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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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