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Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations

Authors: Isabelle Gallagher and Marius Paicu
Journal: Proc. Amer. Math. Soc. 137 (2009), 2075-2083
MSC (2000): Primary 76D05
Published electronically: January 15, 2009
MathSciNet review: 2480289
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Abstract: In a 2001 paper, S. Montgomery-Smith provides a one-dimensional model for the three-dimensional, incompressible Navier-Stokes equations, for which he proves the blow-up of solutions associated with a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this paper the model is adapted to the cases of two and three space dimensions, with the additional feature that the divergence-free condition is preserved. It is checked that a family of initial data constructed by Chemin and Gallagher, which is arbitrarily large yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow-up for the toy model -- meaning that the precise structure of the nonlinear term is crucial to understanding the dynamics of large solutions to the Navier-Stokes equations.

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Additional Information

Isabelle Gallagher
Affiliation: Institut de Mathématiques de Jussieu, UMR 7586, Université Paris 7, 175 rue du Chevaleret, 75013 Paris, France

Marius Paicu
Affiliation: Département de Mathématiques, Université Paris 11, Bâtiment 425, 91405 Orsay Cedex, France

Keywords: Navier-Stokes equations, blow-up
Received by editor(s): May 21, 2008
Published electronically: January 15, 2009
Communicated by: Walter Craig
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.