分层欧拉方程的不稳定流形
Unstable Manifolds of Stratified Euler Equations
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中文总结 AI 辅助
本文针对d维区域上不可压缩分层欧拉方程的谱不稳定定态，利用李雅普诺夫-佩尔顿方法构造其局部稳定与不稳定流形，并探讨了相关应用。
中文摘要 AI 辅助
我们考虑一类d维区域上不可压缩分层欧拉方程的谱不稳定定态(ρ₀,v₀)。假设线性化方程关于背景定态流v₀的最大李雅普诺夫指数具有足够大的谱隙，我们构造(ρ₀,v₀)的局部稳定流形与不稳定流形。该证明基于李雅普诺夫-佩尔顿方法，将欧拉方程重新表述为保体积拉格朗日映射的无限维流形上的常微分方程，其中密度被视为冻结的拉格朗日参数及L²度量中的权重。我们还讨论了其在二维定态流中的若干应用。
英文摘要
We consider a spectrally unstable steady state $(ρ_0,v_0)$ 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 $(ρ_0,v_0)$. 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.
