• Previous Article
    The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law
  • CPAA Home
  • This Issue
  • Next Article
    Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions
July & August  2021, 20(7&8): 2839-2856. doi: 10.3934/cpaa.2021072

Remarks on global weak solutions to a two-fluid type model

School of Mathematics, South China University of Technology, Guangzhou 510641, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the occasion of his 80th birthday

Received  January 2021 Revised  March 2021 Published  July & August 2021 Early access  April 2021

Fund Project: H. Wen was partially supported by the National Natural Science Foundation of China # 12071152 and #11671150. C. Zhu was partially supported by the National Natural Science Foundation of China #11771150, #11831003 and #11926346, and by Guangdong Basic and Applied Basic Research Foundation #2020B1515310015

The present paper aims to give a review of a two-fluid type model mostly on large-data solutions. Some derivations of the model arising in different physical background will be introduced. In addition, we will sketch the proof of global existence of weak solutions to the Dirichlet problem for the model in one dimension with more general pressure law which can be non-monotone, in the context of allowing unconstrained transition to single-phase flow.

Citation: 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
References:
[1]

J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system, hyperbolic problems: theory, numerics, applications, AIMS Series on Applied Mathematics, 8 (2014), 301-308. 

[2]

J. Ballew and K. Trivisa, Weakly dissipative solutions and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system, Nonlinear Anal., 91 (2013), 1-19.  doi: 10.1016/j.na.2013.06.002.

[3]

J. W. BarrettY. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15 (2017), 1265-1323.  doi: 10.4310/CMS.2017.v15.n5.a5.

[4]

S. BerresR. BürgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.

[5]

D. BreschB. DesjardinsJ. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Ration. Mech. Anal., 196 (2010), 599-629.  doi: 10.1007/s00205-009-0261-6.

[6]

D. BreschX. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Commun. Math. Phys., 309 (2012), 737-755.  doi: 10.1007/s00220-011-1379-6.

[7]

D. Bresch and P. E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. Math., 188 (2018), 577-684.  doi: 10.4007/annals.2018.188.2.4.

[8]

D. BreschP. B. Mucha and E. Zatorska, Finite-energy solutions for compressible two-fluid Stokes system, Arch. Ration. Mech. Anal., 232 (2019), 987-1029.  doi: 10.1007/s00205-018-01337-6.

[9]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.

[10]

J. A. CarrilloT. Karper and K. Trivisa, On the dynamics of a fluid-particle interaction model: the bubbling regime, Nonlinear Anal., 74 (2011), 2778-2801.  doi: 10.1016/j.na.2010.12.031.

[11]

Y. S. ChenS. J. Ding and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst A., 36 (2016), 5287-5307.  doi: 10.3934/dcds.2016032.

[12]

H. B. CuiW. J. WangL. 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.

[13]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.

[14]

S. J. DingB. Y. Huang and H. Y. Wen, Global well-posedness of classical solutions to a fluid-particle interaction model in $\mathbb{R}^3$, J. Differ. Equ., 263 (2017), 8666-8717.  doi: 10.1016/j.jde.2017.08.048.

[15]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.

[16]

D. Y. Fang, R. Z. Zi and T. Zhang, Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys., 53 (2012), 033706. doi: 10.1063/1.3693979.

[17]

S. EvjeT. Flåtten and H. A. Friis, Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum, Nonlinear Anal., TMA, 70 (2009), 3864-3886.  doi: 10.1016/j.na.2008.07.043.

[18]

S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Differ. Equ., 245 (2008), 2660-2703.  doi: 10.1016/j.jde.2007.10.032.

[19]

S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Commun. Pure Appl. Anal., 8 (2009), 1867-1894.  doi: 10.3934/cpaa.2009.8.1867.

[20]

S. EvjeW. J. Wang and H. Y. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Ration. Mech. Anal., 221 (2016), 1285-1316.  doi: 10.1007/s00205-016-0984-0.

[21]

S. EvjeH. Y. Wen and C. J. Zhu, On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow, Math. Models Methods Appl. Sci., 27 (2017), 323-346.  doi: 10.1142/S0218202517500038.

[22]

S. EvjeH. 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.

[23]

E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differ. Equ., 184 (2002), 97-108.  doi: 10.1006/jdeq.2001.4137.

