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:
[1]

M. T. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.   Google Scholar

[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.  Google Scholar

[3]

M. T. ChenB. 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.  Google Scholar

[4]

M. T. ChenX. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

F. M. HuangJ. 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.  Google Scholar

[10]

F. M. HuangA. 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.  Google Scholar

[11]

F. M. HuangA. 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.  Google Scholar

[12]

F. M. HuangZ. 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.  Google Scholar

[13]

S. KawashimaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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).  Google Scholar

[26]

T. NakamuraS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. QinT. 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

[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.  Google Scholar

show all references

References:
[1]

M. T. Chen, Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.   Google Scholar

[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.  Google Scholar

[3]

M. T. ChenB. 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.  Google Scholar

[4]

M. T. ChenX. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

F. M. HuangJ. 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.  Google Scholar

[10]

F. M. HuangA. 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.  Google Scholar

[11]

F. M. HuangA. 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.  Google Scholar

[12]

F. M. HuangZ. 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.  Google Scholar

[13]

S. KawashimaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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).  Google Scholar

[26]

T. NakamuraS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. QinT. 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

[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.  Google Scholar

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