Journal of Statistical Physics

, Volume 167, Issue 1, pp 1–13 | Cite as

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

  • Carlo Boldrighini
  • Dong Li
  • Yakov G. Sinai


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.


Navier–Stokes equations Global regularity Blow-up 



We thank S. Frigio and P. Maponi for many discussions and for providing the data of unpublished computer simulations.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Istituto Nazionale di Alta Matematica (INdAM)GNFM, Unità locale Università Roma TreRomeItaly
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Department of MathematicsPrinceton University PrincetonNew JerseyUSA

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