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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

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.

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 and 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.

[2]

Q. L. ChenC. 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.

[3]

Q. L. ChenC. 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. 

[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.

[5]

H. B. CuiW. J. WangL. 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.

[6]

H. B. CuiH. 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.

[7]

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

[8]

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

[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.

[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.

[11]

S. EvjeT. 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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[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.

[21]

H. Y. WenL. 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.

[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.

[23]

L. YaoT. 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.

[24]

L. YaoT. 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.

[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.

[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.

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.

[2]

Q. L. ChenC. 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.

[3]

Q. L. ChenC. 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. 

[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.

[5]

H. B. CuiW. J. WangL. 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.

[6]

H. B. CuiH. 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.

[7]

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

[8]

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

[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.

[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.

[11]

S. EvjeT. 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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[19]

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

[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.

[21]

H. Y. WenL. 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.

[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.

[23]

L. YaoT. 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.

[24]

L. YaoT. 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.

[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.

[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.

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