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Navier–Stokes equations: an analysis of a possible gap to achieve the energy equality

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Abstract

The paper is concerned with the IBVP of the Navier–Stokes equations. The goal is to evaluate the possible gap between the energy equality and the energy inequality deduced for a weak solution. This kind of analysis is new and the result is a natural continuation and improvement of a result obtained by the same authors in Crispo et al. (Some new properties of a suitable weak solution to the Navier–Stokes equations. arXiv:1904.07641).

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Notes

  1. We stress that, for all \(m\in {\mathbb {N}}\), \(meas(T_R^m)\ne 0\). Actually, if for \(2TR>A\) we guess that \(meas(T_R^m)=0\), then

    $$\begin{aligned} A<\int \limits _{0}^{T}2R\le \int \limits _{0}^{T} h^m(t)dt\le A. \end{aligned}$$

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Correspondence to Paolo Maremonti.

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Carlo Romano Grisanti: The research activity of F. Crispo and P. Maremonti is performed under the auspices of GNFM-INdAM. The research activity of F. Crispo is also supported by the Program “Vanvitelli per la Ricerca: VALERE 2019” financed by the University of Campania “L. Vanvitelli”. The research activity of C.R. Grisanti is performed under the auspices of GNAMPA-INdAM and partially supported by the Research Project of the University of Pisa “Energia e regolarità: tecniche innovative per problemi classici di equazioni alle derivate parziali”.

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Maremonti, P., Crispo, F. & Grisanti, C.R. Navier–Stokes equations: an analysis of a possible gap to achieve the energy equality. Ricerche mat 70, 235–249 (2021). https://doi.org/10.1007/s11587-020-00525-5

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  • DOI: https://doi.org/10.1007/s11587-020-00525-5

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