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

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

Jin Lai 1, , Huanyao Wen 2, and  Lei Yao 2,,

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.

Keywords: Non-conservative compressible two-fluid model, global existence and uniqueness, vanishing capillarity limit, energy estimates.
Mathematics Subject Classification: Primary:76T10, 76N10.
Citation: Jin Lai, Huanyao Wen, Lei Yao. Vanishing capillarity limit of the non-conservative compressible two-fluid model. Discrete and 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. 
[2]

J. Bear, Dynamics of Fluids in Porous Media, Environmental Science Series, New York: Elsevier; 1972. reprinted with corrections, New York: Dover; 1988.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[17] M. Ishii, Thremo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, paris, 1975. 
[18]

S. Kawashima, Systems of Hyperbolic-Parabolic Comprosite Type, with Applications to the Equations of Msgnetohydrodynsmics, Kyoto Unvisity, 1983.

[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.

[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.

[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.

[22] A. J. Madjda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. 
[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. 

[24] A. Prosperertti, Computational Methods for Multiphase Flow, Cambridge University Press, 2007. 
[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.

[26]

I. E. Segal, Quantization and dispersion for nonlinear relativistic equations, Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA,, (1996), 79-108. 

[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.

[28] M. E. Taylor, Partial Differential Equations Ⅲ: Nonlinear Equations, Springer, New York, 1997. 
[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.

show all references

References:
[1] R. Adams, Sobolev Spaces, Springer-Verlag, New York, 1985. 
[2]

J. Bear, Dynamics of Fluids in Porous Media, Environmental Science Series, New York: Elsevier; 1972. reprinted with corrections, New York: Dover; 1988.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[17] M. Ishii, Thremo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, paris, 1975. 
[18]

S. Kawashima, Systems of Hyperbolic-Parabolic Comprosite Type, with Applications to the Equations of Msgnetohydrodynsmics, Kyoto Unvisity, 1983.

[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.

[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.

[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.

[22] A. J. Madjda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. 
[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. 

[24] A. Prosperertti, Computational Methods for Multiphase Flow, Cambridge University Press, 2007. 
[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.

[26]

I. E. Segal, Quantization and dispersion for nonlinear relativistic equations, Mathematical Theory of Elementary Particles, MIT Press, Cambridge, MA,, (1996), 79-108. 

[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.

[28] M. E. Taylor, Partial Differential Equations Ⅲ: Nonlinear Equations, Springer, New York, 1997. 
[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.

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