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doi: 10.3934/dcdsb.2020377

Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum

Tong Tang 1,, and  Jianzhu Sun 2,

1. 

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China
2. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

* Corresponding author: Tong Tang

Received  June 2020 Revised  October 2020 Published  December 2020

Fund Project: Tong Tang is partially supported by NSFC (No. 11801138), the special grant of Jiangsu Provincial policy guidance plan for introducing foreign talents–BX2020082 and the Fundamental Research Funds for the Central Universities B200202156

In this paper, we prove the local well-posedness of strong solutions to the density-dependent incompressible magneto-micropolar system with vacuum. There is no assuming compatibility condition on the initial data.

Keywords: LWP, magneto-micropolar, vacuum.
Mathematics Subject Classification: Primary: 35Q35, 35B40, 76N15.
Citation: Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020377
References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.  Google Scholar

[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.  Google Scholar

[3]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.  Google Scholar

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R. Danchin and B. Ducomet, The low Mach number limit for a barotropic model of radiative flow, SIAM J. Math. Anal., 48 (2016), 1025-1053.  doi: 10.1137/15M1009081.  Google Scholar

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R. Danchin and P. B. Mucha, The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385.  doi: 10.1002/cpa.21806.  Google Scholar

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L. Du and Y. Wang, Mass concentration phenomenon in compressible magnetohydrodynamic flows, Nonlinearity, 28 (2015), 2959-2976.  doi: 10.1088/0951-7715/28/8/2959.  Google Scholar

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B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

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M. Durán, J. Ferreira and M. A. Rojas-Medar, Reproductive weak solutions of magneto-micropolar fluid equations in exterior domains, Math. Comput. Modelling, 35 (2002), 779-791.  doi: 10.1016/S0895-7177(02)00049-3.  Google Scholar

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A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[10]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.  Google Scholar

[11]

J. Fan, W. Sun and J. Yin, Blow-up criteria for Boussinesq system and MHD system and Landau-Lifshitz equations in a bounded domain, Bound. Value Probl., 90 (2016), 19 pp. doi: 10.1186/s13661-016-0598-3.  Google Scholar

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J. Fan, B. Samet and Y. Zhou, A regularity criterion for a density-dependent incompressible liquid crystals model with vacuum, Hiroshima Math. J., 49 (2019), 129-138.  doi: 10.32917/hmj/1554516040.  Google Scholar

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G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

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A. V. Kazhikov, Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR, 216 (1974), 1008-1010.   Google Scholar

[15]

J. U. Kim, Weak solutions of an initial boundary-value problems for an incompressible viscous fluid with non-negative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007.  Google Scholar

[16]

J. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.  Google Scholar

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Y. Liu and S. Li, Global well-posedness for magneto-micropolar system in $2\frac12$ dimensions, Appl. Math. Comput., 280 (2016), 72-85.  doi: 10.1016/j.amc.2016.01.002.  Google Scholar

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G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

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G. Łukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55 (2004), 247-257.  doi: 10.1007/s00033-003-1127-7.  Google Scholar

[20]

L. Ma, Global existence of three-dimensional incompressible magneto-micropolar system with mixed partial dissipation, magnetic diffusion and angular viscosity, Comput. Math. Appl., 75 (2018), 170-186.  doi: 10.1016/j.camwa.2017.09.009.  Google Scholar

[21]

L. Ma, On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity, Nonlinear Anal. Real World Appl., 40 (2018), 95-129.  doi: 10.1016/j.nonrwa.2017.08.014.  Google Scholar

[22]

G. Metivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158, (2001), 61–90. doi: 10.1007/PL00004241.  Google Scholar

[23]

C. J. Niche and C. Perusato, Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids, preprint, 2020, arXiv: 2006.14427. Google Scholar

[24]

J. Simon, Nonhomogeneous viscous incompressible fluid: Existence of velocity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar

[25]

Y. F. Wang, L. Du and S. Li, Blowup mechanism for viscous compressible heat-conductive magnetohydrodynamic flows in three dimensions, Sci. China Math., 58 (2015), 1677-1696.  doi: 10.1007/s11425-014-4951-7.  Google Scholar

[26]

Y. F. Wang, Weak Serrin-type blowup criterion for three-dimensional nonhomogeneous viscous incompressible heat conducting flows, Phys. D, 402 (2020), 132203, 8 pp. doi: 10.1016/j.physd.2019.132203.  Google Scholar

[27]

H. Wu, Strong solution to the incompressible MHD equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753.  doi: 10.1016/j.camwa.2011.03.033.  Google Scholar

[28]

J. Yuan, Existence theorem and blow-up criterion for the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130.  doi: 10.1002/mma.967.  Google Scholar

[29]

