Acta Mathematica

Liouville theorems for the Navier–Stokes equations and applications

Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, and Vladimir Šverák

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Abstract

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in Rn × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].

Note

The fourth author was supported in part by NSF Grant DMS-0457061.

Article information

Source
Acta Math., Volume 203, Number 1 (2009), 83-105.

Dates
Received: 5 October 2007
Revised: 15 April 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892409

Digital Object Identifier
doi:10.1007/s11511-009-0039-6

Mathematical Reviews number (MathSciNet)
MR2545826

Zentralblatt MATH identifier
1208.35104

Rights
2009 © Institut Mittag-Leffler

Citation

Koch, Gabriel; Nadirashvili, Nikolai; Seregin, Gregory A.; Šverák, Vladimir. Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203 (2009), no. 1, 83--105. doi:10.1007/s11511-009-0039-6. https://projecteuclid.org/euclid.acta/1485892409


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