Unstable manifolds of Euler equations

TL;DR

This paper constructs unstable manifolds for steady states of the Euler equations in bounded domains, demonstrating nonlinear instability under certain spectral conditions, with applications to both 2D and 3D flows.

Contribution

It develops a method to construct unstable manifolds for Euler equations on infinite-dimensional manifolds, establishing nonlinear instability results.

Findings

Unstable manifolds are constructed under spectral gap conditions.

Finite-dimensional unstable subspaces imply nonlinear instability.

The approach applies to both 2D and 3D Euler flows.

Abstract

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.