Regularity, Convergence, and Uniqueness Problems for Harmonic Map Flows

9706855 王 这个项目的工作涉及到几个分析问题， 从几何变分领域：弱 调和映射，调和映射的弱流到一般 黎曼流形p-调和映射解决方案的光滑度 这样的问题经常会出故障。特别感兴趣的是 调和映射弱流的部分正则性和唯一性 在适当的类中，解空间的序列紧性 包括弱调和映射和调和映射的弱流，Ginzburg-Landau型流和 调和映射，奇点附近解的行为，自相似解的存在性和几何性质， 谐波映射流。这项工作将涉及几何测量 理论和偏微分方程方法的几何时尚。PI还建议 导出亚历山德罗夫空间间调和映射的热流。 偏微分方程是描述 身体问题。调和映射模型极值（或最优） 对象相对于满足共同约束的对象族中的物理自然能量泛函。调和映射的热流研究这类族中物体的长时间动力学行为。 这项研究将加强我们的 理解这些映射，改进控制奇异集的方法，并预测这些问题的解的奇异行为。研究结果将在包括液晶和弹性/塑性的材料科学中得到应用。

9706855 Wang Work on this project concerns several analytic problems arising from the area of geometric variational calculus: weakly harmonic maps, weak flows of harmonic maps into general Riemannian manifolds, p-harmonic maps. Smoothness of solutions to such problems often breaks down. Of particular interest are the partial regularity and uniqueness of weak flows of harmonic maps in suitable classes, sequential compactness of the solution space including weakly harmonic maps and weak flows of harmonic maps, relationships between flows of Ginzburg-Landau type and harmonic maps, behavior of solutions near singular points, and existence and geometric properties of self-similar solutions to harmonic map flows. This work will involve geometric measure theory and PDE methods in geometric fashion. The PI also proposes to formulate heat flow of harmonic maps between Alexandrov spaces. Partial differential equations are the basic tools to describe physical problems. Harmonic maps model extremal (or optimal) objects with respect to physically natural energy functionals in families of objects satisfying common constraints. The heat flow of harmonic maps studies the long-time dynamical behavior of objects in such families. The study will enhance our understanding of these maps, improve methods to control the singular sets, and predict singular behavior of solutions to these problems. The results will have applications to material science including liquid crystals and elasticity/plasticity.