×

Vanishing capillarity limit of the non-conservative compressible two-fluid model. (English) Zbl 1360.76335

Summary: In this paper, we consider the non-conservative compressible two-fluid model with constant viscosity coefficients and unequal pressure function in \(\mathbb{R}^3\), we study the vanishing capillarity limit of the smooth solution to the initial value problem. We first establish the uniform estimates of global smooth solution with respect to the capillary coefficients \(\sigma^+\) and \(\sigma^-\), then by the Lion-Aubin lemma, we can obtain the unique smooth solution of the 3D non-conservative compressible two-fluid model with the capillary coefficients converges globally in time to the smooth solution of the 3D generic two-fluid model as \(\sigma^+\) and \(\sigma^-\) tend to zero. Also, we give the convergence rate estimates with respect to the capillary coefficients \(\sigma^+\) and \(\sigma^-\) for any given positive time.

MSC:

76T10 Liquid-gas two-phase flows, bubbly flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Adams, <em>Sobolev Spaces</em>,, Springer-Verlag (1985)
[2] J. Bear, <em>Dynamics of Fluids in Porous Media, Environmental Science Series</em>,, New York: Elsevier; 1972. reprinted with corrections (1972) · Zbl 1191.76001
[3] D. F. Bian, Vanishing capillarity limit of the compressible fluid models of korteweg type to the Navier-Stokes equations,, SIAM J. Math. Anal., 46, 1633 (2014) · Zbl 1304.35531 · doi:10.1137/130942231
[4] D. Bresch, Global weak solutions to a generic two-fluid model,, Arch. Rational Mech. Anal., 196, 599 (2010) · Zbl 1193.35146 · doi:10.1007/s00205-009-0261-6
[5] D. Bresch, Global weak solutions to one-dimensional non-conservation viscous compressible two-phase system,, Comm. Math. Phys., 309, 737 (2012) · Zbl 1235.76182 · doi:10.1007/s00220-011-1379-6
[6] H. B. Cui, Decay rates for a nonconservative compressible generic two-fluid model,, SIAM J. Math. Anal., 48, 470 (2016) · Zbl 1331.76120 · doi:10.1137/15M1037792
[7] R. Danchin, Existence of solutions for compressible fluid models of Koreteweg type,, Ann. Inst. H. Pincare Anal. Non Lineaire, 18, 97 (2001) · Zbl 1010.76075 · doi:10.1016/S0294-1449(00)00056-1
[8] R. J. Duan, Optimal \(L^p-L^q\) convergence rates for the compressible Navier-Stokes equations with potential force,, J. Differential Equations, 238, 220 (2007) · Zbl 1121.35096 · doi:10.1016/j.jde.2007.03.008
[9] R. J. Duan, Optimal convergence rates for the compressible Navier-Stokes equations with potential force,, Math. Models Methods Appl. Sci., 17, 737 (2007) · Zbl 1122.35093 · doi:10.1142/S021820250700208X
[10] R. J. Duan, Optimal decay rates to conservation laws with diffusion type terms of regularity-gain and regularity-loss,, Math. Models Methods Appl. Sci., 22 (2012) · Zbl 1241.35133 · doi:10.1142/S0218202512500121
[11] S. Evje, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model,, Arch. Rational Mech. Anal., 221, 1285 (2016) · Zbl 1344.35094 · doi:10.1007/s00205-016-0984-0
[12] S. Evje, Global solutions to a one-dimensional non-conservative two-phase model,, Discrete Contin. Dyn. Syst., 36, 1927 (2016) · Zbl 1327.76152 · doi:10.3934/dcds.2016.36.1927
[13] H. Hattori, Solutions for two-dimensional stytem for materials of Korteweg type,, SIAM J. Math. Anal., 25, 85 (1994) · Zbl 0817.35076 · doi:10.1137/S003614109223413X
[14] H. Hattori, Global Solutions of a high-dimensional stytem for Korteweg type materials,, J. Math. Anal. Appl., 198, 84 (1996) · Zbl 0858.35124 · doi:10.1006/jmaa.1996.0069
[15] H. Hattori, The existence of global Solutions to a fluid dynamic model for materials for Korteweg type,, J. Partial Differential Equations, 9, 323 (1996) · Zbl 0881.35095
[16] D. Hoff, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves,, Z. Angew. Math. Phys., 48, 597 (1997) · Zbl 0882.76074 · doi:10.1007/s000330050049
[17] M. Ishii, <em>Thremo-Fluid Dynamic Theory of Two-Phase Flow</em>,, paris: Eyrolles (1975)
[18] S. Kawashima, <em>Systems of Hyperbolic-Parabolic Comprosite Type, with Applications to the Equations of Msgnetohydrodynsmics</em>,, Kyoto Unvisity (1983)
[19] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type,, H. Poincaré Anal. Non Linéaire, 25, 679 (2008) · Zbl 1141.76053 · doi:10.1016/j.anihpc.2007.03.005
[20] D. L. Li, The Green’s function of the Navier-Stokes equations for the gas dynamics in \(\mathbbR^3\),, Comm. Math. Phys., 257, 579 (2005) · Zbl 1075.76053 · doi:10.1007/s00220-005-1351-4
[21] T. P. Liu, The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions,, Comm. Math. Phys., 196, 145 (1998) · Zbl 0912.35122 · doi:10.1007/s002200050418
[22] A. J. Madjda, <em>Vorticity and Incompressible Flow</em>,, Cambridge University Press (2002)
[23] A. Matsumura, The intial value problem for the equation of motion of compressible viscous and heat-conductive gases,, J. Math. Kyoto Univ, 20, 67 (1980)
[24] A. Prosperertti, <em>Computational Methods for Multiphase Flow</em>,, Cambridge University Press (2007)
[25] X. K. Pu, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction,, Kinet. Relat. Models, 9, 165 (2016) · Zbl 1330.35339 · doi:10.3934/krm.2016.9.165
[26] I. E. Segal, Quantization and dispersion for nonlinear relativistic equations,, Mathematical Theory of Elementary Particles, 79 (1996)
[27] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure,, SIAM J. Math. Anal., 21, 1093 (1990) · Zbl 0702.76039 · doi:10.1137/0521061
[28] M. E. Taylor, <em>Partial Differential Equations III: Nonlinear Equations</em>,, Springer (1997)
[29] Y. J. Wang, Optimal decay rates for the compressible fluid models of Korteweg type,, J. Math. Anal. Appl., 379, 256 (2011) · Zbl 1211.35228 · doi:10.1016/j.jmaa.2011.01.006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.