Workshop on Recent developments in incompressible fluid dynamics

Abstract: Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. The presence of vortex stretching is the primary source of a potential finite-time singularity. However, to construct a singularity, the effect of the advection is one of the obstacles. In this talk, we will first show some examples in incompressible fluids about the competition between advection and vortex stretching. Then we will discuss the De Gregorio (DG) model on a circle, which adds a natural advection term to the Constantin-Lax-Majda model to model this competition. The regularity of the DG model on a circle remains an open problem. We consider odd initial data with a specific sign property, which provides the most promising candidate for a potential blowup solution up to now. For this class of initial data, the regularity of the initial data determines the competition between advection and vortex stretching. We show that the solution exists globally for C^1 initial data. On the other hand, for any $0<\alpha < 1$, we construct a finite time blowup solution from some C^{\alpha} initial data.