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Determination of the solutions of the Navier-Stokes equations by a set of nodal values. (English) Zbl 0563.35058

The authors give several very interesting results on the determination of the solutions of the Navier-Stokes equations of incompressible viscous fluids by their values on a finite set. For instance, two stationary solutions in a bounded domain \(\Omega\) of \({\mathbb{R}}^ n\), \(n=2,3\) coincide if they coincide on a finite set sufficiently dense.
In the 2-dimensional case, let f,g be two body forces such that f(t)- g(t)\(\to 0\) in \(L^ 2\), as \(t\to +\infty\). Then if the corresponding strong solutions u and v are such that \(u(x_ j,t)-v(x_ j,t)\to 0\) in \(L^ 2\) as \(t\to \infty\), for every \(x_ j\) of a finite set, sufficiently dense, then u(\(\cdot,t)-v(\cdot,t)\to 0\) in C(\({\bar \Omega}\)).
A similar statement holds for time-periodic solutions. The large time behaviour of the solution is therefore determined by its large time behaviour on a suitable discrete set.
Reviewer: J.-C.Saut

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35B60 Continuation and prolongation of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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