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On the smoothness of the nonlinear spectral manifolds associated to the Navier-Stokes equations. (English) Zbl 0572.35081

The paper deals with the Navier-Stokes equations \[ \partial u/\partial t+(u.\nabla)u-\nu \Delta u+\nabla p=0,\quad \nabla \cdot u=0;\quad u(x,0)=u_ 0(x) \] in \(\Omega\) \(\times (0,\infty)\), \(\Omega \subset {\mathbb{R}}^ n\), \(n=2,3\), where \(\Omega\) is smooth and bounded, or \(\Omega =(0,{\mathcal L})^ n\), \({\mathcal L}>0\). A boundary condition \(u|_{\partial \Omega}=0\), or a periodic condition \(u(x+{\mathcal L}e_ j)=u(x)\), \(x\in {\mathbb{R}}^ n\), \(1\leq j\leq n\) is added. The authors proceed in their study of the smoothness of sets \({\mathcal M}_ k\), \(k=1,2,..\). such that a solution u(t) decays exacty as \(e^{-\nu \Lambda (u_ 0)t}\), where \(\Lambda (u_ 0)\) is an eigenvalue of the Stokes operator whereby \(\Lambda (u_ 0)=\Lambda_ k\)- the \(k^{th}\) distinct eigenvalue of Stokes operator, if and only if \(u_ 0\in {\mathcal M}_{k- 1}\setminus {\mathcal M}_ k\) with \({\mathcal M}_ 0={\mathfrak R}\)- the space of initial data \(u_ 0\) leading to regular solutions. They prove that \({\mathcal M}_ k\) is a smooth analytic manifold, i.e. \({\mathcal M}_ k\) has no singularities.
Reviewer: I.Bock

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
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