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

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

2008-05-21

arxiv.org/abs/0805.3215v2

Mathematics

Analysis of PDEs

Version to be published in the Proceedings of the AMS, typos and corrections according to the recommendations of the referees

Scientific paper

S. Montgomery-Smith provided a one dimensional model for the three dimensional, incompressible Navier-Stokes equations, for which he proved the blow up of solutions associated to a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this note the model is adapted to the case of two and three space dimensions, with the additional feature that the divergence free condition is preserved. It is checked that the family of initial data constructed previously by J.-Y Chemin and I. Gallagher which is arbitrarily large but 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 understand the dynamics of large solutions to the Navier-Stokes equations.

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