Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations

Details Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations

2006-03-06

arxiv.org/abs/math/0603126v1

Mathematics

Analysis of PDEs

18 pages, 0 figures

Scientific paper

We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local 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 which shrinks to a point dynamically as the time, $t$, approaches the singularity time, $T$. The solution outside the inner core region is assumed to be regular. Under the assumption that the local self-similar velocity profile converges to a limiting profile as $t \to T$ in $L^p$ for some $p \in (3,\infty)$, we prove that such local self-similar blow-up is not possible for any finite time.

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