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Norm Inflation for Generalized Navier-Stokes Equations.

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posted on 08.01.2016 by A. Cheskidov, M. Dai
We consider the incompressible Navier-Stokes equation with a fractional power alpha is an element of E [1, infinity) of the Laplacian in the three-dimensional case. We prove the existence of a smooth solution with arbitrarily small initial data in (B) over dot (-alpha)(infinity,p) (2 < p <= infinity) that becomes arbitrarily large in (B) over dot (-s)(infinity,infinity) for all s > 0 in arbitrarily small time. This extends the result of Bourgain and Pavlovic [1] for the classical Navier-Stokes equation, a result which uses the fact that the energy transfer to low modes increases norms with negative smoothness indexes. It is remarkable that the space (B) over dot (-alpha)(infinity,infinity) is supercritical for alpha > 1. Moreover, the norm inflation occurs even in the case alpha >= 5/4 where the global regularity is known.

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Post print version of article may differ from published version. This is an electronic version of an article published in Cheskidov, A. and Dai, M. Norm Inflation for Generalized Navier-Stokes Equations. Indiana University Mathematics Journal. 2014. 63(3): 869-884. is available online at: http://www.informaworld.com/smpp/

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Indiana University Mathematics Journal

issn

0022-2518

Issue date

01/01/2014

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