Invariant Manifolds and Their Zero-Viscosity Limits for Navier-Stokes Equations

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 the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in $H^\ell$ ($\ell = 0,1,2, ...$). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zero-viscosity limits ($ν\ra 0^+$) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the center-stable manifold of a steady state are $O(\sqrtν)$, while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are $O(ν)$, as $ν\ra 0^+$. Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete estimate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other two examples, partial estimates on the eigenvalues are obtained.