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On the Oseen boundary-value problem in exterior domains. (English) Zbl 0783.35047
The Navier-Stokes equations II - theory and numerical methods, Proc. Conf., Oberwolfach/Ger. 1991, Lect. Notes Math. 1530, 111-131 (1992).
[For the entire collection see Zbl 0759.00010.]
Consider the Oseen equations $\Delta v-\text{Re} {\partial v \over \partial x_ 1}=\nabla p+f \text{ in } \Omega,\;\nabla\cdot v=0 \text{ in }\Omega,\;v=v_* \text{ at }\partial \Omega,\;\lim_{| x | \to \infty}v(x)=v_ \infty,$ where $$\Omega \subset \mathbb{R}^ n$$ is the domain exterior to the obstacle $$O$$, Re is the Reynolds number, $$f$$ is the forcing term, $$v_*$$ is the velocity field prescribed on the boundary $$\partial \Omega$$, and $$v_ \infty$$ is the constant vector prescribed at $$\infty$$. For any $$n \geq 2$$, the author obtains various $$L^ q$$ estimates for the solutions in terms of the data, and the author shows the existence and the uniqueness of the weak and strong solutions in the space $$L^ q$$, where the $$q$$ depending on $$n$$. For the special case $$n=2$$, the author observes that the components of the velocity $$v=(v_ 1,v_ 2)$$ belong to different $$L^ q$$ spaces. Applications of the results to the Navier-Stokes equations are given elsewhere by the author.

##### MSC:
 35Q30 Navier-Stokes equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35D05 Existence of generalized solutions of PDE (MSC2000)