[24]

E. FeireislA. Novotn$\acute{y}$ and H. Petzeltov$\acute{a}$, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[25]

S. L. Gavrilyuk and J. Fabre, Lagrangian coordinates for a drift-flux model of a gas-liquid mixture, Int. J. Multiphase Flow, 22 (1996), 453-460. 

[26]

Z. H. Guo, J. Yang and L. Yao, Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum, J. Math. Phys., 52 (2011), 093102. doi: 10.1063/1.3638039.

[27]

C. C. Hao and H. L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332.  doi: 10.1137/110851602.

[28]

X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[29]

X. P. Hu, Global existence for two dimensional compressible magnetohydrodynamic flows with zero magnetic diffusivity, arXiv: 1405.0274v1.

[30]

B. Y. HuangS. J. Ding and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, Discrete Contin. Dyn. Syst. S., 9 (2016), 1717-1752.  doi: 10.3934/dcdss.2016072.

[31]

B. Y. Huang, J. R. Huang and H. Y. Wen, Low Mach number limit of the compressible Navier-Stokes-Smoluchowski equations in multi-dimensions, J. Math. Phys., 60 (2019), 061501. doi: 10.1063/1.5089229.

[32]

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Fluid Flow, Eyrolles, Paris, 1975.

[33]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[34]

S. Jiang and J. W. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612.  doi: 10.1088/1361-6544/aa82f2.

[35]

B. J. Jin and A. Novotný, Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids, J. Differ. Equ., 268 (2019), 204-238.  doi: 10.1016/j.jde.2019.08.025.

[36]

C. Kleinstreuer, Two Phase Flow: Theory and Applications, Taylor & Francis, 2003.

[37]

H. L. LiX. Y. Xu and J. W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.

[38] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. II, Compressible Models, Clarendon Press, Oxford, 1998. 
[39]

Y. Li and Y. Z. Sun, Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids, J. Differ. Equ., 267 (2019), 3827-3851.  doi: 10.1016/j.jde.2019.04.024.

[40]

Y. LiY. Z. Sun and E. Zatorska, Large time behavior for a compressible two-fluid model with algebraic pressure closure and large initial data, Nonlinearity, 33 (2020), 4075-4094.  doi: 10.1088/1361-6544/ab801c.

[41]

Y. Lu and M. Pokorný, Global existence of large data weak solutions for a simplified compressible Oldroyd-B model without stress diffusion, Anal. Theory Appl., 36 (2020), 348-372.  doi: 10.4208/ata.oa-su3.

[42]

Y. Lu and Z. F. Zhang, Relative entropy, weak-strong uniqueness and conditional regularity for a compressible Oldroyd-B model, SIAM J. Math. Anal., 50 (2018), 557-590.  doi: 10.1137/17M1128654.

[43]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063.  doi: 10.1142/S0218202507002194.

[44]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier- Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.

[45]

A. Novotný and M. Pokorný, Weak solutions for some compressible multicomponent fluid models, Arch. Ration. Mech. Anal., 235 (2020), 355-403.  doi: 10.1007/s00205-019-01424-2.

[46]

A. Novotný, Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids, Sci. China Math., 63 (2020), 2399-2414.  doi: 10.1007/s11425-019-9552-1.

[47]

X. K. Pu and B. L. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538.  doi: 10.1007/s00033-012-0245-5.

[48]

W. K. Sartory, Three-component analysis of blood sedimentation by the method of characteristics, Math. Biosci., 33 (1977), 145-165. 

[49]

A. Spannenberg and K. P. Galvin, Continuous differential sedimentation of a binary suspension, Chem. Engrg. Aust., 21 (1996), 7-11. 

[50]

Z. Tan and Y. J. Wang, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal., 50 (2018), 1432-1470.  doi: 10.1137/16M1088156.

[51]

W. J. Wang and H. Y. Wen, The Cauchy problem for an Oldroyd-B model in three dimensions, Math. Models Methods Appl. Sci., 30 (2020), 139-179.  doi: 10.1142/s0218202520500049.

[52]

F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555. 

[53]

F. A. Williams, Combustion Theory, 2nd edn., Benjamin/Cummings, 1985.

[54]

A. VasseurH. Y. Wen and C. Yu, Global weak solution to the viscous two-fluid model with finite energy, J. Math. Pures Appl., 9 (2019), 247-282.  doi: 10.1016/j.matpur.2018.06.019.

