THE Navier-stokes flow around a rotating obstacle with time-dependent body force

Toshiaki Hishida
2009 Nonlocal and Abstract Parabolic Equations and their Applications   unpublished
We study the motion of a viscous incompressible fluid filling the whole threedimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, 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 = div F with F ∈ BU C(R; L 3/2,∞ (D)), we consider this problem in D×R and prove that there exists a unique solution u ∈ BU C(R; L3,∞(D)) when F and |ω| are sufficiently small.
more » ... If, in particular, the external force for the original problem is independent of t, then f is periodic with period 2π/|ω|. 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. 2000 Mathematics Subject Classification: 35Q30, 76D05. [149] c Instytut Matematyczny PAN, 2009 152 T. HISHIDA interpolation, T a (t) can be extended as the semigroup in various solenoidal Lorentz spaces J q,r (D) = (J q0 (D), J q1 (D)) θ,r where q and r satisfy (2.1). Note that it is not of class (C 0 ) in the space J q,∞ (D) = J q/(q−1),1 (D) *
doi:10.4064/bc86-0-9 fatcat:2uils4gf2zhclph2clr36jfux4