Unstable Manifolds of Euler Equations

We consider a steady state v₀ of the Euler equation in a fixed bounded domain in ?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} 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 W^{k, q} perturbations can lead to L² growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.