Proceedings of the American Mathematical Society

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

Author(s): Isabelle Gallagher; Marius Paicu
Journal: Proc. Amer. Math. Soc. 137 (2009), 2075-2083.
MSC (2000): Primary 76D05
Posted: January 15, 2009
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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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Isabelle Gallagher
Affiliation: Institut de Mathématiques de Jussieu, UMR 7586, Université Paris 7, 175 rue du Chevaleret, 75013 Paris, France
Email: Isabelle.Gallagher@math.jussieu.fr

Marius Paicu
Affiliation: Département de Mathématiques, Université Paris 11, Bâtiment 425, 91405 Orsay Cedex, France
Email: marius.paicu@math.u-psud.fr

DOI: 10.1090/S0002-9939-09-09765-2
PII: S 0002-9939(09)09765-2
Keywords: Navier-Stokes equations, blow-up
Received by editor(s): May 21, 2008
Posted: January 15, 2009
Communicated by: Walter Craig
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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