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2009 Liouville theorems for the Navier–Stokes equations and applications
Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, Vladimir Šverák
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Acta Math. 203(1): 83-105 (2009). DOI: 10.1007/s11511-009-0039-6


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].

Funding Statement

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


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Gabriel Koch. Nikolai Nadirashvili. Gregory A. Seregin. Vladimir Šverák. "Liouville theorems for the Navier–Stokes equations and applications." Acta Math. 203 (1) 83 - 105, 2009.


Received: 5 October 2007; Revised: 15 April 2008; Published: 2009
First available in Project Euclid: 31 January 2017

zbMATH: 1208.35104
MathSciNet: MR2545826
Digital Object Identifier: 10.1007/s11511-009-0039-6

Rights: 2009 © Institut Mittag-Leffler


Vol.203 • No. 1 • 2009
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