# zbMATH — the first resource for mathematics

The Stokes operator with rotation effect in exterior domains. (English) Zbl 0938.35114
The paper is a step to the mathematical analysis of the Navier-Stokes flow past a rotating obstacle. Let $$K\subset \mathbb{R}^3$$ be a compact, isolated and rigid obstacle with smooth boundary $$\Gamma$$, $$\Omega=\mathbb{R}^3\setminus K$$ the exterior domain occupied by a viscous imcompressible fluid. The obstacle $$K$$ is rotating about the $$x_3-$$axis with constant angular velocity $$\omega=(0, 0, 1)$$.
The author studies the linearized problem in the coordinate system $$\{x\}$$ attached to the rotating obstacle \begin{aligned} &\frac{\partial v}{\partial t}-\Delta v-(\omega\times x)\cdot\nabla v+ \omega\times v+\nabla p=0,\quad x\in\Omega,\;t>0,\\ &\nabla\cdot v=0, \qquad x\in\Omega,\;t\geqslant 0,\\ &v=0 \quad (x\in\Gamma,\;t>0),\qquad v\to 0\;\text{as }|x|\to \infty,\;t>0,\\ &v(x,0)=a(x). \end{aligned} It is proved that the problem generates a $$(C_0)$$ semigroup on the space $$L^2$$. Some $$L^2$$ estimates for the operator $$\mathcal L=-\mathcal P[\Delta v+ (\omega\times x)\cdot\nabla v-\omega\times v]$$ are obtained too. $$\mathcal P$$ is the projection from $$L^2(\Omega)$$ onto $$L^2_\sigma(\Omega)$$ associated with the Helmholtz decomposition.

##### MSC:
 35Q30 Navier-Stokes equations 47D03 Groups and semigroups of linear operators 76D07 Stokes and related (Oseen, etc.) flows
Full Text: