Unstable Manifolds of Euler Equations

We consider a steady state v 0 of the Euler equation in a fixed bounded domain in ℝ n . 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 W k , q( k > 1 + n / q )the unstable (and stable) manifolds of v 0 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 v 0 in the sense that arbitrarily small W k , q perturbations can lead to L 2 growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.