American Institute of Mathematical Sciences

Advanced
September  2016, 9(3): 429-441. doi: 10.3934/krm.2016001

A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions

Yingshan Chen 1, and  Mei Zhang 1,

1. 

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

Received  November 2015 Revised  December 2015 Published  May 2016

In this paper, we establish a new blowup criterion for the strong solutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In [13], Wen, Yao, and Zhu prove that if the strong solutions blow up at finite time $T^*$, the mass in $L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extend Wen, Yao, and Zhu's work in the sense of the concentration of mass in $BMO(\Omega)$ norm at $T^*$. The method can be applied to study the blow-up criterion in terms of the concentration of density in $BMO(\Omega)$ norm for the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a byproduct, we can also improves the corresponding result about Navier-Stokes equations in [11]. Moreover, the appearance of vacuum is allowed in the paper.
Keywords: strong solution, Liquid-gas two-phase flow model, BMO criterion, vacuum..
Mathematics Subject Classification: 76T10, 76N10, 35L6.
Citation: Yingshan Chen, Mei Zhang. A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions. Kinetic & Related Models, 2016, 9 (3) : 429-441. doi: 10.3934/krm.2016001
References:
[1]

P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269. doi: 10.1007/BF01759640.  Google Scholar

[2]

H. B. Cui, H. Y. Wen and H. Y. Yin, Global classical solutions of viscous liquid-gas two-phase flow model, Math. Meth. Appl. Sci., 36 (2013), 567-583. doi: 10.1002/mma.2614.  Google Scholar

[3]

S. Evje, T. 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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

Z. H. Guo, J. Yang and L. Yao, Global strong solution for a three-dimensional viscous liquid-gas two-phase flowmodelwith vacuum, Journal of Mathematical Physics, 52 (2011), 093102, 14pp.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

O. A. Ladyzenskaja, V. A. Solonikov and N. N. Ural'ceva, Linear and Quasilinear Equation of Parabolic Type, Amer. Math. Soc., Providence RI, 1968.  Google Scholar

[10]

A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, Cambridge, 2009.  Google Scholar

[11]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[12]

V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscosity fluid, Siberian Math. J., 36 (1995), 1108-1141. doi: 10.1007/BF02106835.  Google Scholar

[13]

H. Y. Wen, L. 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.  Google Scholar

[14]

H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Advances in Mathematics, 248 (2013), 534-572. doi: 10.1016/j.aim.2013.07.018.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

L. Yao, T. 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.  Google Scholar

[18]

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

show all references

References:
[1]

P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269. doi: 10.1007/BF01759640.  Google Scholar

[2]

H. B. Cui, H. Y. Wen and H. Y. Yin, Global classical solutions of viscous liquid-gas two-phase flow model, Math. Meth. Appl. Sci., 36 (2013), 567-583. doi: 10.1002/mma.2614.  Google Scholar

[3]

S. Evje, T. 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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

Z. H. Guo, J. Yang and L. Yao, Global strong solution for a three-dimensional viscous liquid-gas two-phase flowmodelwith vacuum, Journal of Mathematical Physics, 52 (2011), 093102, 14pp.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

O. A. Ladyzenskaja, V. A. Solonikov and N. N. Ural'ceva, Linear and Quasilinear Equation of Parabolic Type, Amer. Math. Soc., Providence RI, 1968.  Google Scholar

[10]

A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, Cambridge, 2009.  Google Scholar

[11]

Y. Z. Sun, C. Wang and Z. F. Zhang, A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier-Stokes equations, J. Math. Pures Appl., 95 (2011), 36-47. doi: 10.1016/j.matpur.2010.08.001.  Google Scholar

[12]

V. A. Vaigant and A. V. Kazhikhov, On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscosity fluid, Siberian Math. J., 36 (1995), 1108-1141. doi: 10.1007/BF02106835.  Google Scholar

[13]

H. Y. Wen, L. 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.  Google Scholar

[14]

H. Y. Wen and C. J. Zhu, Blow-up criterions of strong solutions to 3D compressible Navier-Stokes equations with vacuum, Advances in Mathematics, 248 (2013), 534-572. doi: 10.1016/j.aim.2013.07.018.  Google Scholar

[15]

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

[16]

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.  Google Scholar

[17]

L. Yao, T. 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.  Google Scholar

[18]

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

[1]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[2]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021
[3]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[4]

Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146
[5]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[6]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[7]

Qinglong Zhang. Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3235-3258. doi: 10.3934/cpaa.2021104
[8]

T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665
[9]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198
[10]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[11]

Sili Liu, Xinhua Zhao, Yingshan Chen. A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4515-4533. doi: 10.3934/dcdsb.2020110
[12]

Steinar Evje, Kenneth Hvistendahl Karlsen. Global weak solutions for a viscous liquid-gas model with singular pressure law. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1867-1894. doi: 10.3934/cpaa.2009.8.1867
[13]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[14]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
[15]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011
[16]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
[17]

Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016
[18]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010
[19]

Zhen Cheng, Wenjun Wang. The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021151
[20]

Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences