Norm Inflation for Generalized Navier-Stokes Equations

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We consider the incompressible Navier-Stokes equation with a fractional power α∈[1,∞) of the Laplacian in the three dimensional case. We prove the existence of a smooth solution with arbitrarily small in B∞,ₚ⁻α (2<p ≤ ∞) initial data that becomes arbitrarily large in B∞,∞⁻ˢ for all s> 0 in arbitrarily small time. This extends the result of Bourgain and Pavlović for the classical Navier-Stokes equation which utilizes the fact that the energy transfer to low modes increases norms with negative smoothness indexes. It is remarkable that the space B∞,∞⁻α is supercritical for α>1. Moreover, the norm inflation occurs even in the case α≥ 5/4 where the global regularity is known.