|Date||Aug 31, 2021|
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 obtain a singularity, the effect of the advection is one of the obstacles. In this talk, we will first talk about 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 was proposed in 1990, and generalized the Constantin-Lax-Majda model to model this competition. The regularity of the DG model on a circle remains an open problem. For initial data with specific sign and symmetry properties, which provides the most promising candidate for a potential blowup solution up to now, we will show that the DG model develops a finite-time singularity if the advection is "weaker" than the vortex stretching , and it is globally well-posed if the advection is "stronger". In particular, our results rule out the potential blowup from smooth initial data in such a class.