Mathematics – Analysis of PDEs
Scientific paper
2005-05-18
Dynamics of PDE, Vol.2, No.2, (2005), 159-196
Mathematics
Analysis of PDEs
28pp
Scientific paper
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 ($\nu \ra 0^+$) 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 $O(\sqrt{\nu})$, while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are $O(\nu)$, as $\nu \ra 0^+$. 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.
No associations
LandOfFree
Invariant Manifolds and Their Zero-Viscosity Limits for Navier-Stokes Equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Invariant Manifolds and Their Zero-Viscosity Limits for Navier-Stokes Equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Invariant Manifolds and Their Zero-Viscosity Limits for Navier-Stokes Equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-710302