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      Unstable manifolds of Euler equations

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          Abstract

          We consider a steady state \(v_{0}\) of the Euler equation in a fixed bounded domain in \(\mathbf{R}^{n}\). 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 \(W^{k, q}\), \((k>1+\frac{n}{q})\), the unstable (and stable) manifolds of \(v_{0}\) are constructed under certain spectral gap condition which is verified for both 2D and 3D 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.

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          Geometry and a priori estimates for free boundary problems of the Euler's equation

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            Local stability conditions in fluid dynamics

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              Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations

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                Author and article information

                Journal
                19 December 2011
                Article
                1112.4525
                4623d954-f5bc-4cdd-aefd-de40824acdcd

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                28 pages
                math.AP

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