Mathematical Sciences: Nonlinear Partial Differential Equations

该项目的工作继续进行数学研究 应用粘度解决方案解决 非线性偏微分方程理论 一 这项研究的一个特别有趣的方向是 涉及到描述一个表面 在每个点处沿法线方向演化，如果速率 进化的速度与曲率成正比（在点处）。 这些问题的物理对应部分可以在 相变和波前传播的研究。 最近 结果表明，弱解描述了演化到 规律性被打破的地方。 工作还得做 在崩溃开始时分析解决方案。 其他的工作涉及调和映射的规律性。 这里 将努力扩大一个新的规律性结果， 映射到球面到1，对于平稳调和映射到 一般目标流形。 作为规律性失效的例子 对于维度大于2的域存在，不能 在这里期待任何具有完全普遍性的结果。 工作也将 在有涡面的二维欧拉方程上进行 初始数据。 有趣的是弱收敛的发生 二维不可压缩欧拉方程的解序列 对速度和涡量的物理合理假设。 问题是解的弱极限是否仍然是一个 溶液 非正式计算表明，一个易于处理的 解决这个问题是可能的。

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.