This seminar will be presented in hybrid mode.  The speaker will deliver his talk in person.

Title: Euler-Poincaré equations (roughly)

Abstract: Vladimir Arnold’s pioneering work in 1966 clarified that the configuration space for ideal fluids is the Lie group of volume-preserving diffeomorphisms on a given domain and fluid flow is a geodesic on the group endowed with the right invariant Riemannian L2-metric. The system of PDEs that govern the velocity field in ideal fluid flow, discovered by Euler in 1757 are, in fact, the Euler-Poincaré equations, which are the reduced form of the Euler-Lagrange equations discovered by Poincaré for a general Lie group and action-invariant Lagrangian in 1901. Moreover, the right-invariance of the metric, equivalent to the so-called particle relabeling symmetry of the kinetic energy, necessitates a conservation law by Noether’s theorem. This conservation law is the celebrated Kelvin’s circulation theorem. Holm, Ratiu, and Marsden (1998) extended the scope of the geometric and variational approach to fluid dynamics to cast many inviscid geophysical fluid dynamic equations as Euler-Poincaré equations on semidirect product Lie algebras by incorporating additional physical variables advected by the flow and adding potential energy terms to the Lagrangian.

I will discuss joint work with Dan Crisan, Darryl Holm, and Torstein Nilssen, in which we put forward variational principles for fluid dynamics on geometric rough paths. Motivated by a structural scale separation assumption and a homogenization argument, we begin with the premise that the Eulerian velocity that advects physical quantities decomposes into a sum of a smooth and geometric rough-in-time vector field. Critical points of the variational principle are characterized by a system of rough PDEs, which are a perturbed version of the Euler-Poincaré equations derived in Holm, Ratiu, and Marsden (1998). In the ideal fluid setting, I will also discuss the well-posedness and a blowup criterion for the corresponding Euler system of rough partial differential equations on the torus.

The talk will primarily be based on:

• Crisan, D., Holm, D.D., Leahy, J.M. and Nilssen, T., 2022. Variational principles for fluid dynamics on rough paths. Advances in Mathematics, 404, p.108409.

• Crisan, D., Holm, D.D., Leahy, J.M. and Nilssen, T., 2022. Solution properties of the incompressible Euler system with rough path advection. Journal of Functional Analysis, 283(9), p.109632.

The talk will be followed by refreshments in the Huxley Common Room at 4pm.