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June  2017, 37(6): 3423-3434. doi: 10.3934/dcds.2017145

Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain

Jingrui Su †,

Department of Mathematics, Nanjing University, Nanjing 210093, China

† Current address: Department of Mathematics, Taizhou University, Taizhou 225300, China

Received  September 2015 Revised  January 2017 Published  February 2017

This paper is devoted to investigating the global existence of strong solutions to the three-dimensional compressible micropolar fluids model in a bounded domain with small initial data. Furthermore, we present the low Mach number limit to the corresponding problem.

Keywords: Compressible micropolar fluids model, global existence, low Mach number limit, bounded domain.
Mathematics Subject Classification: Primary: 76W05; Secondary: 35B40.
Citation: Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3423-3434. doi: 10.3934/dcds.2017145
References:
[1]

M. Chen, Blow-up criterion for viscous, compressible micropolar fluids with vacuum, Nonlinear Anal. Real World Appl., 13 (2012), 850-859.  doi: 10.1016/j.nonrwa.2011.08.021.

[2]

M. Chen, B. Huang and J. Zhang, Blow-up criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum, Nonlinear Anal., 79 (2013), 1-11.  doi: 10.1016/j.na.2012.10.013.

[3]

M. Chen, Global well-posedness of the 2D incompressible micropolar fluid flows with partial viscosity and angular viscosity, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 929-935.  doi: 10.1016/S0252-9602(13)60051-X.

[4]

M. Chen, X. Xu and J. Zhang, The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect, Z. Angew. Math. Phys., 65 (2014), 687-710.  doi: 10.1007/s00033-013-0345-x.

[5]

C. Dou, S. Jiang and Y. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differential Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[6]

C. Dou, S. Jiang and Q. Ju, Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary, Z. Angew. Math. Phys., 64 (2013), 1661-1678.  doi: 10.1007/s00033-013-0311-7.

[7]

I. Dravić, N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem, Bound. Value Probl. , 2012 (2012), 25pp. doi: 10.1186/1687-2770-2012-69.

[8]

I. Dra$\mathbf{v}$ić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution, J. Math. Anal. Appl., 431 (2015), 545-568.  doi: 10.1016/j.jmaa.2015.06.002.

[9]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. 

[10]

J. Fan, H. Gao and B. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient, Math. Methods Appl. Sci., 34 (2011), 2181-2188.  doi: 10.1002/mma.1515.

[11]

J. Fan, F. Li and G. Nakamura, Uniform well-posedness and singular limits of the isentropic Navier-Stokes-Maxwell system in a bounded domain, Z. Angew. Math. Phys., 66 (2015), 1581-1593.  doi: 10.1007/s00033-014-0484-8.

[12]

J. Fan, F. Li and G. Nakamura, Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain, Dynamical Systems, Differential equations and Applications, Proc. of 10th AIMS Conference, (2015), 387–394. doi: 10.3934/proc.2015.0387.

[13]

E. Feireisl and A. Novotný, The low Mach number limit for the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.

[14]

I. Fërste, On the theory of micropolar fluids, Adv. in Mech., 2 (1979), 81-100. 

[15]

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

[16]

X. Hu and D. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.  doi: 10.1137/080723983.

[17]

S. Jiang, Q. Ju and F. Li, Low Mach limits for the multi-dimensional full Magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[18]

S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach numberlimit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[19]

H. Lange, The existence of instationary flows of incompressible micropolar fluids, Arch. Mech., 29 (1977), 741-744. 

[20]

F. Li and Y. Mu, Low Mach limits of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334-344.  doi: 10.1016/j.jmaa.2013.10.064.

[21]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2 Compressible Models, Oxford University Press, New York, 1998.

[22]

G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Model., 34 (2001), 487-509.  doi: 10.1016/S0895-7177(01)00078-4.

[23]

G. Łukaszewicz, Asymptotic behavior of micropolar fluid flows, Int. J. Eng. Sci., 41 (2003), 259-269.  doi: 10.1016/S0020-7225(02)00208-2.

[24]

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

[25]

G. Lukaszewicz, M. Rojas-Medar and M. Santos, Stationary micropolar fluid with boundary data in $L^2$, J. Math. Anal. Appl., 271 (2002), 91-107.  doi: 10.1016/S0022-247X(02)00100-2.

[26]

M. A. Rojas-Medar, Magneto-Micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.

[27]

F. V. Silva, Leray's problem for a viscous incompressible micropolar fluid, J. Math. Anal. Appl., 306 (2005), 692-713.  doi: 10.1016/j.jmaa.2004.10.007.

[28]

J. Su, Incompressible limit of a compressible micropolar fluid model with general initial data, Nonlinear Anal., 132 (2016), 1-24.  doi: 10.1016/j.na.2015.10.020.

[29]

J. Su, Low Mach number limit of a compressible micropolar fluid model, (submitted).

[30]

X. Yang, Low Mach number limit of the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 25 (2015), 118-126.  doi: 10.1016/j.nonrwa.2015.03.007.

show all references

References:
[1]

M. Chen, Blow-up criterion for viscous, compressible micropolar fluids with vacuum, Nonlinear Anal. Real World Appl., 13 (2012), 850-859.  doi: 10.1016/j.nonrwa.2011.08.021.

