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Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity
1. | College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450045, Henan Province, China, China, China |
References:
[1] |
T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Rational Mech. Anal., 180 (2006), 1-73.
doi: 10.1007/s00205-005-0393-2. |
[2] |
S. Benzoni-Gavage, R. Danchin and S. Descombes, Well-posedness of one-dimensional Korteweg models, Electron. J. Differential Equations, (2006), 35pp. |
[3] |
Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.
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, Comm. Math. Phys., 238 (2003), 211-223.
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, Comm. Partial Differential Equations, 28 (2003), 843-868.
doi: 10.1081/PDE-120020499. |
[6] |
D. Bresch, B. Desjardins and B. Ducomet, Quasi-neutral limit for a viscous capillary model of plasma, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 1-9.
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, J. Math. Pures Appl. (9), 101 (2014), 330-371.
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, SIAM J. Math. Anal., 45 (2013), 469-494.
doi: 10.1137/120861801. |
[9] |
R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 18 (2001), 97-133.
doi: 10.1016/S0294-1449(00)00056-1. |
[10] |
J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.
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, Geosphys. Astrophys. Fluid Dynam., 87 (1998), 1-50.
doi: 10.1080/03091929808208993. |
[12] |
J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung, Phys. Chem., 13 (1894), 657-725. |
[13] |
E. Grenier, Oscillartory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9), 76 (1997), 477-498.
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, J. Differential Equations, 247 (2009), 3117-3135.
doi: 10.1016/j.jde.2009.09.001. |
[15] |
H. Hattori and D. Li, Global solutions of a high-dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97.
doi: 10.1006/jmaa.1996.0069. |
[16] |
B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249.
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, Commun. Math. Phys., 329 (2014), 725-744.
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, J. Math. Pures Appl. (9), 96 (2011), 1-28.
doi: 10.1016/j.matpur.2011.01.004. |
[19] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.
doi: 10.1137/090776068. |
[20] |
T. Kato, Nonstationary flow of viscous and ideal fluids in $\mathbbR^3$, J. Funct. Anal., 9 (1972), 296-305.
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, Arch. Néer. Sci. Exactes Sér. II, 6 (1901), 1-24. |
[22] |
M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 25 (2008), 679-696.
doi: 10.1016/j.anihpc.2007.03.005. |
[23] |
M. Kotschote, Dynamics of compressible non-isothermal fluids of non-Newtonian Korteweg type, SIAM J. Math. Anal., 44 (2012), 74-101.
doi: 10.1137/110821202. |
[24] |
Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232.
doi: 10.1016/j.jmaa.2011.11.006. |
[25] |
P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 77 (1998), 585-627.
doi: 10.1016/S0021-7824(98)80139-6. |
[26] |
N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data, Commun. Pure and Appl. Math., 53 (2000), 432-483. |
[27] |
F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal., 27 (1967), 329-348. |
[28] |
C. Rohde, On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions, ZAMM Z. Angew. Math. Mech., 85 (2005), 839-857.
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, J. Math. Anal. Appl., 390 (2012), 181-187.
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, Arch. Rational Mech. Anal., 180 (2006), 1-73.
doi: 10.1007/s00205-005-0393-2. |
[2] |
S. Benzoni-Gavage, R. Danchin and S. Descombes, Well-posedness of one-dimensional Korteweg models, Electron. J. Differential Equations, (2006), 35pp. |
[3] |
Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.
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, Comm. Math. Phys., 238 (2003), 211-223.
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, Comm. Partial Differential Equations, 28 (2003), 843-868.
doi: 10.1081/PDE-120020499. |
[6] |
D. Bresch, B. Desjardins and B. Ducomet, Quasi-neutral limit for a viscous capillary model of plasma, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 1-9.
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, J. Math. Pures Appl. (9), 101 (2014), 330-371.
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, SIAM J. Math. Anal., 45 (2013), 469-494.
doi: 10.1137/120861801. |
[9] |
R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 18 (2001), 97-133.
doi: 10.1016/S0294-1449(00)00056-1. |
[10] |
J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.
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, Geosphys. Astrophys. Fluid Dynam., 87 (1998), 1-50.
doi: 10.1080/03091929808208993. |
[12] |
J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung, Phys. Chem., 13 (1894), 657-725. |
[13] |
E. Grenier, Oscillartory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9), 76 (1997), 477-498.
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, J. Differential Equations, 247 (2009), 3117-3135.
doi: 10.1016/j.jde.2009.09.001. |
[15] |
H. Hattori and D. Li, Global solutions of a high-dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97.
doi: 10.1006/jmaa.1996.0069. |
[16] |
B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249.
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, Commun. Math. Phys., 329 (2014), 725-744.
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, J. Math. Pures Appl. (9), 96 (2011), 1-28.
doi: 10.1016/j.matpur.2011.01.004. |
[19] |
A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.
doi: 10.1137/090776068. |
[20] |
T. Kato, Nonstationary flow of viscous and ideal fluids in $\mathbbR^3$, J. Funct. Anal., 9 (1972), 296-305.
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, Arch. Néer. Sci. Exactes Sér. II, 6 (1901), 1-24. |
[22] |
M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 25 (2008), 679-696.
doi: 10.1016/j.anihpc.2007.03.005. |
[23] |
M. Kotschote, Dynamics of compressible non-isothermal fluids of non-Newtonian Korteweg type, SIAM J. Math. Anal., 44 (2012), 74-101.
doi: 10.1137/110821202. |
[24] |
Y. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232.
doi: 10.1016/j.jmaa.2011.11.006. |
[25] |
P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), 77 (1998), 585-627.
doi: 10.1016/S0021-7824(98)80139-6. |
[26] |
N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data, Commun. Pure and Appl. Math., 53 (2000), 432-483. |
[27] |
F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal., 27 (1967), 329-348. |
[28] |
C. Rohde, On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions, ZAMM Z. Angew. Math. Mech., 85 (2005), 839-857.
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, J. Math. Anal. Appl., 390 (2012), 181-187.
doi: 10.1016/j.jmaa.2012.01.028. |
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