Eigenvalues of the Linearized 2D Euler Equations via Birman–Schwinger and Lin’s Operators

• Published in: Journal of mathematical fluid mechanics Vol. 20; no. 4; pp. 1667 - 1680
• Main Authors: Latushkin, Yuri, Vasudevan, Shibi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-12-2018; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Eigenvalues; Euler-Lagrange equation; Eulers equations; Existence theorems; Fluid mechanics; Fluid- and Aerodynamics; Linearization; Mathematical analysis; Mathematical Methods in Physics; Operators; Perturbation; Physics; Physics and Astronomy; Stability; Theoretical mathematics; Toruses; Vorticity; Instability; 2D Euler equations; Birman–Schwinger operators; 2-modified perturbation determinants
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-018-0383-4

We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman–Schwinger type operators
K
λ
(
μ
)
and their associated 2-modified perturbation determinants
D
(
λ
,
μ
)
. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator
L
vor
in terms of zeros of the 2-modified Fredholm determinant
D
(
λ
,
0
)
=
det
2
(
I
-
K
λ
(
0
)
)
associated with the Hilbert Schmidt operator
K
λ
(
μ
)
for
μ
=
0
. As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for
L
vor
to the number of negative eigenvalues of a limiting elliptic dispersion operator
A
0
.