Abstract: We consider the nonlinear instability of a steady state
$v_*$ of the Euler equation in an $n$-dim fixed smooth bounded domain. When
considered in $H^s$, $s>1$, at the linear level, the stretching of the
steady fluid trajectories induces unstable essential spectrum which
corresponds to linear instability at small spatial scales and the
corresponding growth rate depends on the choice of the space $H^s$.
More physically interesting linear instability relies on the unstable
eigenvalues which correspond to large spatial scales. In the case when
the linearized Euler equation at $v_*$ has an exponential dichotomy of
center-stable and unstable (from eigenvalues) directions, most of the
previous results obtaining the expected nonlinear instability in $L^2$
(the energy space, large spatial scale) were based on the vorticity
formulation and therefore only work in 2-dim. In this talk, we prove,
in any dimensions, the existence of the unique local unstable manifold
of $v_*$, under certain conditions, and thus its nonlinear
instability. Our approach is based on the observation that the Euler
equation on a fixed domain is an ODE on an infinite dimensional
manifold of volume preserving maps in function spaces. This is a
joint work with Zhiwu Lin.