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# Dynamics of Partial Differential Equations

## Volume 2 (2005)

### Number 2

### Invariant manifolds and their zero-viscosity limits for Navier-Stokes equations

Pages: 159 – 186

DOI: http://dx.doi.org/10.4310/DPDE.2005.v2.n2.a4

#### Author

#### Abstract

First we prove a general spectral theorem for the linear Navier-Stokes(NS) operator in both 2D and 3D. The spectral theorem says that thespectrum consists of only eigenvalues which lie in a parabolic region,and the eigenfunctions (and higher order eigenfunctions) form a completebasis in $H^\ell$ ($\ell = 0,1,2, \cdots$). Then we prove the existenceof invariant manifolds. We are also interested in a more challengingproblem, i.e. studying the zero-viscosity limits ($\nu \ra 0^+$) of theinvariant manifolds. Under an assumption, we can show that the sizes ofthe unstable manifold and the center-stable manifold of a steady stateare $O(\sqrt{\nu})$, while the sizes of the stable manifold, the centermanifold, and the center-unstable manifold are $O(\nu)$, as $\nu \ra 0^+$.Finally, we study three examples. The first example is defined on arectangular periodic domain, and has only one unstable eigenvalue whichis real. A complete estimate on this eigenvalue is obtained. Existenceof an 1D unstable manifold and a codim 1 stable manifold is proved withoutany assumption. For the other two examples, partial estimates on theeigenvalues are obtained.

#### Keywords

invariant manifold, zero-viscosity limit, Navier-Stokes equation

#### 2010 Mathematics Subject Classification

Primary 35-xx, 37-xx, 76-xx. Secondary 34-xx.