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

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 ℓ (ℓ = 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 → 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( √ ν), while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are O(ν), as ν → 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.Contents 1. Introduction 159 2. Formulation of the Problems 161 3. Invariant Manifolds 162 4. Examples