Norm Inflation for Generalized Navier-Stokes Equations.
PublisherIndiana University Mathematics Journal
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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  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.