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Unstable Manifolds of Euler Equations
Author(s) -
Lin Zhiwu,
Zeng Chongchun
Publication year - 2013
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21457
Subject(s) - mathematics , euler equations , exponential dichotomy , bounded function , euler's formula , mathematical analysis , linear subspace , nonlinear system , ode , invariant manifold , manifold (fluid mechanics) , instability , domain (mathematical analysis) , pure mathematics , differential equation , physics , mechanical engineering , quantum mechanics , mechanics , engineering
We consider a steady state v 0 of the Euler equation in a fixed bounded domain in ℝ n . 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 W k , q( k > 1 + n / q )the unstable (and stable) manifolds of v 0 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 v 0 in the sense that arbitrarily small W k , q perturbations can lead to L 2 growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.