*(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($\xb7,t)-v(\xb7,t)\to 0$ in C($\overline{{\Omega}}$).

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.

##### MSC:

35Q30 | Stokes and Navier-Stokes equations |

35B40 | Asymptotic behavior of solutions of PDE |

35B60 | Continuation of solutions of PDE |

76D05 | Navier-Stokes equations (fluid dynamics) |