A geometrical regularity criterion in terms of velocity profiles for the three-dimensional Navier-Stokes equations
Abstract
In this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed t∈(0,T), the ‘large velocity’ region Ω:={(x,t)∣|u(x,t)|>C(q)||u||L3q−6}, for some C(q) appropriately defined, shrinks fast enough as q↗∞, then the solution remains regular beyond T. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.
Citation
Tran , C V & Yu , X 2019 , ' A geometrical regularity criterion in terms of velocity profiles for the three-dimensional Navier-Stokes equations ' , Quarterly Journal of Mechanics & Applied Mathematics , vol. 72 , no. 4 , pp. 545–562 . https://doi.org/10.1093/qjmam/hbz018
Publication
Quarterly Journal of Mechanics & Applied Mathematics
Status
Peer reviewed
ISSN
0033-5614Type
Journal article
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