Georgia Institute of Technology College of Sciences

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 manifold 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. This is a joint work with Zhiwu Lin and Chongchun Zeng.