Energy equality for the compressible Navier-Stokes-Korteweg equations

Hui Fang; Yihan Fan; Yanping Zhou; Hui Fang; Yihan Fan; Yanping Zhou

doi:10.3934/math.2022321

AIMS Mathematics

2022, Volume 7, Issue 4: 5808-5820. doi: 10.3934/math.2022321

Previous Article Next Article

Research article

Energy equality for the compressible Navier-Stokes-Korteweg equations

1.
College of Science, China Three Gorges University, Yichang 443002, China
2.
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China

Received: 28 October 2021 Revised: 22 December 2021 Accepted: 05 January 2022 Published: 11 January 2022
MSC : 35Q35, 76N10

In this paper, we investigate the problem of energy equality of the two and three dimensional compressible Navier-Stokes-Korteweg equations with general pressure law. By using the commutator estimation to deal with the nonlinear terms, we obtain the sufficient conditions for the regularity of weak solutions to conserve the energy.
- Navier-Stokes-Korteweg equations,
- compressible fluids,
- energy equality
Citation: Hui Fang, Yihan Fan, Yanping Zhou. Energy equality for the compressible Navier-Stokes-Korteweg equations[J]. AIMS Mathematics, 2022, 7(4): 5808-5820. doi: 10.3934/math.2022321

Related Papers:

Abstract

In this paper, we investigate the problem of energy equality of the two and three dimensional compressible Navier-Stokes-Korteweg equations with general pressure law. By using the commutator estimation to deal with the nonlinear terms, we obtain the sufficient conditions for the regularity of weak solutions to conserve the energy.

