The Arnold stability theorems for the 2D alpha Euler equations

Abstract: In the 1960's V.I. Arnold introduced a simple and beautiful idea to study nonlinear Lyapunov stability of ideal fluids. This relied on exploiting the underlying Hamiltonian structure that the fluid model possessed together with convexity estimates on the second variation. Expanded later into the so called energy-Casimir method, this has spawned a huge literature and has been applied widely to study stability of various model fluid equations. In this work, we apply Arnold's method to study nonlinear stability of the 2D alpha Euler equations. The alpha Euler equations, introduced by C. Foias, D. Holm and E. Titi in the early 2000's, are an inviscid regularization of the classical Euler equations and have found use in diverse areas such as data assimilation, turbulence modeling and as a possible route to proving well posedness results for the Euler equations. We shall prove the Arnold stability theorems for the 2D alpha Euler equations on different domains such as the multi connected domain, the two torus and the periodic channel. By use of examples, we also indicate differences from the corresponding stability results for the classical 2D Euler equations.  This is joint work with Yuri Latushkin (University of Missouri).