## 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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