P. Butta', P. Negrini
On the Stability Problem of Stationary Solutions for the Euler Equation on a 2-Dimensional Torus
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ABSTRACT.  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.