  We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded
domain in $\mathbf{R}^{n}$. 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 $W^{k, q}$, $(k>1+\frac{n}{q})$, the unstable (and stable)
manifolds of $v_{0}$ are constructed under certain spectral gap condition which
is verified for both 2D and 3D 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.
