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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.