June  2021, 26(6): 2899-2920. doi: 10.3934/dcdsb.2020210

Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line

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

* Corresponding author: Haiyan Yin

Received  March 2020 Revised  May 2020 Published  June 2021 Early access  July 2020

Fund Project: The authors were supported by the National Natural Science Foundation of China(Grant Nos. #11601164, #11601165 and #11971183), the Natural Science Foundation of Fujian Province of China(Grant No. 2017J05007), and the Fundamental Research Funds for the Central Universities(Grant No. ZQN-701), Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-PY602)

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the one-dimensional compressible isentropic micropolar fluid model in a half line $ \mathbb{R}_{+}: = (0, \infty). $ We mainly investigate the unique existence, the asymptotic stability and convergence rates of stationary solutions to the outflow problem for this model. We obtain the convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method by taking into account the effect of the microrotational velocity on the viscous compressible fluid.

Citation: Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210
References:
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M. T. 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.

[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. 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]

Q. L. Chen and C. X. 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.

[6]

H. B. Cui and H. Y. Yin, Stability of the composite wave for the inflow problem on the micropolar fluid model, Commun. Pure Appl. Anal., 16 (2017), 1265-1292.  doi: 10.3934/cpaa.2017062.

[7]

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.

[8]

B. Q. DongJ. N. Li and J. H. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.

[9]

I. Dra$\check{z}$i$\acute{c}$ and N. Mujakovi$\acute{c}$, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: large time behavior of the solution, J. Math. Anal. Appl., 431 (2015), 545-568.  doi: 10.1016/j.jmaa.2015.06.002.

[10]

I. Dra$\check{z}$i$\acute{c}$L. Sim$\check{c}$i$\acute{c}$ and N. Mujakovi$\acute{c}$, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution, J. Math. Anal. Appl., 438 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.

[11]

R. Duan, Global solutions for a one-dimensional compressible micropolar fluid model with zero heat conductivity, J. Math. Anal. Appl., 463 (2018), 477-495.  doi: 10.1016/j.jmaa.2018.03.009.

[12]

R. Duan, Global strong solution for initial-boundary value problem of one-dimensional compressible micropolar fluids with density dependent viscosity and temperature dependent heat conductivity, Nonlinear Anal. Real World Appl., 42 (2018), 71-92.  doi: 10.1016/j.nonrwa.2017.12.006.

[13]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. doi: 10.1512/iumj.1967.16.16001.

[14]

[0-387-98620-0] A. C. Erigen, Microcontinuum Field Theories: I. Foundations and Solids, Springer. New York., 1999. doi: 10.1007/978-1-4612-0555-5.

[15]

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.

[16]

L. Huang and D. Y. Nie, Exponential stability for a one-dimensional compressible viscous micropolar fluid, Math. Methods Appl. Sci., 38 (2015), 5197–5206. doi: 10.1002/mma.3445.

[17]

J. Jin and R. Duan, Stability of rarefaction waves for 1-D compressible viscous micropolar fluid model, J. Math. Anal. Appl., 450 (2017), 1123-1143.  doi: 10.1016/j.jmaa.2016.12.085.

[18]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97–127. doi: 10.1007/BF01212358.

[19]

S. KawashimaT. NakamuraS. Nishibata and P. C. Zhu, Stationary waves to viscous heat-conductive gases in half space: Existence, stability and convergence rate, Math. Models Methods Appl. Sci., 20 (2010), 2201-2235.  doi: 10.1142/S0218202510004908.

[20]

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.

[21]

Q. Q. Liu and H. Y. Yin, Stability of contact discontinuity for 1-D compressible viscous micropolar fluid model, Nonlinear Anal.: Theory, Methods Appl., 149 (2017), 41-55.  doi: 10.1016/j.na.2016.10.009.

[22]

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.

[23]

Q. Q. Liu and P. X. Zhang, Long-time behavior of solution to the compressible micropolar fluids with external force, Nonlinear Anal. Real World Appl., 40 (2018), 361-376.  doi: 10.1016/j.nonrwa.2017.08.007.

