# zbMATH — the first resource for mathematics

A simple proof of $$L^q$$-estimates for the steady-state Oseen and Stokes equations in a rotating frame. I: strong solutions. (English) Zbl 1261.35106
Summary: Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity $$\xi \in \mathbb{R}^3$$ and a non-zero angular velocity $$\omega \in \mathbb R^3\setminus \{0\}$$ that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen ($$\xi\neq 0$$) or Stokes ($$\xi =0$$) equations in a rotating frame of reference. We will consider the corresponding steady-state whole-space problem. Our main result in this first part concerns elliptic estimates of the solutions in terms of data in $$L^q(\mathbb R^3)$$. Such estimates have been established by R. Farwig [Tohoku Math. J. (2) 58, No. 1, 129–147 (2006; Zbl 1136.76340)], for the Oseen case, and R. Farwig et al. [Pac. J. Math. 215, No. 2, 297–312 (2004; Zbl 1057.35028)], for the Stokes case. We introduce a new approach resulting in an elementary proof of these estimates. Moreover, our method yields more details on how the constants in the estimates depend on $$\xi$$ and $$\omega$$. In Part II [Zbl 1260.35111] we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space $$D^{-1,q}_0(\mathbb R^3)$$.

##### MSC:
 35Q30 Navier-Stokes equations 35B45 A priori estimates in context of PDEs 76D07 Stokes and related (Oseen, etc.) flows
Full Text:
##### References:
 [1] Paul L. Butzer and Rolf J. Nessel, Fourier analysis and approximation, Academic Press, New York-London, 1971. Volume 1: One-dimensional theory; Pure and Applied Mathematics, Vol. 40. · Zbl 0217.42603 [2] Reinhard Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Regularity and other aspects of the Navier-Stokes equations, Banach Center Publ., vol. 70, Polish Acad. Sci. Inst. Math., Warsaw, 2005, pp. 73 – 84. · Zbl 1101.35348 · doi:10.4064/bc70-0-5 · doi.org [3] Reinhard Farwig, An \?^\?-analysis of viscous fluid flow past a rotating obstacle, Tohoku Math. J. (2) 58 (2006), no. 1, 129 – 147. · Zbl 1136.76340 [4] Reinhard Farwig, Toshiaki Hishida, and Detlef Müller, \?^\?-theory of a singular ”winding” integral operator arising from fluid dynamics, Pacific J. Math. 215 (2004), no. 2, 297 – 312. · Zbl 1057.35028 · doi:10.2140/pjm.2004.215.297 · doi.org [5] Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II, Springer Tracts in Natural Philosophy, vol. 39, Springer-Verlag, New York, 1994. Nonlinear steady problems. · Zbl 0949.35004 [6] Giovanni 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, North-Holland, Amsterdam, 2002, pp. 653 – 791. · Zbl 1230.76016 [7] Giovanni P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity 71 (2003), no. 1-3, 1 – 31. Essays and papers dedicated to the memory of Clifford Ambrose Truesdell III, Vol. II. · Zbl 1156.76367 · doi:10.1023/B:ELAS.0000005543.00407.5e · doi.org [8] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. · Zbl 1245.35002 [9] Giovanni P. Galdi and Mads Kyed, Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 21 – 58. · Zbl 1229.35176 · doi:10.1007/s00205-010-0350-6 · doi.org [10] Giovanni P. Galdi and Ana L. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Ration. Mech. Anal. 184 (2007), no. 3, 371 – 400. · Zbl 1111.76010 · doi:10.1007/s00205-006-0026-4 · doi.org [11] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. · Zbl 0184.52603 [12] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115 – 162. · Zbl 0088.07601
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.