American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210
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.  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. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] B. Q. Dong, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. doi: 10.1512/iumj.1967.16.16001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] S. Kawashima, T. Nakamura, S. 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.  Google Scholar [20] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [28] N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A local existence theorem, Glas. Mat. Ser. III, 33 (1998), 71–91.  Google Scholar [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.   Google Scholar [30] N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: Regularity of the solution, Rad. Mat., 10 (2001), 181-193.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] 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.  Google Scholar [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.  Google Scholar [40] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107–132.  Google Scholar [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.  Google Scholar [42] 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.  Google Scholar [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.  Google Scholar [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.  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.  doi: 10.4208/jpde.v24.n2.5.  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. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] B. Q. Dong, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. doi: 10.1512/iumj.1967.16.16001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] S. Kawashima, T. Nakamura, S. 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.  Google Scholar [20] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [28] N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A local existence theorem, Glas. Mat. Ser. III, 33 (1998), 71–91.  Google Scholar [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.   Google Scholar [30] N. Mujakovi$\acute{c}$, One-dimensional flow of a compressible viscous micropolar fluid: Regularity of the solution, Rad. Mat., 10 (2001), 181-193.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] 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.  Google Scholar [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.  Google Scholar [40] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107–132.  Google Scholar [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.  Google Scholar [42] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009 [2] Hiroshi Inoue, Kei Matsuura, Mitsuharu Ôtani. Strong solutions of magneto-micropolar fluid equation. Conference Publications, 2003, 2003 (Special) : 439-448. doi: 10.3934/proc.2003.2003.439 [3] Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031 [4] Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021 [5] Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199 [6] Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1 [7] Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090 [8] Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115 [9] Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765 [10] Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinetic & Related Models, 2015, 8 (1) : 29-51. doi: 10.3934/krm.2015.8.29 [11] Lvqiao liu, Lan Zhang. Optimal decay to the non-isentropic compressible micropolar fluids. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4575-4598. doi: 10.3934/cpaa.2020207 [12] Youcef Amirat, Kamel Hamdache. Weak solutions to stationary equations of heat transfer in a magnetic fluid. Communications on Pure & Applied Analysis, 2019, 18 (2) : 709-734. doi: 10.3934/cpaa.2019035 [13] Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048 [14] Davide Guidetti. Convergence to a stationary state of solutions to inverse problems of parabolic type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 711-722. doi: 10.3934/dcdss.2013.6.711 [15] M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827 [16] Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040 [17] Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure & Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49 [18] Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 [19] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [20] Pascal Bégout, Jesús Ildefonso Díaz. A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3371-3382. doi: 10.3934/dcds.2014.34.3371

2019 Impact Factor: 1.27

Article outline