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), 231-269. 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

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