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

Gallagher, Isabelle; Paicu, Marius

doi:10.1090/S0002-9939-09-09765-2

Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations
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by Isabelle Gallagher and Marius Paicu PDF

Proc. Amer. Math. Soc. 137 (2009), 2075-2083 Request permission

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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