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

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January 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

Abstract
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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]	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.
[2]	S. Benzoni-Gavage, R. Danchin and S. Descombes, Well-posedness of one-dimensional Korteweg models,, \emph{Electron. J. Differential Equations, (2006).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung,, \emph{Phys. Chem.}, 13 (1894), 657.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data,, \emph{Commun. Pure and Appl. Math.}, 53 (2000), 432.
[27]	F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids,, \emph{Arch. Rational Mech. Anal.}, 27 (1967), 329.
[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.
[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.

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.
[2]	S. Benzoni-Gavage, R. Danchin and S. Descombes, Well-posedness of one-dimensional Korteweg models,, \emph{Electron. J. Differential Equations, (2006).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung,, \emph{Phys. Chem.}, 13 (1894), 657.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data,, \emph{Commun. Pure and Appl. Math.}, 53 (2000), 432.
[27]	F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids,, \emph{Arch. Rational Mech. Anal.}, 27 (1967), 329.
[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.
[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.

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