Chennai Mathematical Institute


Date: Tuesday, February 9, 2021.
Time: 3:30 PM.
Topic: CMI Mathematics Seminar
Continued fractions, Fredholm determinants and unstable eigenvalues of ideal fluids

Shibi Vasudevan
Chennai Mathematical Institute.


In this talk, I present results on the stability of steady state solutions to the two dimensional incompressible Euler equations on a torus. In particular, I offer two characterizations of unstable eigenvalues (that is, eigenvalues with a positive real part) of the linearized vorticity operator: one in terms of continued fractions and the other in terms of 2-modified Fredholm determinants (which are also known as perturbation determinants). In the first part, I discuss stability of unidirectional flows. These are steady states whose vorticity is given by Fourier modes corresponding to a single vector p in the integer lattice Z^2. We obtain a necessary and sufficient condition for the existence of an unstable eigenvalue in terms of roots of equations involving certain continued fractions. In the second part, I consider spectral instability of steady states via certain Birman–Schwinger type operators $K_{\lambda}(\mu)$ and their associated 2-modified perturbation determinants $\mathcal D(\lambda,\mu)$. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{\rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $\mathcal D(\lambda,0)= {\rm det}_{2} (I-K_{\lambda}(0))$ associated with the Hilbert Schmidt operator $K_\lambda(0)$.