Liouville theorems for the Navier-Stokes equations and applications

Details Liouville theorems for the Navier-Stokes equations and applications Liouville theorems for the Navier-Stokes equations and applications

2007-09-22

arxiv.org/abs/0709.3599v1

Mathematics

Analysis of PDEs

26 pages

Scientific paper

We study bounded ancient solutions of the Navier-Stokes equations. These are the solutions which are defined for all past time. In two space dimensions we prove that such solutions are either constant or functions of time only, depending on the exact definition of admissible solutions. The general three dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axi-symmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions.

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