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Remarques à propos du comportement, lorsque \(t\to +\infty\), des solutions des équations de Navier-Stokes associées à une force nulle. (French) Zbl 0554.35098

The paper improves and extends a result of C. Foias and J. C. Saut on the asymptotics for \(t\to \infty\) of the solutions of Navier-Stokes equations corresponding to a zero exterior force. Two cases are considered: I. the periodic case with respect to the space \({\mathbb{R}}^ N\) equations, of period \(Q=[0,L)^ N\) and II. the equations on a bounded domain \(\Omega\) of \({\mathbb{R}}^ N\) \((N=2,3).\)
In case I the Stokes operator A is selfadjoint in the phase space H of the divergence free, periodic vector fields in \(L^ 2_{loc}({\mathbb{R}}^ N)\) of zero mean value, on the domain \(D(A)=H\cap H^ 2_{loc}({\mathbb{R}}^ N)\). In this framework the following result is proved: for each weak solution u on [0,\(\infty)\), non-zero and very regular on an interval \((t_ 0,\infty)\), there exists an eigenvalue \(\Lambda\) of the operator A and a non-zero eigenvector \(W_{\Lambda}\) corresponding to it such that \(\lim_{t\to \infty}e^{\nu \Lambda t}u(t)=w_{\Lambda}\) in \(D(A^{m/2})\) for all \(m\in {\mathbb{N}}.\)
In case II the phase space H consists of the divergence free tangent to the boundary vector fields in \(L^ 2(\Omega)\) and the Stokes operator A is selfadjoint in H on \(D(A)=H\cap H^ 1_ 0(\Omega)\cap H^ 2(\Omega)\). One shows that for each non-zero, very regular solution on an interval \((t_ 0,\infty)\) there exists an eigenvalue \(\Lambda\) of the operator A and a non-zero eigenvector \(W_{\Lambda}\) such that the following asymptotics hold \(\lim_{t\to \infty}e^{\nu \lambda t}u^{(j)}(t)=(-\nu \Lambda)^ jw_{\Lambda}\) in \(H^ m(\Omega)\cap H\) for all the time derivatives \(u^{(j)}\) and all \(m\in {\mathbb{N}}\).
Reviewer: G.Minea

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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References:

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