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Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum
1. | Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098, China |
2. | Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
doi: 10.1512/iumj.1967.16.16001. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[18] |
G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[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. |
[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. |
[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. |
[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. |
[23] |
C. J. Niche and C. Perusato, Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids, preprint, 2020, arXiv: 2006.14427. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
doi: 10.1512/iumj.1967.16.16001. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[18] |
G. Łukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[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. |
[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. |
[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. |
[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. |
[23] |
C. J. Niche and C. Perusato, Sharp decay estimates and asymptotic behaviour for 3D magneto-micropolar fluids, preprint, 2020, arXiv: 2006.14427. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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