The first part of the talk will cover an introduction to geometric fluid mechanics. Geometric fluid mechanics is an approach to derive and analyze the incompressible Euler equation using infinite dimensional differential geometry and group theory, pioneered by Vladimir Arnold in the 1960s, with further geometric mechanical foundation given by J. Marsden and A. Weinstein. I will first review Lie algebra, and symplectic and Poisson structures. Then I will use these frameworks to derive the Euler equation, and discuss the relationship between the circulation theorem, coadjoint orbits, and Clebsch representations.

 

In the second part of the talk, we present a topological analysis of the vorticity formulation of the incompressible Euler equation. In particular, we elucidate the equations of motion for the often-omitted cohomology component of the velocity on non-simply-connected domains. These equations have nontrivial coupling with the vorticity, which is crucial for characterizing correct vortex motions with presence of tunnels or islands in the domain. The dynamics of fluid’s cohomology reveals the curvature of the commutator subgroup of the Lie group of volume-preserving diffeomorphisms, and it is also associated with new conservation laws as Casimir invariants. Additional results include new analytical solutions of the Euler equations that are related to the Hilbert transform; and the first general vortex-streamfunction-based numerical method on curved surfaces with arbitrary genus and boundaries that is consistent with the Euler equation.

 

WebEx Link: 

• Session 1 (10:00 -12:00) https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m792c57765b0abcc730f7ed0713842f77

• Session 2 (16:00-17:00) https://nationaltaiwanuniversity-ksz.my.webex.com/nationaltaiwanuniversity-ksz.my/j.php?MTID=m8c5d378c6fbbbe72b4a6d55f80f0d288