Gianne Derks attended the Banff International Research Station in Canada the week of 4 November for a workshop on Analysis, Stability and Bifurcation in Modern Nonlinear Physical Systems. Her talk title was “Viscosity induced instability for Euler and averaged Euler equations in a circular domain” with abstract: We consider the infinite time behaviour of a family of stationary solutions of Euler’s equation, which can be described as constrained minima of energy on level sets of enstrophy. For free boundary conditions, this family shadows solutions of 2D Navier-Stokes equations. However, under the no-slip
and under the Navier-slip boundary conditions and in a circular domain, the infinite time Navier-Stokes evolution orbit of a starting point on the family of constrained minima has order 1 distance to the family, however small the viscosity is. The viscosity in the Navier-Stokes equations is a singular perturbation for Euler’s equation and one might suspect that the viscosity-induced instability is related to this singularity. This is not the case: we show that the same phenomenon can be observed for the averaged Euler equations and second grade fluids with Navier-slip boundary conditions in a circular domain.
A video of Gianne’s talk is available here.