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Abstract |
The goal of the talks is to introduce some well-posedness theories for a large class of Hamilton-Jacobi PDEs in space of probability measures (Wasserstein space). A significant part of the issues are better understood if we further generalize by viewing the equation as defined in a complete metric spaces with geodesic (or more generally length) property. The main motivations come from issues in interacting particle systems and variational formulations of continuum mechanics equations. Concrete examples from these areas will be used to develop the theories.
The following is a list of topics to be covered:
0. A quick review on Hamilton-Jacobi equations and viscosity solution in Euclidean spaces.
1. Second order Hamilton-Jacobi equation in Euclidean space as first order HJ equation in Wasserstein space:
2. Stochastic Hamilton-Jacobi equation in Euclidean space as first order Hamilton-Jacobi equation in Wasserstein space
3. A calculus of functions and curves on abstract length metric space
4. Several variational problems in length metric space, and first order Hamilton-Jacobi equations
5. A delicate aspect of Wasserstein space -- linear tangent spaces v.s. geometric tangent cones.
6. First order Hamilton-Jacobi equations in Wasserstein space, well-posedness and its relation with compressible irrotational Euler equation.
7. A renormalized viscosity solution theory in length metric spaces. |
