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June  2018, 23(4): 1395-1410. doi: 10.3934/dcdsb.2018156

Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model

Haibo Cui 1, , Qunyi Bie 2,, and  Zheng-An Yao 3,

1. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
2. 

College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China
3. 

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

* Corresponding author

Received  August 2016 Revised  February 2018 Published  June 2018 Early access  April 2018

Fund Project: Research Supported by the NNSF of China (Grant Nos. 11601164, 11271381 and 11701325), the National Basic Research Program of China (973 Program) (Grant No. 2010CB808002), the Natural Science Foundation of Fujian Province of China (Grant Nos. 2016J05010 and 2017J05007) and the Scientific Research Funds of Huaqiao University (Grant No.15BS201).

This paper is dedicated to the study of the Cauchy problem for a compressible viscous liquid-gas two-phase flow model in $\mathbb{R}^N\,(N≥2)$. We concentrate on the critical Besov spaces based on the $L^p$ setting. We improve the range of Lebesgue exponent $p$, for which the system is locally well-posed, compared to [22]. Applying Lagrangian coordinates is the key to our statements, as it enables us to obtain the result by means of Banach fixed point theorem.

Keywords: Liquid-gas two-phase flow model, local well-posedness, critical Besov spaces, Lagrangian coordinates.
Mathematics Subject Classification: Primary: 35Q35, 76N10.
Citation: 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
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg, 2011.  Google Scholar

[2]

Q. L. Chen, C. X. Miao and Z. F. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.  Google Scholar

[3]

Q. L. Chen, C. X. Miao and Z. F. Zhang, Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26 (2010), 915-946.   Google Scholar

[4]

N. Chikami and R. Danchin, On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces, J. Differential Equations, 258 (2015), 3435-3467.  doi: 10.1016/j.jde.2015.01.012.  Google Scholar

[5]

H. B. Cui, W. J. Wang, L. 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.  Google Scholar

[6]

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

[7]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar

[8]

R. Danchin, Lagrangian approach for the compressible Navier-Stokes equations, Ann. Inst. Fourier, 64 (2014), 753-791.  doi: 10.5802/aif.2865.  Google Scholar

[9]

R. Danchin and P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409.  Google Scholar

[10]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.  doi: 10.1081/PDE-100106132.  Google Scholar

[11]

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

[12]

S. Evje and K. 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

[13]

S. Evje and K. 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

[14]

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

[15]

P. B. Mucha, The cauchy problem for the compressible Navier-Stokes equations in the Lp-framework, Nonlinear Anal., 52 (2003), 1379-1392.  doi: 10.1016/S0362-546X(02)00270-5.  Google Scholar

[16]

J. Nash, Le probléme de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.   Google Scholar

[17]

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

[18]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Volume 3, Walter de Gruyter, Berlin, 1996.  Google Scholar

[19]

A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130 (1982), 197-213.  doi: 10.1007/BF01761495.  Google Scholar

[20]

A. Valli and W. M. Zajaczkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296.  doi: 10.1007/BF01206939.  Google Scholar

[21]

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

[22]

F. Y. Xu and J. Yuan, On the well-posedness for a multi-dimensional compressible viscous liquid-gas two-phase flow model in critical spaces, Z. Angew. Math. Phys., 66 (2015), 2395-2417.  doi: 10.1007/s00033-015-0529-7.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer, Heidelberg, 2011.  Google Scholar

[2]

Q. L. Chen, C. X. Miao and Z. F. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.  Google Scholar

[3]

Q. L. Chen, C. X. Miao and Z. F. Zhang, Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26 (2010), 915-946.   Google Scholar

[4]

N. Chikami and R. Danchin, On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces, J. Differential Equations, 258 (2015), 3435-3467.  doi: 10.1016/j.jde.2015.01.012.  Google Scholar

[5]

H. B. Cui, W. J. Wang, L. 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.  Google Scholar

[6]

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

[7]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.  Google Scholar

[8]

R. Danchin, Lagrangian approach for the compressible Navier-Stokes equations, Ann. Inst. Fourier, 64 (2014), 753-791.  doi: 10.5802/aif.2865.  Google Scholar

[9]

R. Danchin and P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409.  Google Scholar

[10]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.  doi: 10.1081/PDE-100106132.  Google Scholar

[11]

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

[12]

S. Evje and K. 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

[13]

S. Evje and K. 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

[14]

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

[15]

P. B. Mucha, The cauchy problem for the compressible Navier-Stokes equations in the Lp-framework, Nonlinear Anal., 52 (2003), 1379-1392.  doi: 10.1016/S0362-546X(02)00270-5.  Google Scholar

[16]

J. Nash, Le probléme de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.   Google Scholar

[17]

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

[18]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Volume 3, Walter de Gruyter, Berlin, 1996.  Google Scholar

[19]

A. Valli, An existence theorem for compressible viscous fluids, Ann. Mat. Pura Appl., 130 (1982), 197-213.  doi: 10.1007/BF01761495.  Google Scholar

[20]

A. Valli and W. M. Zajaczkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys., 103 (1986), 259-296.  doi: 10.1007/BF01206939.  Google Scholar

[21]

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

[22]

F. Y. Xu and J. Yuan, On the well-posedness for a multi-dimensional compressible viscous liquid-gas two-phase flow model in critical spaces, Z. Angew. Math. Phys., 66 (2015), 2395-2417.  doi: 10.1007/s00033-015-0529-7.  Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

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