# American Institute of Mathematical Sciences

June  2017, 22(4): 1361-1392. doi: 10.3934/dcdsb.2017066

## Vanishing capillarity limit of the non-conservative compressible two-fluid model

 1 School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China 2 School of Mathematics, South China University of Technology, Guangzhou, 510641, China, School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

* Corresponding author: Lei Yao

Received  February 2016 Revised  December 2016 Published  February 2017

Fund Project: Lai and Yao were supported by the National Natural Science Foundation of China #11571280,11331005, FANEDD #201315, Science and Technology Program of Shaanxi Province #2013KJXX-23, and China Scholarship Council. Wen was supported by the National Natural Science Foundation of China #11671150,11301205 and the Fundamental Research Funds for the Central Universities #D2154560

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 $σ^+$ and $σ^-$, 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 $σ^+$ and $σ^-$ tend to zero. Also, we give the convergence rate estimates with respect to the capillary coefficients $σ^+$ and $σ^-$ for any given positive time.

Citation: Jin Lai, Huanyao Wen, Lei Yao. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1361-1392. doi: 10.3934/dcdsb.2017066
##### References:
 [1] R. Adams, Sobolev Spaces, Springer-Verlag, New York, 1985. Google Scholar [2] J. Bear, Dynamics of Fluids in Porous Media, Environmental Science Series, New York: Elsevier; 1972. reprinted with corrections, New York: Dover; 1988.Google Scholar [3] D. F. Bian, L. Yao and C. J. Zhu, Vanishing capillarity limit of the compressible fluid models of korteweg type to the Navier--Stokes equations, SIAM J. Math. Anal., 46 (2014), 1633-1650. doi: 10.1137/130942231. Google Scholar [4] D. Bresch, B. Desjardins, J.-M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629. doi: 10.1007/s00205-009-0261-6. Google Scholar [5] D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservation viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755. doi: 10.1007/s00220-011-1379-6. Google Scholar [6] H. B. Cui, W. J. Wang, L. Yao and C. J. Zhu, Decay rates for a nonconservative compressible generic two-fluid model, SIAM J. Math. Anal., 48 (2016), 470-512. doi: 10.1137/15M1037792. Google Scholar [7] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Koreteweg type, Ann. Inst. H. Pincare Anal. Non Lineaire, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar [8] R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008. Google Scholar [9] R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X. Google Scholar [10] R. J. Duan, L. Z. Ruan and C. J. Zhu, Optimal decay rates to conservation laws with diffusion type terms of regularity-gain and regularity-loss, Math. Models Methods Appl. Sci., 22 (2012), 1250012, 39 pp. doi: 10.1142/S0218202512500121. Google Scholar [11] S. Evje, W. J. Wang and H. Y. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Rational Mech. Anal., 221 (2016), 1285-1316. doi: 10.1007/s00205-016-0984-0. Google Scholar [12] S. Evje, H. Y. Wen and L. Yao, Global solutions to a one-dimensional non-conservative two-phase model, Discrete Contin. Dyn. Syst., 36 (2016), 1927-1955. doi: 10.3934/dcds.2016.36.1927. Google Scholar [13] H. Hattori and D. Li, Solutions for two-dimensional stytem for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X. Google Scholar [14] H. Hattori and D. Li, Global Solutions of a high-dimensional stytem for Korteweg type materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069. Google Scholar [15] H. Hattori and D. Li, The existence of global Solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Differential Equations, 9 (1996), 323-342. Google Scholar [16] D. Hoff and K. Zumbrum, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614. doi: 10.1007/s000330050049. Google Scholar [17] M. Ishii, Thremo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, paris, 1975. Google Scholar [18] S. Kawashima, Systems of Hyperbolic-Parabolic Comprosite Type, with Applications to the Equations of Msgnetohydrodynsmics, Kyoto Unvisity, 1983.Google Scholar [19] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, H. Poincaré Anal. Non Linéaire, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005. Google Scholar [20] D. L. Li, The Green's function of the Navier-Stokes equations for the gas dynamics in $\mathbb{R}^3$, Comm. Math. Phys., 257 (2005), 579-619. doi: 10.1007/s00220-005-1351-4. Google Scholar [21] T. P. Liu and W. K. Wang, The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions, Comm. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418. Google Scholar [22] A. J. Madjda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. Google Scholar [23] A. Matsumura and T. Nishida, The intial value problem for the equation of motion of compressible viscous and heat-conductive gases, J. Math. Kyoto Univ, 20 (1980), 67-104. Google Scholar [24] A. Prosperertti, Computational Methods for Multiphase Flow, Cambridge University Press, 2007. Google Scholar [25] X. K. Pu and B. L. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Relat. Models, 9 (2016), 165-191. doi: 10.3934/krm.2016.9.165. Google Scholar [26] I. E. Segal, Quantization and dispersion for nonlinear relativistic equations, Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA,, (1996), 79-108. Google Scholar [27] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. Google Scholar [28] M. E. Taylor, Partial Differential Equations Ⅲ: Nonlinear Equations, Springer, New York, 1997. Google Scholar [29] Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar

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##### References:
 [1] R. Adams, Sobolev Spaces, Springer-Verlag, New York, 1985. Google Scholar [2] J. Bear, Dynamics of Fluids in Porous Media, Environmental Science Series, New York: Elsevier; 1972. reprinted with corrections, New York: Dover; 1988.Google Scholar [3] D. F. Bian, L. Yao and C. J. Zhu, Vanishing capillarity limit of the compressible fluid models of korteweg type to the Navier--Stokes equations, SIAM J. Math. Anal., 46 (2014), 1633-1650. doi: 10.1137/130942231. Google Scholar [4] D. Bresch, B. Desjardins, J.-M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629. doi: 10.1007/s00205-009-0261-6. Google Scholar [5] D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservation viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755. doi: 10.1007/s00220-011-1379-6. Google Scholar [6] H. B. Cui, W. J. Wang, L. Yao and C. J. Zhu, Decay rates for a nonconservative compressible generic two-fluid model, SIAM J. Math. Anal., 48 (2016), 470-512. doi: 10.1137/15M1037792. Google Scholar [7] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Koreteweg type, Ann. Inst. H. Pincare Anal. Non Lineaire, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar [8] R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p$-$L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008. Google Scholar [9] R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X. Google Scholar [10] R. J. Duan, L. Z. Ruan and C. J. Zhu, Optimal decay rates to conservation laws with diffusion type terms of regularity-gain and regularity-loss, Math. Models Methods Appl. Sci., 22 (2012), 1250012, 39 pp. doi: 10.1142/S0218202512500121. Google Scholar [11] S. Evje, W. J. Wang and H. Y. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Rational Mech. Anal., 221 (2016), 1285-1316. doi: 10.1007/s00205-016-0984-0. Google Scholar [12] S. Evje, H. Y. Wen and L. Yao, Global solutions to a one-dimensional non-conservative two-phase model, Discrete Contin. Dyn. Syst., 36 (2016), 1927-1955. doi: 10.3934/dcds.2016.36.1927. Google Scholar [13] H. Hattori and D. Li, Solutions for two-dimensional stytem for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X. Google Scholar [14] H. Hattori and D. Li, Global Solutions of a high-dimensional stytem for Korteweg type materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069. Google Scholar [15] H. Hattori and D. Li, The existence of global Solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Differential Equations, 9 (1996), 323-342. Google Scholar [16] D. Hoff and K. Zumbrum, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614. doi: 10.1007/s000330050049. Google Scholar [17] M. Ishii, Thremo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, paris, 1975. Google Scholar [18] S. Kawashima, Systems of Hyperbolic-Parabolic Comprosite Type, with Applications to the Equations of Msgnetohydrodynsmics, Kyoto Unvisity, 1983.Google Scholar [19] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, H. Poincaré Anal. Non Linéaire, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005. Google Scholar [20] D. L. Li, The Green's function of the Navier-Stokes equations for the gas dynamics in $\mathbb{R}^3$, Comm. Math. Phys., 257 (2005), 579-619. doi: 10.1007/s00220-005-1351-4. Google Scholar [21] T. P. Liu and W. K. Wang, The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions, Comm. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418. Google Scholar [22] A. J. Madjda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. Google Scholar [23] A. Matsumura and T. Nishida, The intial value problem for the equation of motion of compressible viscous and heat-conductive gases, J. Math. Kyoto Univ, 20 (1980), 67-104. Google Scholar [24] A. Prosperertti, Computational Methods for Multiphase Flow, Cambridge University Press, 2007. Google Scholar [25] X. K. Pu and B. L. Guo, Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction, Kinet. Relat. Models, 9 (2016), 165-191. doi: 10.3934/krm.2016.9.165. Google Scholar [26] I. E. Segal, Quantization and dispersion for nonlinear relativistic equations, Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA,, (1996), 79-108. Google Scholar [27] J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. doi: 10.1137/0521061. Google Scholar [28] M. E. Taylor, Partial Differential Equations Ⅲ: Nonlinear Equations, Springer, New York, 1997. Google Scholar [29] Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar
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