Geometric Analysis of Vortex Sheet Evolution and Value Distribution of Harmonic Maps into Hadamard Surfaces

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Proposal Number: DMS-0103888 Han's proposal comprises of two research projects and an education project. In the first research project Han proposes to adapt recent techniques from geometric analysis to study certain geometrical and analytical problems in the evolution of vortex sheets in two dimensional Euler equations. In particular, the project proposes to study a possible notion of weak solution in terms of rectifiable varifolds, which, if successful, should provide more geometric description to the evolution of singular vortex sheet than the currently available notion of weak solution in terms of integrable vorticity functions. In some prototype situations, Han also proposes to study the more precise geometric behavior in the roll up of intersecting vortex sheets. In the second project, Han proposes to establish a value distribution theory for harmonic maps into negatively curved surfaces. Recent results in this area have suggested very rich geometric behavior of such harmonic maps in terms of induced foliation structures and tree-like structures in the vicinity of infinity. The geometric description of such behavior is closely related to the solvability of asymptotic boundary value problems of the relevant system of partial differential equations. The interaction of geometric analysis, partial differential equations, and their application to some interesting applied analysis problems is the thread connecting the two projects. In the education project, Han proposes to rejuvenate an undergraduate geometry course mostly for math education majors and an undergraduate differential geometry course to better serve the need of a wider audience that has grown out of today's rapidly changing technological environments and emergence of new interdisciplinary fields. A good understanding of the evolution of concentrated vortices, such as vortex sheet, has immensely important practical values. The mathematical equations that govern the evolution of vortex sheets exhibit many features very similar to those that have been successfully studied in geometric analysis in recent years. The PI hopes that the interactions of ideas and tools from across the fields will bring some fruitful results in understanding better the geometric aspects of vortex sheet evolution. In his second project, the PI also hopes to combine closer the geometric and the analytic approaches, the success of which may provide further insight back to purely geometrical or analytical problems. The successful implementation of the third proposed project will will be a positive contribution to the K-12 math education through better training of math education majors.