Title: Instability of the 2D Taylor-Green vortex
Abstract: We will explain a new general criterion that characterizes the unstable eigenvalues of certain Hamiltonian operators as the zeros of an holomorphic function (or Evans function) given by the determinant of a finite-dimensional matrix. In the case of Hamiltonian systems with one negative direction, we show this new function is monotone on the real axis, reducing the study of spectral stability to the sign of a explicit quantity that can be computed by hand. In the case of more than one negative direction (in which complex instabilities can arise), our criterion can still be applied, in combination with computer-assisted techniques.
We apply this method to study the spectral properties of the Taylor-Green vortex in two-dimensional ideal fluids. We use our criterion to rule out instabilities in the presence of odd symmetry, via a “pen & paper” proof. We also use our criterion combined with computer-asssitance to prove a complex instability of the even in x & even in y type.