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Zbl 0949.35111
Maremonti, Paolo; Russo, Remigio; Starita, Giulio
On the Stokes equations: The boundary value problem. (English)
[A] Maremonti, Paolo (ed.), Advances in fluid dynamics. Rome: Aracne. Quad. Mat. 4, 69-140 (1999). ISBN 88-7999-259-6

This paper deals with a boundary value problem (bvp) for the Stokes equation. The corresponding domain $\Omega$ is not obliged to be simply connected and it can be either a bounded or an exterior one. In their investigations the authors use the theory of hydrodynamical potentials. An existence of a unique classical solution is proved in the case of a bounded domain $\Omega$ (\S 5). Similar results are obtained in \S 6, \S 7 assuming $\Omega$ to be an exterior domain. To be more precise, we give the uniqueness result as follows: Let $(u_1,p_1)$, $(u_2,p_2)$ be two classical solutions of the Stokes bvp. Then, if $$u_1- u_2= \cases o(\log r),\quad &n=2,\\ o(1),\quad &n\ge 3,\endcases$$ we have $u_1= u_2$ and $p_1= p_2+\text{const}$.
[P.Popivanov (Sofia)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory
76D07 Stokes flows

Keywords: Stokes equation; hydrodynamical potential; existence of a unique classical solution; bounded domain; exterior domain

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