# zbMATH — the first resource for mathematics

On the spectrum of a Stokes-type operator arising from flow around a rotating body. (English) Zbl 1126.35050
Summary: We present the description of the spectrum of a linear perturbed Stokes-type operator which arises from equations of motion of a viscous incompressible fluid in the exterior of a rotating compact body. Considering the operator in the function space $$L^2_\sigma(\Omega)$$ we prove that the essential spectrum consists of a set of equally spaced half lines parallel to the negative real half line in the complex plane. Our approach is based on a reduction to invariant closed subspaces of $$L^2_\sigma(\Omega)$$ and on a Fourier series expansion with respect to an angular variable in a cylindrical coordinate system attached to the axis of rotation. Moreover, we show that the leading part of the operator is normal if and only if the body is axially symmetric about this axis.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35P20 Asymptotic distributions of eigenvalues in context of PDEs 76D07 Stokes and related (Oseen, etc.) flows
Full Text:
##### References:
 [1] Borchers W. and Sohr H. (1990). On the equations rot V = g and div v = f with zero boundary conditions. Hokkaido Math. J. 19: 67–87 · Zbl 0719.35014 [2] Cumsille P. and Tucsnak M. (2006). Well-posedness for the Navier–Stokes flow in the exterior of a rotating obstacle. Math. Methods Appl. Sci. 29: 595–623 · Zbl 1093.76013 · doi:10.1002/mma.702 [3] Farwig R. (2005). An L p -analysis of viscous fluid flow past a rotating obstacle. Tohoku Math. J. 58: 129–147 · Zbl 1136.76340 · doi:10.2748/tmj/1145390210 [4] Farwig R. (2005). Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Center Publ 70: 73–84 · Zbl 1101.35348 · doi:10.4064/bc70-0-5 [5] Farwig R., Hishida T. and Müller D. (2004). L q -theory of a singular ”winding” integral operator arising from fluid dynamics. Pacific J. Math. 215: 297–312 · Zbl 1057.35028 · doi:10.2140/pjm.2004.215.297 [6] Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, vol. I. Linear steady problems. Springer tracts in natural philosophy 38 (1998) [7] Galdi G.P. (2002). On the motion of a rigid body in a viscous liquid. In: Amathematicalanalysiswithapplications. Friedlander, S. and Serre, D. (eds) Handbook of Mathematical Fluid Mechanics., pp 653–791. Elsevier, Amsterdam · Zbl 1230.76016 [8] Galdi G.P. (2003). Steady flow of a Navier–Stokes fluid around a rotating obstacle. J. Elasticity 71: 1–32 · Zbl 1156.76367 · doi:10.1023/B:ELAS.0000005543.00407.5e [9] Galdi G.P. and Padula M. (1990). A new approach to energy theory in the stability of fluid motion. Arch Rational Mech. Anal. 110: 187–286 · Zbl 0719.76035 · doi:10.1007/BF00375129 [10] Galdi G.P. and Silvestre A. (2005). Strong Solutions to the Navier-Stokes equations around a rotating obstacle. Arch. Rational Mech. Anal. 176: 331–350 · Zbl 1081.35076 · doi:10.1007/s00205-004-0348-z [11] Geissert M., Heck H. and Hieber M. (2006). L p -theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596: 45–62 · Zbl 1102.76015 · doi:10.1515/CRELLE.2006.051 [12] Giga Y. (1981). Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z. 178: 297–329 · Zbl 0461.47019 · doi:10.1007/BF01214869 [13] Giga Y. and Sohr H. (1989). On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sec. IA 36: 103–130 [14] Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Moscow 1963 (Russian). English version: Israel Progr. Sci. Transl., (1965) · Zbl 0143.36505 [15] Hishida T. (1999). An existence theorem for the Navier–Stokes flow in the exterior of a rotating obstacle. Arch. Rational Mech. Anal. 150: 307–348 · Zbl 0949.35106 · doi:10.1007/s002050050190 [16] Hishida T. (1999). The Stokes operator with rotating effect in exterior domains. Analysis 19: 51–67 · Zbl 0938.35114 [17] Hishida, T.: L q estimates of weak solutions to the stationary Stokes equations around a rotating body, Hokkaido University. Preprint series in Math., No. 691 (2004) [18] Kato T. (1959). Growth properties of solutions of the reduced wave equation with a variable coefficient. Comm. Pure Appl. Math. XII: 403–425 · Zbl 0091.09502 · doi:10.1002/cpa.3160120302 [19] Kato T. (1966). Perturbation Theory for Linear Operators. Springer, Berlin · Zbl 0148.12601 [20] Kračmar S., Nečasová Š. and Penel P. (2005). Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations. IASME Trans 6(2): 854–861 [21] Ladyzhenskaya O.A. (1969). The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York · Zbl 0184.52603 [22] Leis, R.: Initial Boundary Value Problems in Mathematical Physics. B.G. Teubner, Chichester/New York (1986) · Zbl 0599.35001 [23] Nečasová Š. (2005). On the problem of the Stokes flow in $$\mathbb {R}^3$$ with Coriolis force arising from fluid dynamics IASME Trans 7(2): 1262–1270 [24] Nečasová Š. (2004). Asymptotic properties of the steady fall of a body in viscous fluids. Math. Meth. Appl. Sci. 27: 1969–1995 · Zbl 1174.76306 · doi:10.1002/mma.467 [25] Schwartz, J.T.: Linear Operators I, II. Interscience Publishers, New York/London, (1958, 1963) · Zbl 0084.10402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.