[55]

Y. H. Wang, H. Y. Wen and L. Yao, On a non-conservative compressible two-fluid model in a bounded domain: global existence and uniqueness, J. Math. Fluid Mech., 23 (2021), 24pp. doi: 10.1007/s00021-020-00531-5.

[56]

H. Y. Wen, On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions, arXiv: 1902.05190.

[57]

H. Y. WenL. Yao and C. J. Zhu, A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum, J. Math. Pures Appl., 97 (2012), 204-229.  doi: 10.1016/j.matpur.2011.09.005.

[58]

J. H. Wu and Y. F. Wu, Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion, Adv. Math., 310 (2017), 759-888.  doi: 10.1016/j.aim.2017.02.013.

[59]

L. YaoT. Zhang and C. J. Zhu, Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 42 (2010), 1874-1897.  doi: 10.1137/100785302.

[60]

L. YaoT. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model, J. Differ. Equ., 250 (2011), 3362-3378.  doi: 10.1016/j.jde.2010.12.006.

[61]

L. Yao and C. J. Zhu, Free boundary value problem for a viscous two-phase model with mass-dependent viscosity, J. Differ. Equ., 247 (2009), 2705-2739.  doi: 10.1016/j.jde.2009.07.013.

[62]

L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2011), 903-928.  doi: 10.1007/s00208-010-0544-0.

[63]

X. P. Zhai, Global well-posedness and large time behavior of solutions to the n-dimensional compressible Oldroyd-B model, arXiv: 1912.00372.

show all references

References:
[1]

J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system, hyperbolic problems: theory, numerics, applications, AIMS Series on Applied Mathematics, 8 (2014), 301-308. 

[2]

J. Ballew and K. Trivisa, Weakly dissipative solutions and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system, Nonlinear Anal., 91 (2013), 1-19.  doi: 10.1016/j.na.2013.06.002.

[3]

J. W. BarrettY. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15 (2017), 1265-1323.  doi: 10.4310/CMS.2017.v15.n5.a5.

[4]

S. BerresR. BürgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.

[5]

D. BreschB. DesjardinsJ. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Ration. Mech. Anal., 196 (2010), 599-629.  doi: 10.1007/s00205-009-0261-6.

[6]

D. BreschX. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Commun. Math. Phys., 309 (2012), 737-755.  doi: 10.1007/s00220-011-1379-6.

[7]

D. Bresch and P. E. Jabin, Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. Math., 188 (2018), 577-684.  doi: 10.4007/annals.2018.188.2.4.

[8]

D. BreschP. B. Mucha and E. Zatorska, Finite-energy solutions for compressible two-fluid Stokes system, Arch. Ration. Mech. Anal., 232 (2019), 987-1029.  doi: 10.1007/s00205-018-01337-6.

[9]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.

[10]

J. A. CarrilloT. Karper and K. Trivisa, On the dynamics of a fluid-particle interaction model: the bubbling regime, Nonlinear Anal., 74 (2011), 2778-2801.  doi: 10.1016/j.na.2010.12.031.

[11]

Y. S. ChenS. J. Ding and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst A., 36 (2016), 5287-5307.  doi: 10.3934/dcds.2016032.

[12]

H. B. CuiW. J. WangL. 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.

[13]

C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal., 70 (1979), 167-179.  doi: 10.1007/BF00250353.

[14]

S. J. DingB. Y. Huang and H. Y. Wen, Global well-posedness of classical solutions to a fluid-particle interaction model in $\mathbb{R}^3$, J. Differ. Equ., 263 (2017), 8666-8717.  doi: 10.1016/j.jde.2017.08.048.

[15]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.

[16]

D. Y. Fang, R. Z. Zi and T. Zhang, Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys., 53 (2012), 033706. doi: 10.1063/1.3693979.

[17]

S. EvjeT. Flåtten and H. A. Friis, Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum, Nonlinear Anal., TMA, 70 (2009), 3864-3886.  doi: 10.1016/j.na.2008.07.043.

[18]

S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Differ. Equ., 245 (2008), 2660-2703.  doi: 10.1016/j.jde.2007.10.032.

[19]

S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Commun. Pure Appl. Anal., 8 (2009), 1867-1894.  doi: 10.3934/cpaa.2009.8.1867.

[20]

S. EvjeW. J. Wang and H. Y. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Ration. Mech. Anal., 221 (2016), 1285-1316.  doi: 10.1007/s00205-016-0984-0.

