Unstable Manifolds of Euler Equations
Unstable Manifolds of Euler Equations
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欧拉方程的不稳定流形
DOI：
10.1002/cpa.21457
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发表时间：
2011
影响因子：
3
通讯作者：
C. Zeng
中科院分区：
文献类型：
--
作者：
Zhiwu Lin;C. Zeng
我们考虑一个稳定状态的欧拉方程的v0在一个固定的有界的区域。假设线性化欧拉方程具有不稳定子空间和中心稳定子空间的指数二分类。通过将欧拉方程改写为Wk， q （k>1+n/q）中保持体积映射的无限维流形上的ODE，在一定的谱隙条件下构造了v0的不稳定（和稳定）流形，该条件对二维和三维例子都是满足的。特别是，当不稳定子空间是有限维时，这意味着v0的非线性不稳定性，即任意小的Wk， q扰动可以导致非线性解的L2增长。©2013 Wiley期刊公司
We consider a steady state v0 of the Euler equation in a fixed bounded domain in ℝn. 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 Wk, q (k>1+n/q) the unstable (and stable) manifolds of v0 are constructed under a certain spectral gap condition that is satisfied for both two‐dimensional and three‐dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v0 in the sense that arbitrarily small Wk, q perturbations can lead to L2 growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.