2005/05/18 by Y. Charles Li
Mathematics · Physics and Astronomy · #math.AP #math.DS #physics.flu-dyn #msc:35 #msc:76 #msc:37 #msc:34
published as Dynamics of PDE, Vol.2, No.2, (2005), 159-196 · 28pp
arxiv created 2005/05/18 · arxiv updated 2009/12/01
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\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(√ν), 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.