Caltech/UCLA/USC Joint Analysis Seminar

In-person held at Caltech, 310 Linde Hall

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, which adds an advection term to the Constantin-Lax-Majda model to model this competition. In an effort to establish singularity formation in incompressible fluids, we develop a novel approach based on dynamic rescaling formulation. Using this approach, we construct finite time singularities of the DG model on the real line from smooth initial data and on a circle from C^{\alpha} initial data with any $0<\alpha < 1$. On the other hand, for $C^1$ initial data with the same sign and symmetry properties as those of the blowup solution, we prove that the solution of the DG model on a circle exists globally.