Title: On the competition between advection and vortex stretching

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 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 study the De Gregorio (DG) model and the generalized Constantin-Lax-Majda (gCLM) model, which model this competition, and several conjectures on these models. Using a novel framework based on dynamic rescaling, we construct finite time singularities of the DG model and gCLM model if the advection is "weaker" than the vortex stretching. On the other hand,  for initial data with the same sign and symmetry properties as the blowup solution, if the advection is "stronger", we show that the solution to the DG model exists globally.