[21]

S. EvjeH. Y. Wen and C. J. Zhu, On global solutions to the viscous liquid-gas model with unconstrained transition to single-phase flow, Math. Models Methods Appl. Sci., 27 (2017), 323-346.  doi: 10.1142/S0218202517500038.

[22]

S. EvjeH. 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.

[23]

E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differ. Equ., 184 (2002), 97-108.  doi: 10.1006/jdeq.2001.4137.

[24]

E. FeireislA. Novotn$\acute{y}$ and H. Petzeltov$\acute{a}$, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[25]

S. L. Gavrilyuk and J. Fabre, Lagrangian coordinates for a drift-flux model of a gas-liquid mixture, Int. J. Multiphase Flow, 22 (1996), 453-460. 

[26]

Z. H. Guo, J. Yang and L. Yao, Global strong solution for a three-dimensional viscous liquid-gas two-phase flow model with vacuum, J. Math. Phys., 52 (2011), 093102. doi: 10.1063/1.3638039.

[27]

C. C. Hao and H. L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332.  doi: 10.1137/110851602.

[28]

X. P. Hu and D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203-238.  doi: 10.1007/s00205-010-0295-9.

[29]

X. P. Hu, Global existence for two dimensional compressible magnetohydrodynamic flows with zero magnetic diffusivity, arXiv: 1405.0274v1.

[30]

B. Y. HuangS. J. Ding and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, Discrete Contin. Dyn. Syst. S., 9 (2016), 1717-1752.  doi: 10.3934/dcdss.2016072.

[31]

B. Y. Huang, J. R. Huang and H. Y. Wen, Low Mach number limit of the compressible Navier-Stokes-Smoluchowski equations in multi-dimensions, J. Math. Phys., 60 (2019), 061501. doi: 10.1063/1.5089229.

[32]

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Fluid Flow, Eyrolles, Paris, 1975.

[33]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Commun. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[34]

S. Jiang and J. W. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612.  doi: 10.1088/1361-6544/aa82f2.

[35]

B. J. Jin and A. Novotný, Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids, J. Differ. Equ., 268 (2019), 204-238.  doi: 10.1016/j.jde.2019.08.025.

[36]

C. Kleinstreuer, Two Phase Flow: Theory and Applications, Taylor & Francis, 2003.

[37]

H. L. LiX. Y. Xu and J. W. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal., 45 (2013), 1356-1387.  doi: 10.1137/120893355.

[38] P. L. Lions, Mathematical Topics in Fluid Mechanics, vol. II, Compressible Models, Clarendon Press, Oxford, 1998. 
[39]

Y. Li and Y. Z. Sun, Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids, J. Differ. Equ., 267 (2019), 3827-3851.  doi: 10.1016/j.jde.2019.04.024.

[40]

Y. LiY. Z. Sun and E. Zatorska, Large time behavior for a compressible two-fluid model with algebraic pressure closure and large initial data, Nonlinearity, 33 (2020), 4075-4094.  doi: 10.1088/1361-6544/ab801c.

[41]

Y. Lu and M. Pokorný, Global existence of large data weak solutions for a simplified compressible Oldroyd-B model without stress diffusion, Anal. Theory Appl., 36 (2020), 348-372.  doi: 10.4208/ata.oa-su3.

[42]

Y. Lu and Z. F. Zhang, Relative entropy, weak-strong uniqueness and conditional regularity for a compressible Oldroyd-B model, SIAM J. Math. Anal., 50 (2018), 557-590.  doi: 10.1137/17M1128654.

[43]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063.  doi: 10.1142/S0218202507002194.

[44]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier- Stokes system of equations, Commun. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.

[45]

A. Novotný and M. Pokorný, Weak solutions for some compressible multicomponent fluid models, Arch. Ration. Mech. Anal., 235 (2020), 355-403.  doi: 10.1007/s00205-019-01424-2.

[46]

A. Novotný, Weak solutions for a bi-fluid model for a mixture of two compressible non interacting fluids, Sci. China Math., 63 (2020), 2399-2414.  doi: 10.1007/s11425-019-9552-1.

[47]

X. K. Pu and B. L. Guo, Global existence and convergence rates of smooth solutions for the full compressible MHD equations, Z. Angew. Math. Phys., 64 (2013), 519-538.  doi: 10.1007/s00033-012-0245-5.

[48]

W. K. Sartory, Three-component analysis of blood sedimentation by the method of characteristics, Math. Biosci., 33 (1977), 145-165. 

