supported by NSF under the NSF FRG grant DMS-0353838 and ITR Grant ACI-0204932.
Submitted - 0601427.pdf