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 \(H^{\frac 12}\).


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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