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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.

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