中科院数学与系统科学研究院

数学研究所

学术报告会

报告人： Prof. Jin Feng(University of Kansas, USA)

题  目：On the metric nature of two Hamilton-Jacobi equations for infinite particles

时  间：2016.07.05（星期二），15:30-16:30

地  点：数学院南楼N913室

Abstract:

In this talk,  a class of Hamilton-Jacobi PDEs in the space of probability measures will be formulated as equations in length metric spaces. The equations arise from both statistical mechanics as well as deterministic continuum mechanics applications. We develop an abstract well-posedness theory by introducing a new notion of viscosity solution.

The "metric nature" in the title reflects an observation that the space of probability measures can be viewed as an infinite dimensional polyhedron, a singular space where a good choice of metric can capture the singularities. In such context, a metric formulation of differential calculus using local Lipschitz constant gives more precise subtle information than the usual smooth calculus. Two situations illustration the above claim better. In the first one, I will use the geometric tangent cone concept, instead of the usual locally linear tangent space structure, in the Wasserstein space setting to handle mass condensation property of some Hamilton-Jacobi equations.  In another one, we make another observation that the local Lipschitz constant is metric dependent. With a change of base metric, the notion of derivative may change, hence re-normalizing the Hamilton-Jacobi PDEs. Also relying upon a few other techniques from the weak KAM theory in Lagrangian dynamics, we give a program for an abstract well-posedness theory for a Hamilton-Jacobi PDE modeling infinite particles with singular attractive pair-wise potentials. Well-posedness for the first problem is a published theorem now, a precise proof for the second problem is still a few lemmas away.