# American Institute of Mathematical Sciences

June  2021, 29(2): 2063-2075. doi: 10.3934/era.2020105

## Asymptotic behavior of the one-dimensional compressible micropolar fluid model

 1 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China 2 The Hubei Key Laboratory of Mathematical Physics, School of Mathematics, and Statistics, Central China Normal University, Wuhan 430079, China 3 School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

* Corresponding author: Haibo Cui

Received  June 2020 Revised  August 2020 Published  June 2021 Early access  September 2020

Fund Project: Cui and Gao were supported by the National Natural Science Foundation of China #11601164, and #11971183, the Fundamental Research Funds for the Central Universities(Grant No. ZQN-701). Yao was supported by the National Natural Science Foundation of China #11931013 and Natural Science Basic Research Program of Shaanxi(Program No. 2019JC-26)

In this paper, we study the large time behavior of the solution for one-dimensional compressible micropolar fluid model with large initial data. This model describes micro-rotational motions and spin inertia which is commonly used in the suspensions, animal blood, and liquid crystal. We get the uniform positive lower and upper bounds of the density and temperature independent of both space and time. In particular, we also obtain the asymptotic behavior of the micro-rotation velocity.

Citation: Haibo Cui, Junpei Gao, Lei Yao. Asymptotic behavior of the one-dimensional compressible micropolar fluid model. Electronic Research Archive, 2021, 29 (2) : 2063-2075. doi: 10.3934/era.2020105
##### References:
 [1] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar [2] M. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.  doi: 10.4208/jpde.v24.n2.5.  Google Scholar [3] Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar [4] M. Chen, X. Xu and J. Zhang, Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum, Commun. Math. Sci., 13 (2015), 225-247.  doi: 10.4310/CMS.2015.v13.n1.a11.  Google Scholar [5] H. Cui and Z.-A. Yao, Asymptotic behavior of compressible $p$-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953.  doi: 10.1016/j.jde.2014.10.011.  Google Scholar [6] H. Cui and H. Yin, Stationary solutions to the one-dimensional micropolar fluid model in a half line: Existence, stability and convergence rate, J. Math. Anal. Appl., 449 (2017), 464-489.  doi: 10.1016/j.jmaa.2016.11.065.  Google Scholar [7] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar [8] A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0555-5.  Google Scholar [9] Z. Feng and C. Zhu, Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum, Discrete Contin. Dyn. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.  Google Scholar [10] S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.  Google Scholar [11] S. Jiang, Remarks on the asymptotic behaviour of solutions to the compressible Navier-Stokes equations in the half-line, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 627-638.  doi: 10.1017/S0308210500001815.  Google Scholar [12] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar [13] J. Li and Z. Liang, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.  Google Scholar [14] Q. Liu and P. Zhang, Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661.  doi: 10.1016/j.jde.2016.01.037.  Google Scholar [15] G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar [16] N. Mujaković, Global in time estimates for one-dimensional compressible viscous micropolar fluid model, Glas. Mat. Ser. Ⅲ, 40 (2005), 103-120.  doi: 10.3336/gm.40.1.10.  Google Scholar [17] N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: Stabilization of the solution, in Proceedings of the Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, 2005,253–262. doi: 10.1007/1-4020-3197-1_18.  Google Scholar [18] Y. Qin, T. Wang and G. Hu, The Cauchy problem for a 1D compressible viscous micropolar fluid model: Analysis of the stabilization and the regularity, Nonlinear Anal. Real World Appl., 13 (2012), 1010-1029.  doi: 10.1016/j.nonrwa.2010.10.023.  Google Scholar [19] H. Yin, Stability of stationary solutions for inflow problem on the micropolar fluid model, Z. Angew. Math. Phys., 68 (2017), 13pp. doi: 10.1007/s00033-017-0789-5.  Google Scholar

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##### References:
 [1] S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar [2] M. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.  doi: 10.4208/jpde.v24.n2.5.  Google Scholar [3] Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar [4] M. Chen, X. Xu and J. Zhang, Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum, Commun. Math. Sci., 13 (2015), 225-247.  doi: 10.4310/CMS.2015.v13.n1.a11.  Google Scholar [5] H. Cui and Z.-A. Yao, Asymptotic behavior of compressible $p$-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953.  doi: 10.1016/j.jde.2014.10.011.  Google Scholar [6] H. Cui and H. Yin, Stationary solutions to the one-dimensional micropolar fluid model in a half line: Existence, stability and convergence rate, J. Math. Anal. Appl., 449 (2017), 464-489.  doi: 10.1016/j.jmaa.2016.11.065.  Google Scholar [7] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar [8] A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0555-5.  Google Scholar [9] Z. Feng and C. Zhu, Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum, Discrete Contin. Dyn. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.  Google Scholar [10] S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.  Google Scholar [11] S. Jiang, Remarks on the asymptotic behaviour of solutions to the compressible Navier-Stokes equations in the half-line, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 627-638.  doi: 10.1017/S0308210500001815.  Google Scholar [12] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar [13] J. Li and Z. Liang, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.  Google Scholar [14] Q. Liu and P. Zhang, Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661.  doi: 10.1016/j.jde.2016.01.037.  Google Scholar [15] G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar [16] N. Mujaković, Global in time estimates for one-dimensional compressible viscous micropolar fluid model, Glas. Mat. Ser. Ⅲ, 40 (2005), 103-120.  doi: 10.3336/gm.40.1.10.  Google Scholar [17] N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: Stabilization of the solution, in Proceedings of the Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, 2005,253–262. doi: 10.1007/1-4020-3197-1_18.  Google Scholar [18] Y. Qin, T. Wang and G. Hu, The Cauchy problem for a 1D compressible viscous micropolar fluid model: Analysis of the stabilization and the regularity, Nonlinear Anal. Real World Appl., 13 (2012), 1010-1029.  doi: 10.1016/j.nonrwa.2010.10.023.  Google Scholar [19] H. Yin, Stability of stationary solutions for inflow problem on the micropolar fluid model, Z. Angew. Math. Phys., 68 (2017), 13pp. doi: 10.1007/s00033-017-0789-5.  Google Scholar
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