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November  2007, 18(4): 637-642. doi: 10.3934/dcds.2007.18.637

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

Thomas Y. Hou 1, and  Ruo Li 2,

1. 

Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States
2. 

LMAM&School of Mathematical Sciences, Peking University, Beijing, 100871, China

Received  March 2007 Published  May 2007

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.
Keywords: locally self-similar, Euler equations, blow-up., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 76B0.
Citation: Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
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