Title: Unstable manifolds and $L^2$ nonlinear instability of Euler equations

Speaker: Prof. Chongchun Zeng (Georgia Institute of Technology)

Venue: Room 09 at Quan-Zhai (全斋), Beijing International Center for Mathematical Research, Peking University

Time: July 24, 2012, Tuesday, 10:00-11:00am

Abstract: We consider the nonlinear instability of a steady state $v_{0}$ of the Euler equation in an $n$-dim fixed 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 $v_0$ 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 $v_0$, 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 in function spaces. This is a joint work with Zhiwu Lin.