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  2015, 22: 20-31. doi: 10.3934/era.2015.22.20

Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity

Jianwei Yang 1, , Peng Cheng 1, and  Yudong Wang 1,

1. 

College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450045, Henan Province, China, China, China

Received  May 2014 Revised  February 2015 Published  July 2015

In this paper, we study a combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes-Korteweg equations for weak solutions. For well prepared initial data, the convergence of solutions of the compressible Navier-Stokes-Korteweg equations to the solutions of the incompressible Navier-Stokes equation are justified rigorously by adapting the modulated energy method. Furthermore, the corresponding convergence rates are also obtained.
Keywords: Incompressible limit, vanishing capillarity limit, modulated energy, incompressible Navier-Stokes equations., Navier-Stokes-Korteweg model.
Mathematics Subject Classification: 35Q30, 35B40, 35C2.
Citation: Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20
References:
[1]

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F. Charve and B. Haspot, Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system,, \emph{SIAM J. Math. Anal.}, 45 (2013), 469. doi: 10.1137/120861801. Google Scholar

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R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type,, \emph{Ann. Inst. H. Poincaré, 18 (2001), 97. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar

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J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working,, \emph{Arch. Rational Mech. Anal.}, 88 (1985), 95. doi: 10.1007/BF00250907. Google Scholar

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P. Embid and A. Majda, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers,, \emph{Geosphys. Astrophys. Fluid Dynam.}, 87 (1998), 1. doi: 10.1080/03091929808208993. Google Scholar

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J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung,, \emph{Phys. Chem.}, 13 (1894), 657. Google Scholar

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E. Grenier, Oscillartory perturbations of the Navier-Stokes equations,, \emph{J. Math. Pures Appl. (9)}, 76 (1997), 477. doi: 10.1016/S0021-7824(97)89959-X. Google Scholar

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I. Gamba, A. Jüngel and A. Vasseur, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations,, \emph{J. Differential Equations}, 247 (2009), 3117. doi: 10.1016/j.jde.2009.09.001. Google Scholar

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[16]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type,, \emph{J. Math. Fluid Mech.}, 13 (2011), 223. doi: 10.1007/s00021-009-0013-2. Google Scholar

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A. Jüngel, C.-K. Lin and K.-C. Wu, An asymptotic limit of a Navier-Stokes system with capillary effects,, \emph{Commun. Math. Phys.}, 329 (2014), 725. doi: 10.1007/s00220-014-1961-9. Google Scholar

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A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, \emph{SIAM J. Math. Anal.}, 42 (2010), 1025. doi: 10.1137/090776068. Google Scholar

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D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité,, \emph{Arch. Néer. Sci. Exactes Sér. II}, 6 (1901), 1. Google Scholar

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M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type,, \emph{Ann. Inst. H. Poincaré, 25 (2008), 679. doi: 10.1016/j.anihpc.2007.03.005. Google Scholar

[23]

M. Kotschote, Dynamics of compressible non-isothermal fluids of non-Newtonian Korteweg type,, \emph{SIAM J. Math. Anal.}, 44 (2012), 74. doi: 10.1137/110821202. Google Scholar

[24]

Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force,, \emph{J. Math. Anal. Appl.}, 388 (2012), 1218. doi: 10.1016/j.jmaa.2011.11.006. Google Scholar

[25]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, \emph{J. Math. Pures Appl. (9)}, 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar

[26]

N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data,, \emph{Commun. Pure and Appl. Math.}, 53 (2000), 432. Google Scholar

[27]

F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids,, \emph{Arch. Rational Mech. Anal.}, 27 (1967), 329. Google Scholar

[28]

C. Rohde, On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions,, \emph{ZAMM Z. Angew. Math. Mech.}, 85 (2005), 839. doi: 10.1002/zamm.200410211. Google Scholar

[29]

Z. Tan, H. Wang and J. Xu, Global existence and optimal $L^2$ decay rate for the strong solutions to the compressible fluid models of Korteweg type,, \emph{J. Math. Anal. Appl.}, 390 (2012), 181. doi: 10.1016/j.jmaa.2012.01.028. Google Scholar

show all references

References:
[1]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations,, \emph{Arch. Rational Mech. Anal.}, 180 (2006), 1. doi: 10.1007/s00205-005-0393-2. Google Scholar

[2]

S. Benzoni-Gavage, R. Danchin and S. Descombes, Well-posedness of one-dimensional Korteweg models,, \emph{Electron. J. Differential Equations, (2006). Google Scholar

[3]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, \emph{Comm. Partial Differential Equations}, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar

[4]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,, \emph{Comm. Math. Phys.}, 238 (2003), 211. doi: 10.1007/s00220-003-0859-8. Google Scholar

[5]

