Oscillations and integrability of the vorticity in the 3D NS flows
Abstract
In the studies of the Navier-Stokes (NS) regularity problem, it has become increasingly clear that a more realistic path to improved a priori bounds is to try to break away from the scaling of the energy-level estimates in the realm of the blow-up-type arguments (the solution in view is regular/smooth up to the possible blow-up time) rather than to try to improve regularity of arbitrary Leray's weak solutions. The present article is a contribution in this direction; more precisely, it is shown--in the context of an algebraic/polynomial-type blow-up profile of arbitrary degree--that a very weak condition on the vorticity direction field (membership in a local $bmo$ space weighted with arbitrary many logarithms)--suffices to break the energy-level scaling in the bounds on the vorticity. At the same time, the obtained bounds transform a 3D NS criticality scenario depicted by the macro-scale long vortex filaments into a no-singularity scenario.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2018
- DOI:
- 10.48550/arXiv.1801.09040
- arXiv:
- arXiv:1801.09040
- Bibcode:
- 2018arXiv180109040D
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35Q30
- E-Print:
- final version