Kazuo Yamazaki, University of Rochester
3:00 PM - 4:15 PM
I will survey two specific examples of PDE in fluid mechanics such that a noise somehow leads to results that seem completely inaccessible in the deterministic case. Firstly, in the deterministic case, when the system is not known to be globally well-posed, an alternative result is typically the global well-posedness for sufficiently small initial data. The current method in the literature to prove this result requires diffusion from every equation in the system. I will give an example of a situation in which, a system forced by Levy noise, still admits this result even if diffusion is missing in some equations. Secondly, in classical fluid mechanics literature there are various formulas and identities such as Kelvin’s conservation of circulation flow and Cauchy’s vorticity transport formula; they hold only for the non-diffusive case and immediately break down as soon as we add diffusion (thus it holds for the Euler equations but not the Navier-Stokes equations). Nevertheless, discovering stochastic Lagrangian formulation, we are able to show that they actually continue to hold, somehow through mathematical expectation (average over earlier times).
Event contact: sevak dot mkrtchyan at rochester dot edu