[24]

[0-8176-4008-8] G. Lukaszewicz, Micropolar fluids. Theory and applications. Modeling and Simulation in Science, Engineering and Technology, Birkh${\rm{\ddot a}}$user, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.

[25]

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. 

[26]

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.

[27]

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.

[28]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A local existence theorem, Glas. Mat. Ser. III, 33 (1998), 71–91.

[29]

N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: A global existence theorem, Glas. Mat. Ser. III, 33 (1998), 199-208. 

[30]

N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: Regularity of the solution, Rad. Mat., 10 (2001), 181-193. 

[31]

N. Mujakovi$\acute{c}$, 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.

[32]

N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: Stabilization of the solution, Proceedings of the Conference on Applied Mathematics and Scientific Computing, 253–262, Springer, Dordrecht, 2005. doi: 10.1007/1-4020-3197-1_18.

[33]

N. Mujakovi$\acute{c}$, Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A local existence theorem, Ann. Univ. Ferrara Sez. VII Sci. Mat., 53 (2007), 361-379.  doi: 10.1007/s11565-007-0023-z.

[34]

N. Mujakovi$\acute{c}$, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution, Bound. Value Probl., 2008 (2008), Article ID 189748, 15pp. doi: 10.1155/2008/189748.

[35]

N. Mujakovi$\acute{c}$, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A global existence theorem, Math. Inequal. Appl., 12 (2009), 651-662.  doi: 10.7153/mia-12-49.

[36]

N. Mujakovi$\acute{c}$, One-dimensional compressible viscous micropolar fluid model: stabilization of the solution for the Cauchy problem, Bound. Value Probl., (2010), Article ID 796065, 21pp. doi: 10.1155/2010/796065.

[37]

N. Mujakovi$\acute{c}$, The existence of a global solution for one dimensional compressible viscous micropolar fluid with non-homogeneous boundary conditions for temperature, Nonlinear Anal. Real World Appl., 19 (2014), 19-30.  doi: 10.1016/j.nonrwa.2014.02.006.

[38]

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.

[39]

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.

[40]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107–132.

[41]

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.

[42]

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.

[43]

Z. G. Wu and W. K. Wang, The pointwise estimates of diffusion wave of the compressible micropolar fluids, J. Differential Equations, 265 (2018), 2544-2576.  doi: 10.1016/j.jde.2018.04.039.

[44]

H. Y. Yin, Stability of stationary solutions for inflow problem on the micropolar fluid model, Z. Angew. Math. Phys., 68 (2017), Paper No. 44, 13 pp. doi: 10.1007/s00033-017-0789-5.

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.  doi: 10.4208/jpde.v24.n2.5.

[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. 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]

Q. L. Chen and C. X. 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.

[6]

H. B. Cui and H. Y. Yin, Stability of the composite wave for the inflow problem on the micropolar fluid model, Commun. Pure Appl. Anal., 16 (2017), 1265-1292.  doi: 10.3934/cpaa.2017062.

[7]

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.

[8]

B. Q. DongJ. N. Li and J. H. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.

[9]

I. Dra$\check{z}$i$\acute{c}$ and N. Mujakovi$\acute{c}$, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: large time behavior of the solution, J. Math. Anal. Appl., 431 (2015), 545-568.  doi: 10.1016/j.jmaa.2015.06.002.

[10]

I. Dra$\check{z}$i$\acute{c}$L. Sim$\check{c}$i$\acute{c}$ and N. Mujakovi$\acute{c}$, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution, J. Math. Anal. Appl., 438 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.

[11]

R. Duan, Global solutions for a one-dimensional compressible micropolar fluid model with zero heat conductivity, J. Math. Anal. Appl., 463 (2018), 477-495.  doi: 10.1016/j.jmaa.2018.03.009.

[12]

R. Duan, Global strong solution for initial-boundary value problem of one-dimensional compressible micropolar fluids with density dependent viscosity and temperature dependent heat conductivity, Nonlinear Anal. Real World Appl., 42 (2018), 71-92.  doi: 10.1016/j.nonrwa.2017.12.006.

[13]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. doi: 10.1512/iumj.1967.16.16001.

