Consider a spectrally unstable steady state $(\rho_0(x), v_0(x))$ of the incompressible stratified Euler equation in certain $d$-dim domain $\Omega$. Assuming the linearized equation satisfies a linear exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of $v_0(x)$, we construct a local unstable manifold of $(\rho_0, v_0)$. The proof is based on the Lyapunov-Perron integral equation method after the Euler equation is reformulated as an ODE on the infinite dimensional manifold of volume-preserving Lagrangian maps where the density is treated as a parameter. Applications to steady states in two space dimensions are also discussed.
Co-authors: Zhiwu Lin and Yanbo Wang
By Stephen Griffies
By Catherine Sulem
By Richard Bamler
By Richard Bamler
By Richard Bamler
By Richard Bamler
By Kenji Nakanishi
