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Published August 2007 | public
Journal Article Open

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


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.

Additional Information

© 2007 American Institute of Mathematical Sciences. Received for publication March 2007. Available Online: May 2007. We would like to thank Profs. Congming Li and Dongho Chae for their useful comments and suggestions. The first author is supported by NSF under the NSF FRG grant DMS-0353838 and ITR Grant ACI-0204932. The second author was partially supported by the National Basic Research Program of China under the grant 2005CB321701.

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August 19, 2023
August 19, 2023