THE Navier-stokes flow around a rotating obstacle with time-dependent body force
Volume 86 / 2009
Abstract
We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity $\omega$, under the action of external force $f$. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain $D$. Given $f=\mathop{\rm div} F$ with $F\in BUC(\mathbb R; L_{3/2,\infty}(D))$, we consider this problem in $D\times \mathbb R$ and prove that there exists a unique solution $u\in BUC(\mathbb R; L_{3,\infty}(D))$ when $F$ and $|\omega|$ are sufficiently small. If, in particular, the external force for the original problem is independent of $t$, then $f$ is periodic with period $2\pi/|\omega|$. In this situation, as a corollary of our result, we obtain a periodic solution with the same period. Stability of our solution is also discussed.
