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.

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