We observe that known regularity results imply weak solutions of the 3D Navier-Stokes equations are regular if their energy spectrum decays as k^s, s<-3, for large wavenumbers k. We allow for weak solutions having infinite total energy by localizing the equations. We then consider certain modified Leray backwards self-similar solutions and show that their energy spectrum decays exactly at the critical rate. Therefore, this rate of decay is consistent with the appearance of a self-similar singularity.

Lecture #12211

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