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

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.
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. doi: 10.1007/BF01759640.

[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. doi: 10.1002/mma.2614.

[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., 70 (2009), 3864. doi: 10.1016/j.na.2008.07.043.

[4]

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

[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. doi: 10.3934/cpaa.2009.8.1867.

[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).

[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. doi: 10.1137/110851602.

[8]

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow,, Eyrolles, (1975).

[9]

O. A. Ladyzenskaja, V. A. Solonikov and N. N. Ural'ceva, Linear and Quasilinear Equation of Parabolic Type,, Amer. Math. Soc., (1968).

[10]

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

[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. doi: 10.1016/j.matpur.2010.08.001.

[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. doi: 10.1007/BF02106835.

[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. doi: 10.1016/j.matpur.2011.09.005.

[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. doi: 10.1016/j.aim.2013.07.018.

[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. doi: 10.1016/j.jde.2009.07.013.

[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. doi: 10.1007/s00208-010-0544-0.

[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. doi: 10.1137/100785302.

[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. doi: 10.1016/j.jde.2010.12.006.

show all references

References:
[1]

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

[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. doi: 10.1002/mma.2614.

[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., 70 (2009), 3864. doi: 10.1016/j.na.2008.07.043.

[4]

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

[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. doi: 10.3934/cpaa.2009.8.1867.

[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).

[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. doi: 10.1137/110851602.

[8]

M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow,, Eyrolles, (1975).

[9]

O. A. Ladyzenskaja, V. A. Solonikov and N. N. Ural'ceva, Linear and Quasilinear Equation of Parabolic Type,, Amer. Math. Soc., (1968).

[10]

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

[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. doi: 10.1016/j.matpur.2010.08.001.

[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. doi: 10.1007/BF02106835.

[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. doi: 10.1016/j.matpur.2011.09.005.

[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. doi: 10.1016/j.aim.2013.07.018.

[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. doi: 10.1016/j.jde.2009.07.013.

[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. doi: 10.1007/s00208-010-0544-0.

[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. doi: 10.1137/100785302.

[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. doi: 10.1016/j.jde.2010.12.006.

[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 - A, 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 - A, 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]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Theodore Tachim Medjo. Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088

[16]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1

[17]

Tong Tang, Yongfu Wang. Strong solutions to compressible barotropic viscoelastic flow with vacuum. Kinetic & Related Models, 2015, 8 (4) : 765-775. doi: 10.3934/krm.2015.8.765

[18]

Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007

[19]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006

[20]

Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]