Piazzale Aldo Moro 2, 00185 Roma, Italy
Dipartimento di Matematica, Universita` di Roma La Sapienza
Butta P., Negrini P.
On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus
2010, vol. 15, no. 6, pp. 637-645
We study the linear stability problem of the stationary solution $\psi^* = −\cos y$ for the Euler equation on a 2-dimensional flat torus of sides $2\pi L$ and $2\pi$. We show that $\psi^*$ is stable if $L \in (0, 1)$ and that exponentially unstable modes occur in a right neighborhood of $L = n$ for any integer $n$. As a corollary, we gain exponentially instability for any $L$ large enough and an unbounded growth of the number of unstable modes as $L$ diverges.