Geometric constrains for global regularity of 2D quasi-geostrophic flows

We study the two-dimensional quasi-geostrophic equations (2D QG) in Sobolev spaces. We first prove a new analytic condition for global regularity which is both sufficient and necessary. We then prove several new results on the geometric constraints on the 2D QG active scalar which suppress the development of singularity from the nonlinear stretching mechanism. We focus mainly on the case with critical dissipation. Our results are also relevant to the inviscid case.