Unstable manifolds of euler equations

Abstract

We consider a steady state v0 of the Euler equation in a fixed bounded domain in ℝn. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler equation as an ODE on an infinite-dimensional manifold of volume-preserving maps in Wk, q (k>1+n/q) the unstable (and stable) manifolds of v0 are constructed under a certain spectral gap condition that is satisfied for both two-dimensional and three-dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v0 in the sense that arbitrarily small Wk, q perturbations can lead to L2 growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.