[49]

A. Spannenberg and K. P. Galvin, Continuous differential sedimentation of a binary suspension, Chem. Engrg. Aust., 21 (1996), 7-11. 

[50]

Z. Tan and Y. J. Wang, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal., 50 (2018), 1432-1470.  doi: 10.1137/16M1088156.

[51]

W. J. Wang and H. Y. Wen, The Cauchy problem for an Oldroyd-B model in three dimensions, Math. Models Methods Appl. Sci., 30 (2020), 139-179.  doi: 10.1142/s0218202520500049.

[52]

F. A. Williams, Spray combustion and atomization, Phys. Fluids, 1 (1958), 541-555. 

[53]

F. A. Williams, Combustion Theory, 2nd edn., Benjamin/Cummings, 1985.

[54]

A. VasseurH. Y. Wen and C. Yu, Global weak solution to the viscous two-fluid model with finite energy, J. Math. Pures Appl., 9 (2019), 247-282.  doi: 10.1016/j.matpur.2018.06.019.

[55]

Y. H. Wang, H. Y. Wen and L. Yao, On a non-conservative compressible two-fluid model in a bounded domain: global existence and uniqueness, J. Math. Fluid Mech., 23 (2021), 24pp. doi: 10.1007/s00021-020-00531-5.

[56]

H. Y. Wen, On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions, arXiv: 1902.05190.

[57]

H. Y. WenL. Yao and C. J. Zhu, A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum, J. Math. Pures Appl., 97 (2012), 204-229.  doi: 10.1016/j.matpur.2011.09.005.

[58]

J. H. Wu and Y. F. Wu, Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion, Adv. Math., 310 (2017), 759-888.  doi: 10.1016/j.aim.2017.02.013.

[59]

L. YaoT. Zhang and C. J. Zhu, Existence and asymptotic behavior of global weak solutions to a 2D viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 42 (2010), 1874-1897.  doi: 10.1137/100785302.

[60]

L. YaoT. Zhang and C. J. Zhu, A blow-up criterion for a 2D viscous liquid-gas two-phase flow model, J. Differ. Equ., 250 (2011), 3362-3378.  doi: 10.1016/j.jde.2010.12.006.

[61]

L. Yao and C. J. Zhu, Free boundary value problem for a viscous two-phase model with mass-dependent viscosity, J. Differ. Equ., 247 (2009), 2705-2739.  doi: 10.1016/j.jde.2009.07.013.

[62]

L. Yao and C. J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum, Math. Ann., 349 (2011), 903-928.  doi: 10.1007/s00208-010-0544-0.

[63]

X. P. Zhai, Global well-posedness and large time behavior of solutions to the n-dimensional compressible Oldroyd-B model, arXiv: 1912.00372.

[1]

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

[2]

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

[3]

B. Wiwatanapataphee, Theeradech Mookum, Yong Hong Wu. Numerical simulation of two-fluid flow and meniscus interface movement in the electromagnetic continuous steel casting process. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1171-1183. doi: 10.3934/dcdsb.2011.16.1171

[4]

Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838

[5]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure and Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011

[6]

Hannes Eberlein, Michael Růžička. Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 4093-4140. doi: 10.3934/dcdss.2020419

[7]

Kota Kumazaki. Periodic solutions for non-isothermal phase transition models. Conference Publications, 2011, 2011 (Special) : 891-902. doi: 10.3934/proc.2011.2011.891

[8]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157

[9]

Alain Miranville. Some mathematical models in phase transition. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271

[10]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1217-1251. doi: 10.3934/dcdsb.2013.18.1217

[11]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198

[12]

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

[13]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212

[14]

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

[15]

Mauro Garavello, Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary. Networks and Heterogeneous Media, 2013, 8 (3) : 649-661. doi: 10.3934/nhm.2013.8.649

[16]

Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741

[17]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[18]

Qiaolin He, Chang Liu, Xiaoding Shi. Numerical study of phase transition in van der Waals fluid. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4519-4540. doi: 10.3934/dcdsb.2018174

[19]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[20]

Hengrong Du, Changyou Wang. Global weak solutions to the stochastic Ericksen–Leslie system in dimension two. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2175-2197. doi: 10.3934/dcds.2021187

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (282)
  • HTML views (176)
  • Cited by (0)

Other articles
by authors

[Back to Top]