The question of global regularity v.s. finite time blow-up remains open for many fluid equations, such as the 3D Euler and Navier-Stokes equation. In two dimensions more is known, although the picture is far from complete. For the 2D incompressible Euler equation, global regularity of solutions has been known since the 1930s. On the other hand, this question is still open for the more singular surface quasi-geostrophic (SQG) equation, which arises in atmospheric science.

In this talk, I will discuss a family of equations which interpolate between the 2D Euler equation and the SQG equation. We focus on the patch dynamics for this family of equation in the half-plane, and obtain the following results: For the 2D Euler patch model, the patches remain globally regular even if they initially touch the boundary of the half-plane; whereas for the family of equations that are slightly more singular than the 2D Euler equation, the patches can develop a finite-time singularity. In this sense, the 2D Euler equation is indeed "critical". To the best of our knowledge, this is the first proof of singularity formation in this type of fluid dynamics models.

### Penn Mathematics Colloquium

Wednesday, November 1, 2017 - 3:30pm

#### Yao Yao

Georgia Tech