Tohoku Mathematical Journal

An $L^{\lowercase{q}}$-analysis of viscous fluid flow past a rotating obstacle

Reinhard Farwig

Full-text: Open access

Abstract

Consider the problem of time-periodic strong solutions of the Stokes and Navier-Stokes system modelling viscous incompressible fluid flow past or around a rotating obstacle in Euclidean three-space. Introducing a rotating coordinate system attached to the body, a linearization yields a system of partial differential equations of second order involving an angular derivative not subordinate to the Laplacian. In this paper we find an explicit solution for the linear whole space problem when the axis of rotation is parallel to the velocity of the fluid at infinity. For the analysis of this solution in $L^q$-spaces, $1<q<\ue$, we will use tools from harmonic analysis and a special maximal operator reflecting paths of fluid particles past or around the obstacle.

Article information

Source
Tohoku Math. J. (2), Volume 58, Number 1 (2006), 129-147.

Dates
First available in Project Euclid: 18 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1145390210

Digital Object Identifier
doi:10.2748/tmj/1145390210

Mathematical Reviews number (MathSciNet)
MR2221796

Zentralblatt MATH identifier
1136.76340

Subjects
Primary: 76D05: Navier-Stokes equations [See also 35Q30]
Secondary: 35C15: Integral representations of solutions 35Q35: PDEs in connection with fluid mechanics 76D99: None of the above, but in this section 76U05: Rotating fluids

Keywords
Littlewood-Paley theory maximal operators Oseen flow rotating obstacles singular integral operator Stokes flow

Citation

Farwig, Reinhard. An $L^{\lowercase{q}}$-analysis of viscous fluid flow past a rotating obstacle. Tohoku Math. J. (2) 58 (2006), no. 1, 129--147. doi:10.2748/tmj/1145390210. https://projecteuclid.org/euclid.tmj/1145390210


Export citation

References

  • W. Borchers, Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitation Thesis, Univ. of Paderborn, 1992.
  • Z. M. Chen and T. Miyakawa, Decay properties of weak solutions to a perturbed Navier-Stokes system in $\R^n$, Adv. Math. Sci. Appl. 7 (1997), 741--770.
  • 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, The stationary Navier-Stokes equations in a $3D$-exterior domain, Recent topics on mathematical theory of viscous incompressible fluid (Tsukuba, 1996), 53--115, Lecture Notes in Numer. Appl. Anal. 16, Kinokuniya, Tokyo, 1998.
  • R. Farwig and H. Sohr, Weighted estimates for the Oseen equations and the Navier-Stokes equations in exterior domains, Theory of Navier-Stokes equations, 11--30, Ser. Adv. Math. Appl. Sci. 47, World Sci. Publishing, River Edge, N. J., 1998.
  • G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I, Linearized steady problems, Springer Tracts Nat. Philos. 38, Springer-Verlag New York, 1994.
  • G. P. Galdi, On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications, Handbook of mathematical fluid dynamics, Vol. I, 653--791, North-Holland, Amsterdam, 2002.
  • G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity 71 (2003), 1--31.
  • 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, The Stokes operator with rotation effect in exterior domains, Analysis (Munich) 19 (1999), 51--67.
  • Š. Nečasova, Some remarks on the steady fall of a body in Stokes and Oseen flow, Acad. Sciences Czech Republic, Math. Institute, Preprint 143 (2001).
  • Š. Nečasova, Asymptotic properties of the steady fall of a body in viscous fluids, Math. Methods Appl. Sci. 27 (2004), 1969--1995.
  • O. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the whole space, Darmstadt University of Technology, Department of Mathematics, Preprint no. 2288 (2003).
  • H. Sohr, The Navier-Stokes Equations. An elementary functional analytic approach, Birkhäuser Adv. Texts, Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, N. J., 1970.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser. 43, Monographs in Harmonic Analysis III, Princeton Univ. Press, Princeton, N. J., 1993.
  • E. A. Thomann and R. B. Guenther, The fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions---time dependent case, J. Math. Fluid Mech., to appear.