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A note on energy equality for the fractional Navier-Stokes equations

Part of: Equations of mathematical physics and other areas of application Partial differential equations Incompressible viscous fluids

Published online by Cambridge University Press:  03 February 2023

Fan Wu
Fan Wu*
Affiliation:
College of Science, Nanchang Institute of Technology, Nanchang, Jiangxi 330099, China (wufan0319@yeah.net)
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Abstract

This paper proves the energy equality for distributional solutions to fractional Navier-Stokes equations, which gives a new proof and covers the classical result of Galdi [Proc. Amer. Math. Soc. 147 (2019), 785–792].

Keywords

Fractional Navier-Stokes equationsenergy equalitydistributional solutions

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 76D03: Existence, uniqueness, and regularity theory 35B65: Smoothness and regularity of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 1 , February 2024 , pp. 201 - 208
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Amann, H.. On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2 (2000), 16–98.CrossRefGoogle Scholar
Chen, Y. and Wei, C.. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete Contin. Dyn. Syst. - Ser. A 36 (2016), 5309.CrossRefGoogle Scholar
Fabes, E. B., Jones, B. F. and Riviere, N. M.. The initial value problem for the Navier-Stokes equations with data in $L^p$. Arch. Ration. Mech. Anal. 45 (1972), 222–240.CrossRefGoogle Scholar
Foias, C.. Une remarque sur l'unicité des solutions des équations de Navier-Stokes en dimension $n$. Bull. Soc. Math. Fr. 89 (1961), 1–8.Google Scholar
Galdi, G.. On the energy equality for distributional solutions to Navier-Stokes equations. Proc. Am. Math. Soc. 147 (2019), 785–792.CrossRefGoogle Scholar
Hopf, E.. Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen. Erhard schmidt zu seinem 75. Geburtstag gewidmet. Math. Nachr. 4 (1950), 213–231.CrossRefGoogle Scholar
Katz, N. and Pavlović, N.. A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation. Geom. Funct. Anal. GAFA 12 (2002), 355–379.CrossRefGoogle Scholar
Ladyzhenskaia, O. A., Solonnikov, V. A. and Ural'tseva, N. N.. Linear and quasi-linear equations of parabolic type (Providence: American Mathematical Society, 1988).Google Scholar
Leray, J.. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), 193–248.CrossRefGoogle Scholar
Lions, J.. Quelques méthodes de résolution des problémes aux limites non linéaires (Paris: Dunod, Gauthier-Villars, 1969).Google Scholar
Lions, J.. Sur la régularité et l'unicité des solutions turbulentes des équations de Navier-Stokes. Rend. Sem. Mat. Univ. Padova 30 (1960), 16–23.Google Scholar
Lions, J.. Mathematical topics in fluid mechanics, vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, vol. 3 (New York: Oxford Science Publications, The Clarendon Press, Oxford University Press, 1996).Google Scholar
Luo, T. and Titi, E. S.. Non-uniqueness of weak solutions to hyperviscous Navier-Stokes equations: on sharpness of J.-L. Lions exponent. Calc. Var. Partial Differ. Equ. 59 (2020), 1–15.CrossRefGoogle Scholar
Nirenberg, L.. On elliptic partial differential equations. Principio di minimo e sue applicazioni alle equazioni funzionali (Berlin, Heidelberg: Springer, 2011), pp. 1–48.Google Scholar
Ren, W., Wang, Y. and Wu, G.. Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations. Commun. Contemp. Math. 18 (2016), 1650018.CrossRefGoogle Scholar
Stein, E.. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, vol. 30 (Princeton: Princeton University Press, 1970).Google Scholar
Tang, L. and Yu, Y.. Partial regularity of suitable weak solutions to the fractional Navier-Stokes equations. Commun. Math. Phys. 334 (2015), 1455–1482.CrossRefGoogle Scholar
Shinbrot, M.. The energy equation for the Navier-Stokes system. SIAM J. Math. Anal. 5 (1974), 948–954.CrossRefGoogle Scholar
Wu, J.. Generalized MHD equations. J. Differ. Equ. 195 (2003), 284–312.CrossRefGoogle Scholar
Yu, C.. A new proof to the energy conservation for the Navier-Stokes equations. arXiv:1604.05697, 2016.Google Scholar