On Local Type I Singularities of the Navier–Stokes Equations and Liouville Theorems

  • Dallas AlbrittonEmail author
  • Tobias Barker


We prove that suitable weak solutions of the Navier–Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier–Stokes equations bounded in \(L^3\) along a backward sequence of times.



The authors would like to thank Gregory Seregin, Vladimır Šverák, and Julien Guillod for helpful suggestions on a preliminary version of the paper. DA was supported by the NDSEG Graduate Fellowship and a travel grant from the Council of Graduate Students at the University of Minnesota.

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Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.DMA, École Normale Supérieure CNRSPSL Research UniversityParisFrance

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