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

Haibo Cui; Haiyan Yin

doi:10.3934/cpaa.2017062

Article Contents

2017, Volume 16, Issue 4: 1265-1292. Doi: 10.3934/cpaa.2017062

This issue Previous Article Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space Next Article Robin problems with indefinite linear part and competition phenomena

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

Haibo Cui^1, and
Haiyan Yin^2, ,

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
^* Corresponding author: yinhaiyan2000@aliyun.com

Received: July 2016

Revised: February 2017

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35, 76D33; Secondary: 35M33, 35B35.

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References

[1]	M. T. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.
[2]	M. T. Chen, Blowup 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.
[3]	M. T. Chen, B. Huang and J. W. Zhang, Blowup 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.
[4]	M. T. Chen, X. Y. Xu and J. W. 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.
[5]	H. B. Cui and H. Y. 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.
[6]	A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
[7]	J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840.
[8]	F. M. Huang and X. H. Qin, Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation, J. Differential Equations, 246 (2009), 4077-4096. doi: 10.1016/j.jde.2009.01.017.
[9]	F. M. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0.
[10]	F. M. Huang, A. Matsumura and X. D. Shi, A gas-solid free boundary problem for compressible viscous gas, SIAM J. Math. Anal., 34 (2003), 1331-1355. doi: 10.1137/S0036141002403730.
[11]	F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77. doi: 10.1007/s00205-005-0380-7.
[12]	F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014.
[13]	S. Kawashima, S. Nishibata and P. C. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2.
[14]	Q. Q. Liu and H. Y. Yin, Stability of contact discontinuity for 1-D compressible viscous micropolar fluid model, Nonlinear Anal., 149 (2017), 41-55. doi: 10.1016/j.na.2016.10.009.
[15]	Q. Q. Liu and P. X. Zhang, Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661. doi: 10.1016/j.jde.2016.01.037.
[16]	T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.
[17]	[0-8176-4008-8] G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Baston, Inc. , Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.
[18]	A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666. doi: 10.4310/MAA.2001.v8.n4.a14.
[19]	A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22. doi: 10.1007/s002050050134.
[20]	A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), 449-474. doi: 10.1007/s002200100517.
[21]	N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.
[22]	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.
[23]	N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: stabilization of the solution, Proceedings of the Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, 2005,253-262. doi: 10.1007/1-4020-3197-1_18.
[24]	N. Mujaković, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: regularity of the solution, Bound. Value Probl. , Article ID 189748 (2008).
[25]	N. Mujaković, One-dimensional compressible viscous micropolar fluid model: stabilization of the solution for the Cauchy problem, Bound. Value Probl. , Article ID 796065 (2010).
[26]	T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016.
[27]	T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyperbolic Differ. Equ., 8 (2011), 651-670. doi: 10.1142/S0219891611002524.
[28]	B. Nowakowski, Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal., Real World Appl., 14 (2013), 635-660. doi: 10.1016/j.nonrwa.2012.07.023.
[29]	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.
[30]	X. H. Qin and Y. Wang, Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 5 (2009), 2057-2087. doi: 10.1137/09075425X.