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Complex Singular Solutions of the 3-d Navier–Stokes Equations and Related Real Solutions

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Abstract

By applying methods of statistical physics Li and Sinai (J Eur Math Soc 10:267–313, 2008) proved that there are complex solutions of the Navier–Stokes equations in the whole space \({\mathbb R}^{3}\) which blow up at a finite time. We present a review of the results obtained so far, by theoretical work and computer simulations, for the singular complex solutions, and compare with the behavior of related real solutions. We also discuss the possible application of the techniques introduced in (J Eur Math Soc 10:267–313, 2008) to the study of the real ones.

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Acknowledgements

We thank S. Frigio and P. Maponi for many discussions and for providing the data of unpublished computer simulations.

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Authors and Affiliations

  1. Istituto Nazionale di Alta Matematica (INdAM), GNFM, Unità locale Università Roma Tre, Largo S. Leonardo Murialdo, 1, 00146, Rome, Italy

    Carlo Boldrighini

  2. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Dong Li

  3. Department of Mathematics, Princeton University Princeton, New Jersey, NJ, 08544, USA

    Yakov G. Sinai

Authors
  1. Carlo Boldrighini
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  2. Dong Li
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  3. Yakov G. Sinai
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Corresponding author

Correspondence to Carlo Boldrighini.

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Partially supported by INdAM (G.N.F.M.) and M.U.R.S.T. research funds.

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Boldrighini, C., Li, D. & Sinai, Y.G. Complex Singular Solutions of the 3-d Navier–Stokes Equations and Related Real Solutions. J Stat Phys 167, 1–13 (2017). https://doi.org/10.1007/s10955-017-1730-1

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  • DOI: https://doi.org/10.1007/s10955-017-1730-1

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