D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems,, \emph{Comm. Partial Differential Equations}, 28 (2003), 843. doi: 10.1081/PDE-120020499. Google Scholar

[6]

D. Bresch, B. Desjardins and B. Ducomet, Quasi-neutral limit for a viscous capillary model of plasma,, \emph{Ann. Inst. H. Poincaré Anal. Non Linéaire}, 22 (2005), 1. doi: 10.1016/j.anihpc.2004.02.001. Google Scholar

[7]

Z. Chen and H. Zhao, Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system,, \emph{J. Math. Pures Appl. (9)}, 101 (2014), 330. doi: 10.1016/j.matpur.2013.06.005. Google Scholar

[8]

F. Charve and B. Haspot, Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system,, \emph{SIAM J. Math. Anal.}, 45 (2013), 469. doi: 10.1137/120861801. Google Scholar

[9]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type,, \emph{Ann. Inst. H. Poincaré, 18 (2001), 97. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar

[10]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working,, \emph{Arch. Rational Mech. Anal.}, 88 (1985), 95. doi: 10.1007/BF00250907. Google Scholar

[11]

P. Embid and A. Majda, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers,, \emph{Geosphys. Astrophys. Fluid Dynam.}, 87 (1998), 1. doi: 10.1080/03091929808208993. Google Scholar

[12]

J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung,, \emph{Phys. Chem.}, 13 (1894), 657. Google Scholar

[13]

E. Grenier, Oscillartory perturbations of the Navier-Stokes equations,, \emph{J. Math. Pures Appl. (9)}, 76 (1997), 477. doi: 10.1016/S0021-7824(97)89959-X. Google Scholar

[14]

I. Gamba, A. Jüngel and A. Vasseur, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations,, \emph{J. Differential Equations}, 247 (2009), 3117. doi: 10.1016/j.jde.2009.09.001. Google Scholar

[15]

H. Hattori and D. Li, Global solutions of a high-dimensional system for Korteweg materials,, \emph{J. Math. Anal. Appl.}, 198 (1996), 84. doi: 10.1006/jmaa.1996.0069. Google Scholar

[16]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type,, \emph{J. Math. Fluid Mech.}, 13 (2011), 223. doi: 10.1007/s00021-009-0013-2. Google Scholar

[17]

A. Jüngel, C.-K. Lin and K.-C. Wu, An asymptotic limit of a Navier-Stokes system with capillary effects,, \emph{Commun. Math. Phys.}, 329 (2014), 725. doi: 10.1007/s00220-014-1961-9. Google Scholar

[18]

S. Jiang and Y. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with well-prepared initial data in three-dimensional bounded domains,, \emph{J. Math. Pures Appl. (9)}, 96 (2011), 1. doi: 10.1016/j.matpur.2011.01.004. Google Scholar

[19]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids,, \emph{SIAM J. Math. Anal.}, 42 (2010), 1025. doi: 10.1137/090776068. Google Scholar

[20]

T. Kato, Nonstationary flow of viscous and ideal fluids in $\mathbbR^3$,, \emph{J. Funct. Anal.}, 9 (1972), 296. doi: 10.1016/0022-1236(72)90003-1. Google Scholar

[21]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité,, \emph{Arch. Néer. Sci. Exactes Sér. II}, 6 (1901), 1. Google Scholar

[22]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type,, \emph{Ann. Inst. H. Poincaré, 25 (2008), 679. doi: 10.1016/j.anihpc.2007.03.005. Google Scholar

[23]

M. Kotschote, Dynamics of compressible non-isothermal fluids of non-Newtonian Korteweg type,, \emph{SIAM J. Math. Anal.}, 44 (2012), 74. doi: 10.1137/110821202. Google Scholar

[24]

Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force,, \emph{J. Math. Anal. Appl.}, 388 (2012), 1218. doi: 10.1016/j.jmaa.2011.11.006. Google Scholar

[25]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid,, \emph{J. Math. Pures Appl. (9)}, 77 (1998), 585. doi: 10.1016/S0021-7824(98)80139-6. Google Scholar

[26]

N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data,, \emph{Commun. Pure and Appl. Math.}, 53 (2000), 432. Google Scholar

[27]

F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids,, \emph{Arch. Rational Mech. Anal.}, 27 (1967), 329. Google Scholar

[28]

C. Rohde, On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions,, \emph{ZAMM Z. Angew. Math. Mech.}, 85 (2005), 839. doi: 10.1002/zamm.200410211. Google Scholar

[29]

Z. Tan, H. Wang and J. Xu, Global existence and optimal $L^2$ decay rate for the strong solutions to the compressible fluid models of Korteweg type,, \emph{J. Math. Anal. Appl.}, 390 (2012), 181. doi: 10.1016/j.jmaa.2012.01.028. Google Scholar

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