A Hamilton-Jacobi theory for the hydrodynamic limit large deviation of nonlinear heat equation from stochastic Carleman particles

Wednesday, June 20, 2018 - 10:00am - 11:00am
Lind 409
Jin Feng (University of Kansas)
The deterministic Carleman equation can be considered as an one dimensional two speed fictitious gas model. Its associated hydrodynamic limit gives a nonlinear heat equation. The rigorous derivation of such limit was known since the 1970th. In this talk, starting from a more refined stochastic model giving the Carleman equation as the mean field, we derive a fluctuation structure associated with the hydrodynamic limit.

The large deviation result is established through an abstract Hamilton-Jacobi method by Kurtz and myself applied to this specific setting. The principal idea is to identify a two scale averaging structure in the context of Hamiltonian convergence in the space of probability measures. This is achieved through a change of coordinate to the density-flux description of the problem. We also extend a method in the weak KAM theory to the infinite particle context for explicitly identifying the effective Hamiltonian. In the end, we conclude by establishing a comparison principle for a set of Hamilton-Jacobi equation in the space of measures.

At the present time, there is still a gap between the final large deviation theory and the completed result. I will present the subtle issues involved and also put the method in perspective regarding challenges we face when applying the method to other hydrodynamic limit issues.

This is a joint work with Toshio Mikami and Johannes Zimmer