Abstract
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.
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References
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
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
Farwig R. (2005). An L p-analysis of viscous fluid flow past a rotating obstacle. Tohoku Math. J. 58: 129–147
Farwig R. (2005). Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Center Publ 70: 73–84
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
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)
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
Galdi G.P. (2003). Steady flow of a Navier–Stokes fluid around a rotating obstacle. J. Elasticity 71: 1–32
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
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
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
Giga Y. (1981). Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math. Z. 178: 297–329
Giga Y. and Sohr H. (1989). On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sec. IA 36: 103–130
Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Moscow 1963 (Russian). English version: Israel Progr. Sci. Transl., (1965)
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
Hishida T. (1999). The Stokes operator with rotating effect in exterior domains. Analysis 19: 51–67
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)
Kato T. (1959). Growth properties of solutions of the reduced wave equation with a variable coefficient. Comm. Pure Appl. Math. XII: 403–425
Kato T. (1966). Perturbation Theory for Linear Operators. Springer, Berlin
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
Ladyzhenskaya O.A. (1969). The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York
Leis, R.: Initial Boundary Value Problems in Mathematical Physics. B.G. Teubner, Chichester/New York (1986)
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
Nečasová Š. (2004). Asymptotic properties of the steady fall of a body in viscous fluids. Math. Meth. Appl. Sci. 27: 1969–1995
Schwartz, J.T.: Linear Operators I, II. Interscience Publishers, New York/London, (1958, 1963)
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Farwig, R., Neustupa, J. On the spectrum of a Stokes-type operator arising from flow around a rotating body. manuscripta math. 122, 419–437 (2007). https://doi.org/10.1007/s00229-007-0078-2
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DOI: https://doi.org/10.1007/s00229-007-0078-2