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On the Stokes equations: The boundary value problem. (English) Zbl 0949.35111
Maremonti, Paolo (ed.), Advances in fluid dynamics. Rome: Aracne. Quad. Mat. 4, 69-140 (1999).
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 $\Omega$ (§ 5). Similar results are obtained in § 6, § 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 $p_1= p_2+\text{const}$. For the entire collection see [Zbl 0934.00018].

35Q30Stokes and Navier-Stokes equations
76D03Existence, uniqueness, and regularity theory
76D07Stokes and related (Oseen, etc.) flows