**Mokameeting, September 20th 2017, 10:00-12:00 AM, room A415 **

**Andrea Natale ** (Imperial College, London)

*Title: *Structure-preserving finite elements for perfect fluids

*Abstract: *Perfect fluids with constant density are governed by the incompressible Euler equations. These equations can be formulated as a Lagrangian system with symmetries on the group of volume-preserving diffeomorphisms. In this talk, I will show how this interpretation can be used to design finite element discretisations which share a similar structure. In particular, this leads to an energy-conserving scheme which also possesses a discrete ver- sion of Kelvin’s circulation theorem [A. Natale and C. J. Cotter. A variational H(div) finite element discretisation for perfect incompressible fluids. IMA Journal of Numerical Analysis. doi: 10.1093/imanum/drx033, 2017].

**Athanasios E. Tzavaras** (KAUST):

*Title: *The relative entropy method for a class of Euler flows

*Abstract: *The relative entropy method is a calculation originally developed for hyperbolic conservation laws

by Dafermos and DiPerna, which exploits the thermodynamical entropy structure of hyperbolic systems in order to

compare two appropriate solutions of the same or related thermomechanical systems.

In this talk I will survey applications of this methodology to a class of dispersive systems that can be written as

Euler flows generated by a variational structure induced by an energy functional.

This class admits as examples the Euler-Korteweg system, the quantum hydrodynamics system, and the Euler-Poisson system.

For these problems we develop a relative energy identity which in turn yields various asymptotic convergence results again for smooth solutions.

One application is the justification of high-friction limits and derivation of diffusion approximations for such models.