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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.
[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.
[10]	A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, Cambridge, 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), 36-47. 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), 1108-1141. 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), 204-229. 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), 534-572. 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), 2705-2739. 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), 903-928. 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), 1874-1897. 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), 3362-3378. doi: 10.1016/j.jde.2010.12.006.

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References:

[1]	P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269. 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), 567-583. 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., TMA, 70 (2009), 3864-3886. 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), 2660-2703. 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), 1867-1894. 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), 093102, 14pp.
[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.
[8]	M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975.
[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.
[10]	A. Prosperetti and G. Tryggvason (Editors), Computational Methods for Multiphase Flow, Cambridge University Press, Cambridge, 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), 36-47. 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), 1108-1141. 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), 204-229. 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), 534-572. 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), 2705-2739. 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), 903-928. 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), 1874-1897. 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), 3362-3378. doi: 10.1016/j.jde.2010.12.006.

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