[2]

M. Chen, B. Huang and J. Zhang, Blow-up criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum, Nonlinear Anal., 79 (2013), 1-11.  doi: 10.1016/j.na.2012.10.013.

[3]

M. Chen, Global well-posedness of the 2D incompressible micropolar fluid flows with partial viscosity and angular viscosity, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 929-935.  doi: 10.1016/S0252-9602(13)60051-X.

[4]

M. Chen, X. Xu and J. Zhang, The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect, Z. Angew. Math. Phys., 65 (2014), 687-710.  doi: 10.1007/s00033-013-0345-x.

[5]

C. Dou, S. Jiang and Y. Ou, Low Mach number limit of full Navier-Stokes equations in a 3D bounded domain, J. Differential Equations, 258 (2015), 379-398.  doi: 10.1016/j.jde.2014.09.017.

[6]

C. Dou, S. Jiang and Q. Ju, Global existence and the low Mach number limit for the compressible magnetohydrodynamic equations in a bounded domain with perfectly conducting boundary, Z. Angew. Math. Phys., 64 (2013), 1661-1678.  doi: 10.1007/s00033-013-0311-7.

[7]

I. Dravić, N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem, Bound. Value Probl. , 2012 (2012), 25pp. doi: 10.1186/1687-2770-2012-69.

[8]

I. Dra$\mathbf{v}$ić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution, J. Math. Anal. Appl., 431 (2015), 545-568.  doi: 10.1016/j.jmaa.2015.06.002.

[9]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18. 

[10]

J. Fan, H. Gao and B. Guo, Low Mach number limit of the compressible magnetohydrodynamic equations with zero thermal conductivity coefficient, Math. Methods Appl. Sci., 34 (2011), 2181-2188.  doi: 10.1002/mma.1515.

[11]

J. Fan, F. Li and G. Nakamura, Uniform well-posedness and singular limits of the isentropic Navier-Stokes-Maxwell system in a bounded domain, Z. Angew. Math. Phys., 66 (2015), 1581-1593.  doi: 10.1007/s00033-014-0484-8.

[12]

J. Fan, F. Li and G. Nakamura, Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain, Dynamical Systems, Differential equations and Applications, Proc. of 10th AIMS Conference, (2015), 387–394. doi: 10.3934/proc.2015.0387.

[13]

E. Feireisl and A. Novotný, The low Mach number limit for the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal., 186 (2007), 77-107.  doi: 10.1007/s00205-007-0066-4.

[14]

I. Fërste, On the theory of micropolar fluids, Adv. in Mech., 2 (1979), 81-100. 

[15]

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

[16]

X. Hu and D. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294.  doi: 10.1137/080723983.

[17]

S. Jiang, Q. Ju and F. Li, Low Mach limits for the multi-dimensional full Magnetohydrodynamic equations, Nonlinearity, 25 (2012), 1351-1365.  doi: 10.1088/0951-7715/25/5/1351.

[18]

S. Jiang, Q. Ju, F. Li and Z. Xin, Low Mach numberlimit for the full compressible magnetohydrodynamic equations with general initial data, Adv. Math., 259 (2014), 384-420.  doi: 10.1016/j.aim.2014.03.022.

[19]

H. Lange, The existence of instationary flows of incompressible micropolar fluids, Arch. Mech., 29 (1977), 741-744. 

[20]

F. Li and Y. Mu, Low Mach limits of the full compressible Navier-Stokes-Maxwell system, J. Math. Anal. Appl., 412 (2014), 334-344.  doi: 10.1016/j.jmaa.2013.10.064.

[21]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2 Compressible Models, Oxford University Press, New York, 1998.

[22]

G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Math. Comput. Model., 34 (2001), 487-509.  doi: 10.1016/S0895-7177(01)00078-4.

[23]

G. Łukaszewicz, Asymptotic behavior of micropolar fluid flows, Int. J. Eng. Sci., 41 (2003), 259-269.  doi: 10.1016/S0020-7225(02)00208-2.

[24]

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

[25]

G. Lukaszewicz, M. Rojas-Medar and M. Santos, Stationary micropolar fluid with boundary data in $L^2$, J. Math. Anal. Appl., 271 (2002), 91-107.  doi: 10.1016/S0022-247X(02)00100-2.

[26]

M. A. Rojas-Medar, Magneto-Micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.  doi: 10.1002/mana.19971880116.

[27]

F. V. Silva, Leray's problem for a viscous incompressible micropolar fluid, J. Math. Anal. Appl., 306 (2005), 692-713.  doi: 10.1016/j.jmaa.2004.10.007.

[28]

J. Su, Incompressible limit of a compressible micropolar fluid model with general initial data, Nonlinear Anal., 132 (2016), 1-24.  doi: 10.1016/j.na.2015.10.020.

[29]

J. Su, Low Mach number limit of a compressible micropolar fluid model, (submitted).

[30]

X. Yang, Low Mach number limit of the compressible Hall-magnetohydrodynamic system, Nonlinear Anal. Real World Appl., 25 (2015), 118-126.  doi: 10.1016/j.nonrwa.2015.03.007.

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