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 and 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. 

[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. 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.

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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. 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.

[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. 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.

[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.

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. 

[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. 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.

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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. 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.

[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. 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.

[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.

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