Discrete Schrödinger equation and ill-posedness for the Euler equation

Abstract

We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by \begin{document}$U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$ \end{document} can be well approximated by the linear Schrödinger equation, at least for a short period of time.