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

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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.
    Mathematics Subject Classification: 35Q30, 35B40, 35C20.


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  • [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.


    S. Benzoni-Gavage, R. Danchin and S. Descombes, Well-posedness of one-dimensional Korteweg models, Electron. J. Differential Equations, (2006), 35pp.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.doi: 10.1007/BF00250907.


    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.


    J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung, Phys. Chem., 13 (1894), 657-725.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    N. Masmoudi, Ekman layers of rotating fluids: The case of general initial data, Commun. Pure and Appl. Math., 53 (2000), 432-483.


    F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal., 27 (1967), 329-348.


    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.


    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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