# American Institute of Mathematical Sciences

July  2017, 16(4): 1265-1292. doi: 10.3934/cpaa.2017062

## Stability of the composite wave for the inflow problem on the micropolar fluid model

 1 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China 2 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

* Corresponding author: yinhaiyan2000@aliyun.com

Received  July 2016 Revised  February 2017 Published  April 2017

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in half line $\mathbb{R}_{+}:=(0, \infty).$ Inspired by the relationship between the micropolar fluid model and Navier-Stokes system, we can prove that the composite wave consisting of the subsonic BL-solution, the contact wave, and the rarefaction wave for the inflow problem on micropolar fluid model is time-asymptotically stable. Meanwhile, we obtain the global existence of solutions based on the basic energy method.

Citation: Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062
##### References:

show all references

##### References:
 [1] Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985 [2] Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048 [3] Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439 [4] Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations & Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019 [5] Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595 [6] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424 [7] M. Ángeles Rodríguez-Bellido, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On a distributed control problem for a coupled chemotaxis-fluid model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 557-571. doi: 10.3934/dcdsb.2017208 [8] Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113 [9] Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735 [10] Jonathan E. Rubin. A nonlocal eigenvalue problem for the stability of a traveling wave in a neuronal medium. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 925-940. doi: 10.3934/dcds.2004.10.925 [11] Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi. Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinetic & Related Models, 2009, 2 (1) : 109-134. doi: 10.3934/krm.2009.2.109 [12] Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193 [13] Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090 [14] Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583 [15] Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765 [16] Giovanni Cupini, Eugenio Vecchi. Faber-Krahn and Lieb-type inequalities for the composite membrane problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2679-2691. doi: 10.3934/cpaa.2019119 [17] Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096 [18] Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024 [19] Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150 [20] Chien Hsun Tseng. Analytical modeling of laminated composite plates using Kirchhoff circuit and wave digital filters. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-40. doi: 10.3934/jimo.2019051

2018 Impact Factor: 0.925