Z. Zhang, Z. A. Yao and X. Wang, A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel-Lizorkin spaces, Nonlinear Anal., 74 (2011), 2220-2225.  doi: 10.1016/j.na.2010.11.026.  Google Scholar

show all references

References:
[1]

G. Ahmadi and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.  doi: 10.1016/0020-7225(74)90042-1.  Google Scholar

[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.  Google Scholar

[3]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.  Google Scholar

[4]

R. Danchin and B. Ducomet, The low Mach number limit for a barotropic model of radiative flow, SIAM J. Math. Anal., 48 (2016), 1025-1053.  doi: 10.1137/15M1009081.  Google Scholar

[5]

R. Danchin and P. B. Mucha, The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385.  doi: 10.1002/cpa.21806.  Google Scholar

[6]

L. Du and Y. Wang, Mass concentration phenomenon in compressible magnetohydrodynamic flows, Nonlinearity, 28 (2015), 2959-2976.  doi: 10.1088/0951-7715/28/8/2959.  Google Scholar

[7]

B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys., 266 (2006), 595-629.  doi: 10.1007/s00220-006-0052-y.  Google Scholar

[8]

M. Durán, J. Ferreira and M. A. Rojas-Medar, Reproductive weak solutions of magneto-micropolar fluid equations in exterior domains, Math. Comput. Modelling, 35 (2002), 779-791.  doi: 10.1016/S0895-7177(02)00049-3.  Google Scholar

[9]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[10]

J. Fan and W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.  doi: 10.1016/j.nonrwa.2007.10.001.  Google Scholar

[11]

J. Fan, W. Sun and J. Yin, Blow-up criteria for Boussinesq system and MHD system and Landau-Lifshitz equations in a bounded domain, Bound. Value Probl., 90 (2016), 19 pp. doi: 10.1186/s13661-016-0598-3.  Google Scholar

[12]

J. Fan, B. Samet and Y. Zhou, A regularity criterion for a density-dependent incompressible liquid crystals model with vacuum, Hiroshima Math. J., 49 (2019), 129-138.  doi: 10.32917/hmj/1554516040.  Google Scholar

[13]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[14]

A. V. Kazhikov, Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR, 216 (1974), 1008-1010.   Google Scholar

[15]

J. U. Kim, Weak solutions of an initial boundary-value problems for an incompressible viscous fluid with non-negative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007.  Google Scholar

[16]

J. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.  Google Scholar

[17]

Y. Liu and S. Li, Global well-posedness for magneto-micropolar system in $2\frac12$ dimensions, Appl. Math. Comput., 280 (2016), 72-85.  doi: 10.1016/j.amc.2016.01.002.  Google Scholar

[18]

G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[19]

G. Łukaszewicz and W. Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys., 55 (2004), 247-257.  doi: 10.1007/s00033-003-1127-7.  Google Scholar

[20]

L. Ma, Global existence of three-dimensional incompressible magneto-micropolar system with mixed partial dissipation, magnetic diffusion and angular viscosity, Comput. Math. Appl., 75 (2018), 170-186.  doi: 10.1016/j.camwa.2017.09.009.  Google Scholar

[21]

L. Ma, On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity, Nonlinear Anal. Real World Appl., 40 (2018), 95-129.  doi: 10.1016/j.nonrwa.2017.08.014.  Google Scholar

[22]

G. Metivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158, (2001), 61–90. doi: 10.1007/PL00004241.  Google Scholar

[23]

C. J. Niche and C. Perusato, Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids, preprint, 2020, arXiv: 2006.14427. Google Scholar

[24]

J. Simon, Nonhomogeneous viscous incompressible fluid: Existence of velocity, density and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.  Google Scholar

[25]

Y. F. Wang, L. Du and S. Li, Blowup mechanism for viscous compressible heat-conductive magnetohydrodynamic flows in three dimensions, Sci. China Math., 58 (2015), 1677-1696.  doi: 10.1007/s11425-014-4951-7.  Google Scholar

[26]

Y. F. Wang, Weak Serrin-type blowup criterion for three-dimensional nonhomogeneous viscous incompressible heat conducting flows, Phys. D, 402 (2020), 132203, 8 pp. doi: 10.1016/j.physd.2019.132203.  Google Scholar

[27]

H. Wu, Strong solution to the incompressible MHD equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753.  doi: 10.1016/j.camwa.2011.03.033.  Google Scholar

[28]

J. Yuan, Existence theorem and blow-up criterion for the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130.  doi: 10.1002/mma.967.  Google Scholar

[29]

Z. Zhang, Z. A. Yao and X. Wang, A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel-Lizorkin spaces, Nonlinear Anal., 74 (2011), 2220-2225.  doi: 10.1016/j.na.2010.11.026.  Google Scholar

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