Academy of Mathematics and Systems Science, CAS Colloquia & Seminars
Prof. Feng Jin,Department of Math, University of Kansas, Lawrence, KS. USA
On a class of Hamilton-Jacobi equations in metric spaces
Time & Venue:
2015.8.11 16:00-17:00 N913
Another title could have been: the metric nature of a Hamilton-Jacobi in space of probability measures. There are two parts of the talk. In the first part, we set up a class of Hamilton-Jacobi equation in general geodesic metric spaces, devise a notion of viscosity solution, and prove a well-posedness result. In the second part, we focus on a particular case where the metric space is the Wasserstein (order-2) space of probability measures. Informally, such equation describes canonical transforms for a class of Lagrangian dynamics which turns out to be a class of compressible Euler equations. Its well posedness has been open previously. To understand why previous attempts failed, we introduce yet another notion of viscosity solution which is more geometric based. A metric nature of the problem will emerge -- we need to augment previous formulations of the Hamiltonian using a notion of geometric tangent cone, in order to be compatible with the previous metric formulation. Such cone can be explicitly identified with a class of Markov transition kernels, giving an interesting probability link. Such cone structure/probability connection is only needed when evolution for mass of particles becomes supported on small sets. This is a joint work with Luigi Ambrosio.
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