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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A simple proof of $L^{q}$-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions
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by Giovanni P. Galdi and Mads Kyed PDF
Proc. Amer. Math. Soc. 141 (2013), 573-583 Request permission

Abstract:

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 in Tohoku Math. J., Vol. 58, 2006, for the Oseen case, and R. Farwig, T. Hishida, and D. Müller in Pacific J. Math., Vol. 215 (2), 2004, 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 we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space $D^{-1,q}_0(\mathbb {R}^3)$.
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Additional Information
  • Giovanni P. Galdi
  • Affiliation: Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
  • MR Author ID: 70660
  • Email: galdi@pitt.edu
  • Mads Kyed
  • Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany
  • MR Author ID: 832297
  • Email: kyed@mathematik.tu-darmstadt.de
  • Received by editor(s): July 1, 2011
  • Published electronically: June 18, 2012
  • Additional Notes: The first author was partially supported by NSF grant DMS-1062381
    The second author was supported by the DFG and JSPS as a member of the International Research Training Group Darmstadt-Tokyo IRTG 1529.
  • Communicated by: Walter Craig
  • © Copyright 2012 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 573-583
  • MSC (2010): Primary 35Q30, 35B45, 76D07
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11638-7
  • MathSciNet review: 2996962