Mathematical Sciences: Nonlinear Partial Differential Equations

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Work on this project continues mathematical research on the application of viscosity solutions to problems arising in the theory of nonlinear partial differential equations. One particularly interesting direction the research has led to involves the geometric problem of describing how a surface evolves in the direction of the normal at each point if the rate of evolution is proportional to the curvature (at the point). The physical counterparts of such questions can be found in studies of phase transition and wave front propagation. Recent results show that weak solutions describe the evolution to the point where regularity breaks down. Work still must be done analyzing solutions at the onset of breakdown. Other work concerns the regularity of harmonic maps. Here efforts will be made to expand a new regularity result valid for maps to the sphere to one for stationary harmonic maps into general target manifolds. As examples of regularity failure exist for domains in dimension greater than two, one cannot expect any results of complete generality here. Work will also be done on two-dimensional Euler equations with vortex sheet initial data. Of interest is the occurrence of weak convergence of sequences of solutions to the 2-D incompressible Euler under physically reasonable assumptions on the velocity and vorticity. The issue is whether or not a weak limit of solutions is still a solution. Informal calculations suggest that a tractable solution to this question is possible.