References

[1]	D. Korteweg, Sur la forme que prennent les équations du mouvements des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais connues et sur la théorie de la capillarité dans l'hypothse d'une variation continue de la densité, Archives Néerlandaises des Sciences exactes et naturelles, 6 (1901), 1–24.
[2]	J. Dunn, J. Serrin, On the thermomechanics of interstitial working, Arch. Ration. Mech. An., 88 (1985), 95–133. https://doi.org/10.1007/BF00250907 doi: 10.1007/BF00250907
[3]	H. Hattori, D. Li, Solutions for two-dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85–98. https://doi.org/10.1137/S003614109223413X doi: 10.1137/S003614109223413X
[4]	H. Hattori, D. Li, Global solutions of a high-dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84–97. https://doi.org/10.1006/jmaa.1996.0069 doi: 10.1006/jmaa.1996.0069
[5]	M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2008), 679–696. https://doi.org/10.1016/j.anihpc.2007.03.005 doi: 10.1016/j.anihpc.2007.03.005
[6]	B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223–249. https://doi.org/10.1007/s00021-009-0013-2 doi: 10.1007/s00021-009-0013-2
[7]	D. Bresch, B. Desjardins, C. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Part. Diff. Eq., 28 (2003), 843–868. https://doi.org/10.1081/PDE-120020499 doi: 10.1081/PDE-120020499
[8]	B. Haspot, Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension, Math. Meth. Appl. Sci., 20 (2012), 141–164.
[9]	Y. Wang, Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256–271. https://doi.org/10.1016/j.jmaa.2011.01.006 doi: 10.1016/j.jmaa.2011.01.006
[10]	Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2011), 1218–1232. https://doi.org/10.1016/j.jmaa.2011.11.006 doi: 10.1016/j.jmaa.2011.11.006
[11]	Z. Tan, R. Zhang, Optimal decay rates for the compressible fluid models of Korteweg type, Z. Angew. Math. Phys., 65 (2014), 279–300. https://doi.org/10.1007/s00033-013-0331-3 doi: 10.1007/s00033-013-0331-3
[12]	W. Wang, W. Wang, Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces, Discrete Cont. Dyn.-A, 35 (2015), 513–536. http://dx.doi.org/10.3934/dcds.2015.35.513 doi: http://dx.doi.org/10.3934/dcds.2015.35.513
[13]	Y. Z. Wang, Y. Wang, Optimal decay estimate of mild solutions to the compressible Navier-Stokes-Korteweg system in the critical Besov space, Math. Meth. Appl. Sci., 41 (2018), 9592–9606. https://doi.org/10.1002/mma.5316 doi: 10.1002/mma.5316
[14]	T. Kobayashi, K. Tsuda, Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition, Asymptotic Anal., 121 (2020), 1–13. https://doi.org/10.3233/ASY-201600 doi: 10.3233/ASY-201600
[15]	Y. Zhang, Z. Tan, Blow-up of smooth solutions to the compressible fluid models of Korteweg type, Acta Math. Sin., 28 (2012), 645–652. https://doi.org/10.1007/s10114-012-9042-5 doi: 10.1007/s10114-012-9042-5
[16]	T. Tang, J. Kuang, Blow-up of compressible Naiver-Stokes-Korteweg equations, Acta Appl. Math., 130 (2014), 1–7. https://doi.org/10.1007/s10440-013-9836-1 doi: 10.1007/s10440-013-9836-1
[17]	T. Tang, Blow-up of smooth solutions to the compressible barotropic Navier-Stokes-Korteweg equations on bounded domains, Acta Appl. Math., 136 (2015), 55–61. https://doi.org/10.1007/s10440-014-9884-1 doi: 10.1007/s10440-014-9884-1
[18]	T. Dȩbiec, P. Gwiazda, A. Świerczewska-Gwiazda, A. Tzavaras, Conservation of energy for the Euler-Korteweg equations, Calc. Var., 57 (2018), 160. https://doi.org/10.1007/s00526-018-1441-8 doi: 10.1007/s00526-018-1441-8
[19]	P. Constantin, E. Weinan, E. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., 165 (1994), 207–209. https://doi.org/10.1007/BF02099744 doi: 10.1007/BF02099744
[20]	E. Feireisl, P. Gwiazda, A. Świerczewska-Gwiazda, Regularity and energy conservation for the compressible Euler equations, Arch. Ration. Mech. Anal., 261 (2017), 1375–1395. https://doi.org/10.1007/s00205-016-1060-5 doi: 10.1007/s00205-016-1060-5
[21]	J. Serrin, On the uniqueness of compressible fluid motion, Arch. Ration. Mech. Anal., 3 (1959), 271–288. https://doi.org/10.1007/BF00284180 doi: 10.1007/BF00284180
[22]	J. Nash, Le problme de Cauchy pour leséquations différentielles d'un fluide général, B. Soc. Math. Fr., 90 (1962), 487–497.
[23]	X. Huang, J. Li, Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equaitons, Commun. Pur. Appl. Math., 65 (2012), 549–585. https://doi.org/10.1002/cpa.21382 doi: 10.1002/cpa.21382
[24]	Z. Xin, Blow up of smooth solutions to the compressible Navier-Stokes equation with compact density, Commun. Pur. Appl. Math., 51 (1998), 229–240.
[25]	Z. Xin, W. Yan, On blow up of classical solutions to the compressible Navier-Stokes equations, Commun. Math. Phys., 321 (2013), 529–541. https://doi.org/10.1007/s00220-012-1610-0 doi: 10.1007/s00220-012-1610-0
[26]	M. Okita, Optimal decay rate for strong solutions in critial spaces to the compressible Navier-Stokes equations, J. Differ. Equ., 257 (2014), 3850–3867. https://doi.org/10.1016/j.jde.2014.07.011 doi: 10.1016/j.jde.2014.07.011
[27]	R. Danchin, J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 224 (2017), 53–90. https://doi.org/10.1007/s00205-016-1067-y doi: 10.1007/s00205-016-1067-y
[28]	J. Serrin, The initial value problem for the Navier-Stokes equations, University of Wisconsin Press, Madison, 1963.
[29]	M. Shinbrot, The energy equation for the Navier-Stokes system, SIAM J. Math. Anal., 5 (1974), 948–954. https://doi.org/10.1137/0505092 doi: 10.1137/0505092
[30]	C. Yu, Energy conservation for the weak solutions of the compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 225 (2017), 1073–1087. https://doi.org/10.1007/s00205-017-1121-4 doi: 10.1007/s00205-017-1121-4
[31]	I. Akramov, T. Dbiec, J. Skipper, E. Wiedemann, Energy conservation for the compressible Euler and Navier-Stokes equations with vacuum, Anal. PDE, 13 (2020), 789–811. https://doi.org/10.2140/apde.2020.13.789 doi: 10.2140/apde.2020.13.789
[32]	Q. Nguyen, P. Nguyen, Q. Bao, Energy equalities for compressible Navier-Stokes equations, Nonlinearity, 32 (2019), 4206–4231. https://doi.org/10.1088/1361-6544/ab28ae doi: 10.1088/1361-6544/ab28ae
[33]	Z. Liang, Regularity criterion on the energy conservation for the compressible Navier-Stokes equations, P. Roy. Soc. Edinb. A, 225 (2020), 1–18. https://doi.org/10.1017/prm.2020.87 doi: 10.1017/prm.2020.87
[34]	L. Berselli, E. Chiodaroli, On the energy equality for the 3D Navier-Stokes equations, Nonlinear Anal., 192 (2020), 111704. https://doi.org/10.1016/j.na.2019.111704 doi: 10.1016/j.na.2019.111704
[35]	T. Leslie, R. Shvydkoy, The energy balance relation for weak solutions of the density-dependent Navier-Stokes equations, J. Differ. Equ., 261 (2016), 3719–3733. https://doi.org/10.1016/j.jde.2016.06.001 doi: 10.1016/j.jde.2016.06.001

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)