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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 Ω of 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)0 in L 2 , as t+. Then if the corresponding strong solutions u and v are such that u(x j ,t)-v(x j ,t)0 in L 2 as t, for every x j of a finite set, sufficiently dense, then u(·,t)-v(·,t)0 in C(Ω ¯).

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:
35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE
35B60Continuation of solutions of PDE
76D05Navier-Stokes equations (fluid dynamics)