## An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle.(English)Zbl 0949.35106

Let $$K\subset \mathbb{R}^3$$ be a compact, isolated rigid obstacle which is bounded by a smooth surface $$\Gamma$$. The exterior domain $$\Omega= \mathbb{R}^3\setminus K$$ is occupied by a viscous incompressible fluid. The obstacle $$K$$ is rotating about the $$x_3$$-axis with angular velocity $$\omega$$. Let $$\Omega(t)$$ be the volume occupied by a fluid at the moment $$t$$, $$\Gamma(t)=\partial\Omega(t)$$. The velocity $$v$$ and the pressure $$p$$ of the fluid satisfy the boundary value problem for the Navier-Stokes equations $\frac{\partial v}{\partial t}-\Delta v+v\cdot\nabla v+\nabla p=0, \qquad \text{div }v=0, \quad x\in\Omega(t),$
$v=\omega\times x, \quad x\in\Gamma(t), \qquad v\to 0\quad \text{as } |x|\to\infty, \quad v(x,0)=a(x).$ It is proved that a unique mild solution to the problem exists locally in time if the initial velocity $$a$$ possesses the regularity $$H^{\frac 12}$$.

### MSC:

 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35D05 Existence of generalized solutions of PDE (MSC2000)
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