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Stationary Navier-Stokes flow around a rotating obstacle. (English) Zbl 1180.35408
The authors deal with the study of the nonlinear stationary problem in exterior domains: \[ \mathbb{L} u+\nabla p+u\cdot\nabla u=f,\text{ div}\, u=0\text{ in }\mathbb{D}\tag{1} \] subject to \[ u|_{\partial\mathbb{D}}=\omega \wedge x,\;u\to 0 \text{ as }| x|\to\infty,\tag{2} \] where \(\mathbb{L}:= -\Delta-(\omega\wedge x)\cdot\nabla+\omega\wedge\), \(\omega\) is a constant angular velocity, and \(\mathbb{D}\) is an exterior domain in \(\mathbb{R}^3\) with smooth boundary \(\Gamma= \partial\mathbb{D}\). The authors show in the class \(L_{3/2,\infty}\) the existence of a unique solution \((\nabla u,p)\) to (1), (2) with force \(f\in\dot W^{-1}_{3/2, \infty}\) when both \(f\) and \(\omega\) are small enough; here \(L_{3/2,\infty}\) is the weak \(-L_{3/2}\) space, one of the Lorenz spaces. Note that \(f\in\dot W^{-1}_{3/2,\infty}\) if and only if \(f=\text{div}\,F\) with \(F\in L_{3/2,\infty}\).

MSC:
35Q30 Navier-Stokes equations
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