The incompressible Euler system with rough path advection

Abstract: The incompressible Euler’s equations are a mathematical model of an incompressible inviscid fluid. We will discuss some aspects of a perturbation of the Euler system by a rough-in-time, divergence-free, Lie-advecting vector field. We are inspired by the problem of parametrizing unmodelled phenomena and representing sources of uncertainty in mathematical fluid dynamics. We will begin by presenting a geometric fluid dynamics inspired variational principle for the equations and the corresponding Kelvin balance law. Then we will give sufficient conditions on the data to obtain i) local well-posedness of the system in any dimension in $L^2$-Sobolev spaces and ii) a Beale-Kato-Majda (BKM) blow-up criterion in terms of the $L_t^1L^\infty_x$-norm of the vorticity. The $L^p$-norms of the vorticity are conserved in two dimensions, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation in two dimensions.

This talk is based on joint work with Dan Crisan, Darryl Holm and Torstein Nilssen.

differential geometryprobability

Audience: advanced learners

Young Researchers between Geometry and Stochastic Analysis 2021