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A local smoothness criterion for solutions of the 3D Navier-Stokes equations
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Robinson, James C. and Sadowski, Witold (2014) A local smoothness criterion for solutions of the 3D Navier-Stokes equations. Rendiconti del Seminario Matematico della Universita di Padova, Volume 131 . pp. 159-178. doi:10.4171/RSMUP/131-9 ISSN 0041-8994.
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Official URL: http://dx.doi.org/10.4171/RSMUP/131-9
Abstract
We consider the three-dimensional Navier–Stokes equations on the whole space R3 and on the three-dimensional torus T3. We give a simple proof of the local existence of (finite energy) solutions in L3 for initial data u0∈L2∩L3, based on energy estimates and regularisation of the initial data with the heat semigroup.
We also provide a lower bound on the existence time of a strong solution in terms of the solution v(t) of the heat equation with such initial data: there is an absolute constant ε>0 such that solutions remain regular on [0,T] if ∥u0∥3L3∫T0∫R3|∇v(s)|2|v(s)|dxdt≤ε. This implies the u∈C0([0,T];L3) regularity criterion due to von Wahl. We also derive simple a priori estimates in Lp for p>3 that recover the well known lower bound ∥u(T−t)∥Lp≥ct−(p−3)/2p on any solution that blows up in Lp at time T.
The key ingredients are a calculus inequality ∥u∥pL3p≤c∫|u|p−2|∇u|2 (valid on R3 and for functions on bounded domains with zero average) and the bound on the pressure ∥p∥Lr≤cr∥u∥2L2r, valid both on the whole space and for periodic boundary conditions.
Keywords: Navier-Stokes equations, critical spaces, calculus inequalities
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Rendiconti del Seminario Matematico della Universita di Padova | ||||
Publisher: | C E D A M | ||||
ISSN: | 0041-8994 | ||||
Official Date: | 2014 | ||||
Dates: |
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Volume: | Volume 131 | ||||
Page Range: | pp. 159-178 | ||||
DOI: | 10.4171/RSMUP/131-9 | ||||
Status: | Peer Reviewed | ||||
Access rights to Published version: | Restricted or Subscription Access |
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