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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

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