The vortex sheet problem serves as a prototype for the evolution of vorticity in fluid flows. One can think for example of the wake of an airfoil as a typical problem of this type. This problem can be described by the incompressible Euler equations, where the initial vorticity is ideally a finite Radon measure supported on a curve. The issue is to determine the evolution of this curve. A further assumption that the vortex sheet remains a curve at a later time leads to the Birkhoff-Rott equation. The PI's initial study shows that a vortex sheet in general can not be a curve of reasonable regularity. On the other hand, Delort's result shows that vortex sheet fits as a weak solution of the Euler equation (for initially non-negative vorticity). However weak solutions seem to be a class too big to describe the specific nature of the vortex sheet evolution. The proposed research focuses on further pin point the nature of the vortex sheet evolution, through studying similarity spiral solutions, understanding the viscosity effects and the evolution of vortex layers.
Water wave is one of our most familiar experiences in daily life. A mathematical description is the incompressible, irrotational Euler equation, defined in the moving water domain. Study of water wave can be traced back to more than 150 years, in which the PI recently established the well-posedness of the problem locally in time, that is, the wave will evolute without breaking for a finite time period, from any initially non-self intersecting wave surface. The proposed research focuses on the large time behavior: the global existence of smooth solutions, the wave breaking-- the mechanisms that cause the wave breaking and breaking profiles. The proposal is to initiate from existing theories on limit equations. Through comparisons of the full water wave equation with the limit equations, the PI aims at developing enough machinery and understanding that lead to further research with greater generality.
The proposed research will further our understanding of the nature phenomena such as the water wave motion and wave breaking, the mixing of fluids, separation of boundary layers, generation of sounds and coherent structures in turbulence models. It will have a direct impact on the science and technology that influence our daily life.
