Journal of Nonlinear Science

, Volume 29, Issue 1, pp 215–228 | Cite as

Weak and Strong Solutions of the 3D Navier–Stokes Equations and Their Relation to a Chessboard of Convergent Inverse Length Scales

  • J. D. GibbonEmail author


Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the 3D Navier–Stokes equations, on a periodic domain \(\mathcal {V} =[0,\,L]^{3}\) an infinite ‘chessboard’ of estimates for these inverse length scales is displayed in terms of labels \((n,\,m)\) corresponding to n derivatives of the velocity field in \(L^{2m}(\mathcal {V})\). The \((1,\,1)\) position corresponds to the inverse Kolmogorov length \(Re^{3/4}\). These estimates ultimately converge to a finite limit, \(Re^3\), as \(n,\,m\rightarrow \infty \), although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by \((n,\,m)\). In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by \((n,\,m)\), the only difference being a factor of 2 in the exponent. This appears to be a generalization of the Prodi–Serrin conditions for \(n\ge 1\).


Navier-stokes Length scales Weak and strong solutions 

Mathematics Subject Classification

76D03 76D05 35Q30 



My thanks go to Vlad Vicol of Princeton University for suggesting the method of proof of Theorem 2 and to Darryl Holm for discussions.


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Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

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