-
Previous Article
Robin problems with indefinite linear part and competition phenomena
- CPAA Home
- This Issue
-
Next Article
Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space
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 |
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.
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. |
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. 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. |
[1] |
Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021021 |
[2] |
Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189 |
[3] |
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 |
[4] |
Kengo Nakai, Yoshitaka Saiki. Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1079-1092. doi: 10.3934/dcdss.2020352 |
[5] |
Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004 |
[6] |
Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021028 |
[7] |
Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021009 |
[8] |
Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159 |
[9] |
Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387 |
[10] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[11] |
Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227 |
[12] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
[13] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[14] |
Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 |
[15] |
Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020466 |
[16] |
Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 |
[17] |
Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020401 |
[18] |
A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020441 |
[19] |
Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275 |
[20] |
Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020050 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]