Stationary Navier-Stokes Flow Around a Rotating Obstacle

Consider a viscous incompressible fluid filling the whole 3-dimensional space exterior to a rotating body with constant angular velocity ω. By using a coordinate system attached to the body, the problem is reduced to an equivalent one in a fixed exterior domain. The reduced equation involves the crucial drift operator (ω ∧ x) · ∇, which is not subordinate to the usual Stokes operator. This paper addresses stationary flows to the reduced problem with an external force f = div F, that is, time-periodic flows to the original one. Generalizing previous results of G. P. Galdi [19] we show the existence of a unique solution (∇u, p) in the class L3/2,∞ when both F ∈ L3/2,∞ and ω are small enough; here L3/2,∞ is the weak-L3/2 space.

[1]  Y. Shibata,et al.  Lp-Lq Estimate of the Stokes Operator and Navier–Stokes Flows in the Exterior of a Rotating Obstacle , 2009 .

[2]  R. Farwig,et al.  A weighted Lq-approach to Stokes flow around a rotating body , 2008 .

[3]  G. Galdi,et al.  The Steady Motion of a Navier–Stokes Liquid Around a Rigid Body , 2007 .

[4]  Matthias Geissert,et al.  L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle , 2006 .

[5]  Reinhard Farwig,et al.  An $L^q$-analysis of viscous fluid flow past a rotating obstacle , 2006 .

[6]  Marius Tucsnak,et al.  Wellposedness for the Navier–Stokes flow in the exterior of a rotating obstacle , 2006 .

[7]  R. Farwig,et al.  On the spectrum of a Stokes-type operator arising from flow around a rotating body , 2007 .

[8]  Toshiaki Hishida,et al.  $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body , 2006 .

[9]  Y. Shibata,et al.  Uniform estimates in the velocity at infinity for stationary solutions to the Navier-Stokes exterior problem , 2005 .

[10]  Giovanni P. Galdi,et al.  Strong Solutions to the Navier-Stokes Equations Around a Rotating Obstacle , 2005 .

[11]  R. Farwig.  Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle , 2005 .

[12]  Š. Nečasová.  Asymptotic properties of the steady fall of a body in viscous fluids , 2004 .

[13]  Ana L. Silvestre,et al.  On the existence of steady flows of a Navier–Stokes liquid around a moving rigid body , 2004 .

[14]  R. Farwig,et al.  Lq-theory of a singular "winding'' integral operator arising from fluid dynamics , 2004 .

[15]  Giovanni P. Galdi,et al.  Steady Flow of a Navier-Stokes Fluid Around a Rotating Obstacle , 2003 .

[16]  Giovanni P. Galdi,et al.  Chapter 7 – On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications , 2002 .

[17]  T. Hishida.  On the Navier–Stokes flow around a rigid body with a prescribed rotation , 2001 .

[18]  T. Hishida.  $L^{2}$ Theory for the Operator $\Delta+(k\times x)\cdot\nabla$ in Exterior Domains , 2000 .

[19]  Toshiaki Hishida,et al.  An Existence Theorem¶for the Navier-Stokes Flow¶in the Exterior of a Rotating Obstacle , 1999 .

[20]  T. Hishida.  THE STOKES OPERATOR WITH ROTATION EFFECT IN EXTERIOR DOMAINS , 1999 .

[21]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[22]  H. Kozono,et al.  Exterior problem for the stationary Navier-Stokes equations in the Lorentz space , 1998 .

[23]  R. Farwig.  The stationary Navier-Stokes equations in a 3D-exterior domain , 1997 .

[24]  Zhimin Chen,et al.  Decay properties of weak solutions to a perturbed Navier-Stokes system in Rn , 1997 .

[25]  W. Borchers,et al.  On stability of exterior stationary Navier-Stokes flows , 1994 .

[26]  H. Kozono,et al.  On a new class of generalized solutions for the Stokes equations in exterior domains , 1992 .

[27]  H. Kozono,et al.  New a priori estimates for the Stokes equations in exterior domains , 1991 .

[28]  W. Borchers,et al.  AlgebraicL2 decay for Navier-Stokes flows in exterior domains , 1990 .

[29]  G. Galdi,et al.  Existence, uniqueness and Lq-estimates for the stokes problem in an exterior domain , 1990 .

[30]  W. Borchers,et al.  On the equations rot v=g and div u=f with zero boundary conditions , 1990 .

[31]  D. Serre,et al.  Chute libre d’un solide dans un fluide visqueux incompressible. existence , 1987 .

[32]  M. E. Bogovskii.  Solution of the first boundary value problem for the equation of continuity of an incompressible medium , 1979 .

[33]  V. A. Solonnikov,et al.  Estimates for solutions of nonstationary Navier-Stokes equations , 1977 .

[34]  Lamberto Cattabriga,et al.  Su un problema al contorno relativo al sistema di equazioni di Stokes , 1961 .