Abstract
Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber \({\Lambda(t)}\) that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided \({\Lambda \in L^{5/2}}\) , which improves our previous \({\Lambda \in L^\infty}\) condition. We also show that \({\Lambda \in L^1}\) for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.
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The work of A. Cheskidov is partially supported by NSF Grant DMS–1108864.
R. Shvydkoy acknowledges the support of NSF grants DMS–0907812 and DMS–1210896.
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Cheskidov, A., Shvydkoy, R. A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range. J. Math. Fluid Mech. 16, 263–273 (2014). https://doi.org/10.1007/s00021-014-0167-4
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DOI: https://doi.org/10.1007/s00021-014-0167-4