Unstable Manifolds of Stratified Euler Equations

Zhiwu Lin; Yanbo Wang; Chongchun Zeng

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arXiv · 2608.05370

Unstable Manifolds of Stratified Euler Equations

Zhiwu Lin·Yanbo Wang·Chongchun Zeng

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Abstract

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifolds of $(\rho_0,v_0)$. The proof is based on the Lyapunov--Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows.

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Zhiwu Lin, Yanbo Wang, Chongchun Zeng. 2026-08-05. Unstable Manifolds of Stratified Euler Equations. https://arxiv.org/abs/2608.05370

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