-
Previous Article
Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays
- DCDS-B Home
- This Issue
-
Next Article
Competition in periodic media:Ⅰ-Existence of pulsating fronts
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 |
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.
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. |
[1] |
Fuyi Xu, Meiling Chi, Lishan Liu, Yonghong Wu. On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2515-2559. doi: 10.3934/dcds.2020140 |
[2] |
Steinar Evje, Huanyao Wen, Lei Yao. Global solutions to a one-dimensional non-conservative two-phase model. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1927-1955. doi: 10.3934/dcds.2016.36.1927 |
[3] |
Huanyao Wen, Changjiang Zhu. Remarks on global weak solutions to a two-fluid type model. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2839-2856. doi: 10.3934/cpaa.2021072 |
[4] |
Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046 |
[5] |
Xuemei Li, Zaijiu Shang. On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4225-4257. doi: 10.3934/dcds.2019171 |
[6] |
Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009 |
[7] |
Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239 |
[8] |
Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415 |
[9] |
Long Fan, Cheng-Jie Liu, Lizhi Ruan. Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow. Electronic Research Archive, 2021, 29 (6) : 4009-4050. doi: 10.3934/era.2021070 |
[10] |
Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577 |
[11] |
Wenjun Wang, Lei Yao. Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2331-2350. doi: 10.3934/cpaa.2014.13.2331 |
[12] |
Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 |
[13] |
Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3423-3434. doi: 10.3934/dcds.2017145 |
[14] |
Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, Nikolaos Sfakianakis. Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4397-4431. doi: 10.3934/dcdsb.2018169 |
[15] |
Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Erratum to: Investigating the steady state of multicellular sheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 697-697. doi: 10.3934/mbe.2012.9.697 |
[16] |
Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039 |
[17] |
Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic and Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043 |
[18] |
Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure and Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010 |
[19] |
Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101 |
[20] |
J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]