Analysis Seminar

The issue of loss of regularity of unique solutions to the 3D incompressible Euler equations is an important open question of fluid mechanics, and is closely related to the emergence of turbulence. We will discuss recent results regarding loss of regularity of solutions of the 2D and 3D Euler equations, and of the surface quasi-geostrophic equations (SQG), which is a well-established 2D model equation of the 3D Euler equations. We will discuss a result of continuous-in-time loss of Sobolev regularity of solutions to the SQG equation. Namely, given $s\in (3/2,2)$ and $\varepsilon >0$, we will describe a construction of a compactly supported initial data $\theta_0$ such that $\| \theta_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $\theta$ of the SQG equation such that $ \theta (\cdot ,t )$ belongs to ${H^{s/(1+ct)}}$ and does not belong to any other ${H^\beta }$, where $\beta > s/(1+ct)$. Moreover $\theta$ is continuous and differentiable on $\R^2\times [0,T]$, and is unique among all solutions with initial condition $\theta_0$ which belong to $C([0,T];H^{1+\alpha })$ for any $\alpha >0$.

This is the first result of this kind in incompressible fluid mechanics. It is also the first ill-posedness result in the supercritical regime which has compact support in space.