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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 ${\bbfR}\sp n$, $n=2,3$ 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 $L\sp 2$, as $t\to +\infty$. Then if the corresponding strong solutions u and v are such that $u(x\sb j,t)-v(x\sb j,t)\to 0$ in $L\sp 2$ as $t\to \infty$, for every $x\sb j$ 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

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