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The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted spaces. (English) Zbl 0727.35106
Consider the stationary Navier-Stokes equations of the viscous incompressible fluid in an exterior domain of \({\mathbb{R}}^ 3\). The velocity is assumed to approximate a constant vector \(u_{\infty}\neq 0\) at infinity. Linearizing the Navier-Stokes equations we are led to the Oseen equations which we study in weighted Sobolev spaces. The weights are anisotropic and reflect the decay properties of the fundamental solution. The main tool is the method of hydrodynamical potentials demanding estimates of singular and weakly singular integral operators in anisotropically weighted spaces. Further we refer to a previous paper of the author about an elliptic model problem related to the Oseen equations. Proving imbedding theorems in anisotropically weighted Sobolev spaces we construct a solution of the Navier-Stokes equations for sufficiently small data by Banach’s fixed point theorem.

MSC:
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
47H10 Fixed-point theorems
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