Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations

Abstract

We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, t, approaches a possible singularity time, T. The solution outside the inner core region is assumed to be regular, but it does not satisfy self-similar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as t → T in L^p for some p ϵ (3,∞), we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.