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On the Navier–Stokes Problem in Exterior Domains with Non Decaying Initial Data

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Abstract

We study the existence and uniqueness of regular solutions to the Navier–Stokes initial-boundary value problem with non-decaying bounded initial data, in a smooth exterior domain of \({{\mathbb R}^n, n\ge3}\) . The pressure field, p, associated to these solutions may grow, for large |x|, as O(|x|γ), for some \({\gamma\in (0,1)}\) . Our class of existence is sharp for well posedeness, in that we show that uniqueness fails if p has a linear growth at infinity. We also provide a sufficient condition on the spatial growth of \({\nabla p}\) for the boundedness of v, at all times. Also this latter result is shown to be sharp.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, USA

    Giovanni P. Galdi

  2. Dipartimento di Matematica, Seconda Università degli Studi di Napoli, via Vivaldi 43, 81100, Caserta, Italy

    Paolo Maremonti

  3. Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

    Yong Zhou

Authors
  1. Giovanni P. Galdi
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  2. Paolo Maremonti
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  3. Yong Zhou
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Correspondence to Giovanni P. Galdi.

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Galdi, G.P., Maremonti, P. & Zhou, Y. On the Navier–Stokes Problem in Exterior Domains with Non Decaying Initial Data. J. Math. Fluid Mech. 14, 633–652 (2012). https://doi.org/10.1007/s00021-011-0083-9

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  • DOI: https://doi.org/10.1007/s00021-011-0083-9

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