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Regularity criterion on the energy conservation for the compressible Navier–Stokes equations

Part of: Incompressible inviscid fluids Compressible fluids and gas dynamics, general Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  11 December 2020

Zhilei Liang Open the ORCID record for Zhilei Liang [Opens in a new window]
Zhilei Liang*
Affiliation:
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China ([email protected])
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Abstract

This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a Lp–Ls type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.

Keywords

Compressible Navier–Stokesenergy conservationweak solutions

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 76B03: Existence, uniqueness, and regularity theory 76N10: Existence, uniqueness, and regularity theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1954 - 1971
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press.

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References

Akramov, I., Debiec, T., Skipper, J. and Wiedemann, E.. Energy conservation for the compressible Euler and Navier–Stokes equations with vacuum, arXiv: 1808.05029.Google Scholar
Bardos, C. and Titi, E. S.. Onsager's conjecture for the incompressible Euler equations in bounded domains. Arch. Ration. Mech. Anal. 228 (2018), 197–207.10.1007/s00205-017-1189-xCrossRefGoogle Scholar
Bardos, C. and Titi, E. S.. Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit. Commun. Math. Phys. 370 (2019), 291–310.10.1007/s00220-019-03493-6CrossRefGoogle Scholar
Berselli, L-C.. On a regularity criterion for the solutjions to the 3D Navier–Stokes equations. Differ. Integral Equ. 15 (2002), 1129–1137.Google Scholar
Berselli, L-C. and Chiodaroli, E.. On the energy equality for the 3D Navier–Stokes equations. Nonlinear Anal. 192 (2020), 111704.10.1016/j.na.2019.111704CrossRefGoogle Scholar
Buckmaster, T., De Lellis, C., Szekelyhidi, Jr. L. Vicol, V.. Onsager's conjecture for admissible weak solutions. Commun. Pure Appl. Math. 72 (2019), 229–274.10.1002/cpa.21781CrossRefGoogle Scholar
Chen, R.-M. and Yu, C.. Onsager's energy conservation for inhomogeneous Euler equations. J. Math. Pures Appl. 131 (2019), 1–16.10.1016/j.matpur.2019.02.003CrossRefGoogle Scholar
Chen, R.-M., Liang, Z, Wang, D. and Xu, R.. Energy equality in compressible fluids with physical boundaries. SIAM J. Math. Anal. 52 (2020), 1363–1385.10.1137/19M1287213CrossRefGoogle Scholar
Cheskidov, A. and Luo, X. On the energy equality for Navier–Stokes equations in weak-in-time Onsager spaces. Nonlinearity 33 (2020), 1388–1403.10.1088/1361-6544/ab60d3CrossRefGoogle Scholar
Cheskidov, A., Constantin, P., Friedlander, S. and Shvydkoy, R.. Energy conservation and Onsagers conjecture for the Euler equations. Nonlinearity 21 (2008), 1233–1252.CrossRefGoogle Scholar
Constantin, P., Wiedemann, E. and Titi, E. S.. Onsager's conjecture on the energy conservation for solutions of Euler's equation. Commun. Math. Phys. 165 (1994), 207–209.CrossRefGoogle Scholar
DiPerna, R. J. and Lions, P.-L.. Ordinary differential equations transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547.CrossRefGoogle Scholar
Drivas, T. D. and Eyink, G.. An Onsager singularity theorem for turbulent solutions of compressible Euler equations. Commun. Math. Phys. 359 (2018), 733–763.CrossRefGoogle Scholar
Drivas, T. D. and Nguyen, H. Q.. Onsager's conjecture and anomalous dissipation on domains with boundary. SIAM J. Math. Anal. 50 (2018), 4785–4811.10.1137/18M1178864CrossRefGoogle Scholar
