Singularity formation in incompressible fluids and related models
Jiajie Chen (California Institute of Technology)
Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In these lectures, we will describe some recent progress on singularity formation in incompressible fluids and related models. We will begin with some properties of the 3D Euler equations useful for studying singularity formation and the dynamic rescaling formulation of the 3D Euler equations. Then we will discuss some ideas to overcome some difficulties in singularity formation and study finite time blowup based on the stability of an approximate blowup profile. We will compare this stability and another notion of stability of blowup. Lastly, we will discuss some ideas for constructing finite time blowup from smooth initial data, particularly in 1D models of the Euler equations, which can be helpful in studying the singularity formation of 3D Euler with smooth data.