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

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  • 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 $t \rightarrow T$ in $L^p$ for some $p \in (3,\infty )$, 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.
    Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 76B03.

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