Duchon, J. and Robert, R.. Inertial energy dissipation for weak solutions of incompressible Euler and Navier–Stokes equations. Nonlinearity 13 (2000), 249–255.CrossRefGoogle Scholar
Evans, L. C., Ed. Partial differential equations. In Graduate Studies in Mathematics, vol. 19, 2nd edn, (Providence, RI: American Mathematical Society, 2010).Google Scholar
Feireisl, E., Novotnśy, A. and Petzeltovśa, H.. On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3 (2001), 358–392.10.1007/PL00000976CrossRefGoogle Scholar
Feireisl, E., Gwiazda, P., Swierczewska-Gwiazda, A. and Wiedemann, E.. Regularity and energy conservation for the compressible Euler equations. Arch. Ration. Mech. Anal. 223 (2017), 1375–1395.10.1007/s00205-016-1060-5CrossRefGoogle Scholar
Isett, P.. A proof of Onsager's conjecture. Ann. Math. 188 (2018), 871–963.10.4007/annals.2018.188.3.4CrossRefGoogle Scholar
Kolmogoroff, A. N.. Dissipation of energy in the locally isotropic turbulence. C. R. (Doklady) Acad. Sci. URSS (N.S.) 32 (1941), 16–18.Google Scholar
Ladyz̆zenskaja, O. A., Solonnikov, V. A. and Ural'ceva, N. N., Eds. Linear and quasilinear equations of parabolic type, translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23, (Providence, RI: American Mathematical Society, 1968).Google Scholar
Leslie, T. M. and Shvydkoy, R.: The energy balance relation for weak solutions of the density-dependent Navier–Stokes equations, J. Differ. Equ. 261 (2016), 3719–3733.CrossRefGoogle Scholar
Leslie, T. M. and Shvydkoy, R.: Conditions implying energy equality for weak solutions of the Navier–Stokes equations, SIAM J. Math. Anal. 50 (2018), 870–890.10.1137/16M1104147CrossRefGoogle Scholar
Lions, J.-L.. Sur la réegularitśe et l'unicitśe des solutions turbulentes des śequations de Navier–Stokes. Rend. Sem. Mat. Univ. Padova 30 (1960), 16–23.Google Scholar
Lions, P.-L.. Mathematical topics in fluid mechanics. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, vol. 1, (New York: 3 Oxford Science Publications, The Clarendon Press, Oxford University Press, 1996).Google Scholar
Lions, P.-L.. Mathematical topics in fluid mechanics. Compressible models. Oxford Lecture Series in Mathematics and its Applications, vol. 2, (New York: 10 Oxford University Press, Oxford Science Publications. The Clarendon Press, 1998).Google Scholar
Nguyen, Q.-H., Nguyen, P.-T. and Tang, B. Energy equalities for compressible Navier–Stokes equations. Nonlinearity 32 (2019), 4206–4231.CrossRefGoogle Scholar
Onsager, L.. Statistical Hydrodynamics. Nuovo Cimento (Supplemento) 6 (1949), 279–287.CrossRefGoogle Scholar
Prodi, G.. Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48 (1959), 173–182.10.1007/BF02410664CrossRefGoogle Scholar
Serrin, J.. The initial value problem for the Navier–Stokes equations. In Proc. Symp. 1963 Nonlinear Problems, Madison, WI, pp. 69–98 (University of Wisconsin Press, Madison, WI, 1962).Google Scholar
Shinbrot, M.. The energy equation for the Navier–Stokes system. SIAM J. Math. Anal. 5 (1974), 948–954.10.1137/0505092CrossRefGoogle Scholar
Yu, C.. Energy conservation for the weak solutions of the compressible Navier–Stokes equations. Arch. Ration. Mech. Anal. 225 (2017), 1073–1087.10.1007/s00205-017-1121-4CrossRefGoogle Scholar
Yu, C.. A new proof to the energy conservation for the Navier–Stokes equations, arXiv: 1604.05697v1.Google Scholar
Yu, C.. The energy conservation for the Navier–Stokes equations in bounded domains. arXiv:1802.07661v1.Google Scholar