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Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. (English) Zbl 1332.35270

Summary: 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.

MSC:

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
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[1] T. Alazard, Low Mach number limit of the full Navier-Stokes equations,, Arch. Rational Mech. Anal., 180, 1 (2006) · Zbl 1108.76061 · doi:10.1007/s00205-005-0393-2
[2] S. Benzoni-Gavage, Well-posedness of one-dimensional Korteweg models,, Electron. J. Differential Equations (2006) · Zbl 1114.76058
[3] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25, 737 (2000) · Zbl 0970.35110 · doi:10.1080/03605300008821529
[4] D. Bresch, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,, Comm. Math. Phys., 238, 211 (2003) · Zbl 1037.76012 · doi:10.1007/s00220-003-0859-8
[5] D. Bresch, On some compressible fluid models: Korteweg, lubrication and shallow water systems,, Comm. Partial Differential Equations, 28, 843 (2003) · Zbl 1106.76436 · doi:10.1081/PDE-120020499
[6] D. Bresch, Quasi-neutral limit for a viscous capillary model of plasma,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22, 1 (2005) · Zbl 1062.35061 · doi:10.1016/j.anihpc.2004.02.001
[7] Z. Chen, Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system,, J. Math. Pures Appl. (9), 101, 330 (2014) · Zbl 1288.35387 · doi:10.1016/j.matpur.2013.06.005
[8] F. Charve, Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system,, SIAM J. Math. Anal., 45, 469 (2013) · Zbl 1294.35091 · doi:10.1137/120861801
[9] R. Danchin, Existence of solutions for compressible fluid models of Korteweg type,, Ann. Inst. H. Poincaré, 18, 97 (2001) · Zbl 1010.76075 · doi:10.1016/S0294-1449(00)00056-1
[10] J. E. Dunn, On the thermomechanics of interstitial working,, Arch. Rational Mech. Anal., 88, 95 (1985) · Zbl 0582.73004 · doi:10.1007/BF00250907
[11] P. Embid, Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers,, Geosphys. Astrophys. Fluid Dynam., 87, 1 (1998) · doi:10.1080/03091929808208993
[12] J. D. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung,, Phys. Chem., 13, 657 (1894)
[13] E. Grenier, Oscillartory perturbations of the Navier-Stokes equations,, J. Math. Pures Appl. (9), 76, 477 (1997) · Zbl 0885.35090 · doi:10.1016/S0021-7824(97)89959-X
[14] I. Gamba, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations,, J. Differential Equations, 247, 3117 (2009) · Zbl 1181.35209 · doi:10.1016/j.jde.2009.09.001
[15] H. Hattori, Global solutions of a high-dimensional system for Korteweg materials,, J. Math. Anal. Appl., 198, 84 (1996) · Zbl 0858.35124 · 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, 223 (2011) · Zbl 1270.35366 · doi:10.1007/s00021-009-0013-2
[17] A. Jüngel, An asymptotic limit of a Navier-Stokes system with capillary effects,, Commun. Math. Phys., 329, 725 (2014) · Zbl 1297.35169 · doi:10.1007/s00220-014-1961-9
[18] S. Jiang, 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, 1 (2011) · Zbl 1283.35063 · 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, 1025 (2010) · Zbl 1228.35083 · doi:10.1137/090776068
[20] T. Kato, Nonstationary flow of viscous and ideal fluids in \(\mathbbR^3\),, J. Funct. Anal., 9, 296 (1972) · Zbl 0229.76018 · 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, 1 (1901) · JFM 32.0756.02
[22] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type,, Ann. Inst. H. Poincaré, 25, 679 (2008) · Zbl 1141.76053 · 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, 74 (2012) · Zbl 1332.76047 · 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, 1218 (2012) · Zbl 1231.35173 · doi:10.1016/j.jmaa.2011.11.006
[25] P.-L. Lions, Incompressible limit for a viscous compressible fluid,, J. Math. Pures Appl. (9), 77, 585 (1998) · Zbl 0909.35101 · 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, 432 (2000) · Zbl 1047.76124
[27] F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids,, Arch. Rational Mech. Anal., 27, 329 (1967) · Zbl 0187.49508
[28] C. Rohde, On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions,, ZAMM Z. Angew. Math. Mech., 85, 839 (2005) · Zbl 1099.76072 · doi:10.1002/zamm.200410211
[29] Z. Tan, 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, 181 (2012) · Zbl 1238.35089 · doi:10.1016/j.jmaa.2012.01.028
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