[14]

[0-387-98620-0] A. C. Erigen, Microcontinuum Field Theories: I. Foundations and Solids, Springer. New York., 1999. doi: 10.1007/978-1-4612-0555-5.

[15]

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.

[16]

L. Huang and D. Y. Nie, Exponential stability for a one-dimensional compressible viscous micropolar fluid, Math. Methods Appl. Sci., 38 (2015), 5197–5206. doi: 10.1002/mma.3445.

[17]

J. Jin and R. Duan, Stability of rarefaction waves for 1-D compressible viscous micropolar fluid model, J. Math. Anal. Appl., 450 (2017), 1123-1143.  doi: 10.1016/j.jmaa.2016.12.085.

[18]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97–127. doi: 10.1007/BF01212358.

[19]

S. KawashimaT. NakamuraS. Nishibata and P. C. Zhu, Stationary waves to viscous heat-conductive gases in half space: Existence, stability and convergence rate, Math. Models Methods Appl. Sci., 20 (2010), 2201-2235.  doi: 10.1142/S0218202510004908.

[20]

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.

[21]

Q. Q. Liu and H. Y. Yin, Stability of contact discontinuity for 1-D compressible viscous micropolar fluid model, Nonlinear Anal.: Theory, Methods Appl., 149 (2017), 41-55.  doi: 10.1016/j.na.2016.10.009.

[22]

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.

[23]

Q. Q. Liu and P. X. Zhang, Long-time behavior of solution to the compressible micropolar fluids with external force, Nonlinear Anal. Real World Appl., 40 (2018), 361-376.  doi: 10.1016/j.nonrwa.2017.08.007.

[24]

[0-8176-4008-8] G. Lukaszewicz, Micropolar fluids. Theory and applications. Modeling and Simulation in Science, Engineering and Technology, Birkh${\rm{\ddot a}}$user, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.

[25]

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. 

[26]

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.

[27]

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.

[28]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A local existence theorem, Glas. Mat. Ser. III, 33 (1998), 71–91.

[29]

N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: A global existence theorem, Glas. Mat. Ser. III, 33 (1998), 199-208. 

[30]

N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: Regularity of the solution, Rad. Mat., 10 (2001), 181-193. 

[31]

N. Mujakovi$\acute{c}$, 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.

[32]

N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: Stabilization of the solution, Proceedings of the Conference on Applied Mathematics and Scientific Computing, 253–262, Springer, Dordrecht, 2005. doi: 10.1007/1-4020-3197-1_18.

[33]

N. Mujakovi$\acute{c}$, Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A local existence theorem, Ann. Univ. Ferrara Sez. VII Sci. Mat., 53 (2007), 361-379.  doi: 10.1007/s11565-007-0023-z.

[34]

N. Mujakovi$\acute{c}$, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution, Bound. Value Probl., 2008 (2008), Article ID 189748, 15pp. doi: 10.1155/2008/189748.

[35]

N. Mujakovi$\acute{c}$, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A global existence theorem, Math. Inequal. Appl., 12 (2009), 651-662.  doi: 10.7153/mia-12-49.

[36]

N. Mujakovi$\acute{c}$, One-dimensional compressible viscous micropolar fluid model: stabilization of the solution for the Cauchy problem, Bound. Value Probl., (2010), Article ID 796065, 21pp. doi: 10.1155/2010/796065.

[37]

N. Mujakovi$\acute{c}$, The existence of a global solution for one dimensional compressible viscous micropolar fluid with non-homogeneous boundary conditions for temperature, Nonlinear Anal. Real World Appl., 19 (2014), 19-30.  doi: 10.1016/j.nonrwa.2014.02.006.

[38]

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.

[39]

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.

[40]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107–132.

[41]

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.

[42]

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.

[43]

Z. G. Wu and W. K. Wang, The pointwise estimates of diffusion wave of the compressible micropolar fluids, J. Differential Equations, 265 (2018), 2544-2576.  doi: 10.1016/j.jde.2018.04.039.

[44]

H. Y. Yin, Stability of stationary solutions for inflow problem on the micropolar fluid model, Z. Angew. Math. Phys., 68 (2017), Paper No. 44, 13 pp. doi: 10.1007/s00033-017-0789-5.

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Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

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