Journal of the Mathematical Society of Japan

Spectral analysis of a Stokes-type operator arising from flow around a rotating body

Reinhard FARWIG, Šárka NEČASOVÁ, and Jiří NEUSTUPA

Full-text: Open access

Abstract

We consider the spectrum of the Stokes operator with and without rotation effect for the whole space and exterior domains in $L^q$-spaces. Based on similar results for the Dirichlet-Laplacian on $mathbf{R}^n$, $n \geq 2$, we prove in the whole space case that the spectrum as a set in $\mathbf{c}$ does not change with $q \in (1,\infty)$, but it changes its type from the residual to the continuous or to the point spectrum with $q$. The results for exterior domains are less complete, but the spectrum of the Stokes operator with rotation mainly is an essential one, consisting of infinitely many equidistant parallel half-lines in the left complex half-plane. The tools are strongly based on Fourier analysis in the whole space case and on stability properties of the essential spectrum for exterior domains.

Article information

Source
J. Math. Soc. Japan, Volume 63, Number 1 (2011), 163-194.

Dates
First available in Project Euclid: 27 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1296138348

Digital Object Identifier
doi:10.2969/jmsj/06310163

Mathematical Reviews number (MathSciNet)
MR2752436

Zentralblatt MATH identifier
1223.35257

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35P99: None of the above, but in this section 47A10: Spectrum, resolvent 76D07: Stokes and related (Oseen, etc.) flows

Keywords
Stokes operator Stokes operator with rotation spectrum essential spectrum point spectrum L^q-theory

Citation

FARWIG, Reinhard; NEČASOVÁ, Šárka; NEUSTUPA, Jiří. Spectral analysis of a Stokes-type operator arising from flow around a rotating body. J. Math. Soc. Japan 63 (2011), no. 1, 163--194. doi:10.2969/jmsj/06310163. https://projecteuclid.org/euclid.jmsj/1296138348


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References

  • J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, New York, 1976.
  • W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitation Thesis, Univ. of Paderborn, 1992.
  • P. Deuring, S. Kračmar and Š. Nečasová, A representation formula for linearized stationary incompressible viscous flows around rotating and translating bodies, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 237–253.
  • P. Deuring, S. Kračmar and Š. Nečasová, On pointwise deacy of linearized stationary incompressible viscous flow around rotating and translating bodies, Université du Littoral, preprint no.,405, 2009.
  • E. Dintelmann, M. Geissert and M. Hieber M, Strong $L^p$-solutions to the Navier-Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity, Trans. Amer. Math. Soc., 361 (2009), 653–669.
  • R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409–447.
  • R. Farwig, An $L^q$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J., 58 (2005), 129–147.
  • R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Banach Center Publications, 70, Warszawa, 2005, pp.,73–84.
  • R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacle, Funkcial. Ekvac., 50 (2007), 371–403.
  • R. Farwig, T. Hishida and D. Müller, $L^q$–theory of a singular “winding” integral operator arising from fluid dynamics, Pacific J. Math., 215 (2004), 297–312.
  • R. Farwig, M. Krbec and Š. Nečasová, A weighted $L^q$-approach to Stokes flow around a rotating body, Ann. Univ. Ferrara Sez. VII Sci. Mat., 54 (2008), 61–84.
  • R. Farwig and J. Neustupa, On the spectrum of a Stokes-type operator arising from flow around a rotating body, Manuscripta Math., 122 (2007), 419–437.
  • R. Farwig, Š. Nečasová and J. Neustupa, On the essential spectrum of a Stokes-type operator arising from flow around a rotating body in the $L^q$–Framework, RIMS Kyoto University, RIMS Kokyuroku Bessatsu, B1 (2007), 93–105.
  • R. Farwig and J. Neustupa, Spectral properties in $L^q$ of an Oseen operator modelling fluid flow past a rotating body, Tohoku Math. J., 62 (2010), 287–309.
  • G. P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, 1, (Eds. S. Friedlander and D. Serre), Elsevier, 2002.
  • G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity, 71 (2003), 1–31.
  • G. P. Galdi and A. L. Silvestre, Strong solutions to the Navier-Stokes equations around a rotating obstacle, Arch. Rational Mech. Anal., 176 (2005), 331–350.
  • G. P. Galdi and A. L. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rational Mech. Anal., 184 (2007), 371–400.
  • M. Geissert, H. Heck and M. Hieber, $L^p$–theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45–62.
  • L. Grafakos, Classical Fourier Analysis, Springer, New York, 2008.
  • R. B. Guenther, R. T. Hudspeth and E. A. Thomann, Hydrodynamic flows on submerged rigid bodies – steady flow, J. Math. Fluid Mech., 4 (2002), 187–202.
  • T. Hishida, $L^q$ estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58 (2006), 743–767.
  • T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 150 (1999), 307–348.
  • T. Hishida and Y. Shibata, $L_p-L_q$ estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle, Arch. Rational Mech. Anal., 193 (2009), 339–421.
  • T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966.
  • S. Kračmar, Š. Nečasová and P. Penel, Estimates of weak solutions in anisotropically weighted Sobolev spaces to the stationary rotating Oseen equations, IASME Transactions, 2 (2005), 854–861.
  • S. Kračmar, Š. Nečasová and P. Penel, $L^q$ approach of weak solutions of Oseen flow around a rotating body, Banach Center Publications, 81, Warszawa, 2008, pp.,259–276.
  • S. Kračmar, Š. Nečasová and P. Penel, $L^q$ approach to weak solutions of Oseen flow around a rotating body in exterior domains, Quarterly J. Math., Oxford, to appear.
  • J. A. San Martín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Rational Mech. Anal., 161 (2002), 113–147.
  • Š. Nečasová, On the problem of the Stokes flow and Oseen flow in $\R^3$ with Coriolis force arising from fluid dynamics, IASME Transactions, 2 (2005), 1262–1270.
  • Š. Nečasová, Asymptotic properties of the steady fall of a body in viscous fluids, Math. Meth. Appl. Sci., 27 (2004), 1969–1995.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, N.J., 1993.
  • G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge University Press, Cambridge, 1980.