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Abstract:  Further results are developed for showing the critical nature of the Navier-Stokes equation.  It is proved that global-in-time smooth solutions exist for the Navier-Stokes equations with hyper-dissipation, i. e. the equations \[ \partial_t u+(-\Delta)^\beta u+ u\cdot\nabla u+\nabla p=0 , \quad \textrm{div}\ u=0, \qquad \beta >1 \] with the hypothesis that there is only one singular point at a possible blow-up time.

Global-in-Time Regularity of the Navier-Stokes Equations with Hyper-Dissipation