| Abstract: |
| A Hamiltonian operator is a linear operator which can be written as JH with J skew-adjoint and H self-adjoint. We will discuss a new criterion to characterize stability and instability of Hamiltonian linear operators when H has finitely many negative directions. This criterion can be used with pen and paper mathematics to prove real instabilities, or combined with computer-assistance to prove complex instabilities.
We will focus on the latter case, proving instability of certain stationary solutions of the 2-d Euler equations. More concretely, we prove the conjectured instability of the Taylor-Green vortex (or cellular flow). Our proof relies on finding zeros of a holomorphic function, using an interval arithmetic version of the Cauchy integral formula. Moreover, the computation of our holomorphic function involves a resolvent, where we capitalize on the fact that the Taylor-Green linear dynamics are local in frequency and employ a primal-dual bound to control the distance between the true resolvent and a numerical guess.
We expect part of our framework to be applicable to a wider class of linear stability problems where the operator can be decomposed as the sum of a stable infinite-dimensional part plus a finite rank term causing instability. |
|