Regularity, Convergence, and Uniqueness Problems for Harmonic Map Flows

本课题的工作涉及几何变分微积分领域中出现的几个解析问题：弱调和映射，调和映射到一般黎曼流形的弱流，p调和映射。解决这类问题的平顺性往往会崩溃。特别感兴趣的是调和映射的弱流在适当类中的部分正则性和唯一性，包括弱调和映射和调和映射的弱流在内的解空间的序紧性，Ginzburg-Landau型流和调和映射之间的关系，奇异点附近解的行为，调和映射流自相似解的存在性和几何性质。这项工作将涉及几何测量理论和几何方式的PDE方法。PI还建议制定亚历山德罗夫空间之间的调和映射的热流。偏微分方程是描述物理问题的基本工具。调和映射根据满足共同约束的对象族中的物理自然能量泛函对极值（或最优）对象进行建模。谐波映射的热流研究的是这类物体的长期动力学行为。该研究将增强我们对这些映射的理解，改进控制奇异集的方法，并预测这些问题解的奇异行为。研究结果将应用于材料科学，包括液晶和弹性/塑性。

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.