Gianne Derks gives a talk at IMACS, Georgia, USA

Gianne Derks attended the Eighth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory in Athens, GA, USA from 25-28 March.  She reported on joint work with Tudor Ratiu (EPFL, Dragos & Universite de Lyon) . Her talk was on the inviscid limit of solutions of the 2D Navier-Stokes equations on a bounded domain.  This limit is well studied on a finite time interval: the 2D Navier-Stokes equations converge to solutions of Euler’s equations in the inviscid limit. This holds under various boundary conditions like the Navier slip, the no-slip, and the free boundary condition.  In this talk, the focus is on 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, an infinite time  orbit,  of a solution of the Navier-Stokes equations, of a starting point on the family of constrained minina, has order one distance to the original 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.