Regularity, Convergence, and Uniqueness Problems for Harmonic Map Flows

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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.