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On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems
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 |
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 [
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. 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. |
[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.
|
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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.
|
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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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