June  2020, 28(2): 861-878. doi: 10.3934/era.2020045

Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry

1. 

College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

2. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

* Corresponding author: Lan Huang

Received  January 2020 Revised  April 2020 Published  June 2020

Fund Project: The first author is supported by the NSFC (No. 11871212 and No. 11501199) and Natural Science Foundation of Henan Province (No. 20A110026)

This paper is concerned with three-dimensional compressible viscous and heat-conducting micropolar fluid in the domain to the subset of $ R^3 $ bounded with two coaxial cylinders that present the solid thermoinsulated walls, being in a thermodynamical sense perfect and polytropic. We prove that the regularity and the exponential stability in $ H^2 $.

Citation: Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045
References:
[1]

S. Antontscv, A. Kazhikhov and V. Monakhov, Boundary Problems in Mechanics of Nonhomogeneous Fluids, Amsterdam, New York, 1990. Google Scholar

[2]

A. BorrelliG. Giantesio and M. C. Patria, An exact solution for the 3D MHD stagnation-point flow of a micropolar fluid, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 121-135.  doi: 10.1016/j.cnsns.2014.04.011.  Google Scholar

[3]

J. ChenC. L. Liang and J. D. Lee, Numerical simulation for unsteady compressible micropolar fluid flow, Comput. Fluids, 66 (2012), 1-9.  doi: 10.1016/j.compfluid.2012.05.015.  Google Scholar

[4]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  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]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A local existence theorem, Bound. Value Probl., 2012 (2012), 25 pp. doi: 10.1186/1687-2770-2012-69.  Google Scholar

[7]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Uniqueness of a generalized solution, Bound. Value Probl., 2014 (2014), 17 pp. doi: 10.1186/s13661-014-0226-z.  Google Scholar

[8]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A global existence theorem, Bound. Value Probl., 2015 (2015), 21 pp. doi: 10.1186/s13661-015-0357-x.  Google Scholar

[9]

I. Dražić and N. Mujaković, 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. DražićL. Simčić and N. Mujaković, 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]

I. DražićN. Mujaković and N. Črnjarić-Žic, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Derivation of the model and a numerical solution, Math. Comput. Simul., 140 (2017), 107-124.  doi: 10.1016/j.matcom.2017.03.006.  Google Scholar

[12]

I. Dražić, 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: A global existence theorem, Math. Meth. Appl. Sci., 40 (2017), 4785-4801.   Google Scholar

[13]

I. DražićL. Simčić and N. Mujaković, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Uniqueness of a generalized solution, Math. Meth. Appl. Sci., 40 (2017), 2686-2701.  doi: 10.1002/mma.4191.  Google Scholar

[14]

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

[15]

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. RWA, 42 (2018), 71-92.  doi: 10.1016/j.nonrwa.2017.12.006.  Google Scholar

[16]

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

[17]

Z. F. Feng and C. J. Zhu, Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum, Discrete Contin. Dyn. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.  Google Scholar

[18]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar

[19]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[20]

B. L. Guo and P. C. Zhu, Asymptotic behavior of the solution to the system for a viscous reactive gas, J. Differential Equstions, 155 (1999), 177-202.  doi: 10.1006/jdeq.1998.3578.  Google Scholar

[21]

L. Huang and I. Dražić, Large-time behavior of solutions to the 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry, Math. Meth. Appl. Sci., 41 (2018), 7888-7905.  doi: 10.1002/mma.5250.  Google Scholar

[22]

L. Huang and I. Drazic, Exponential stability for the compressible micropolar fluid with cylinder symmetry in $R^3$, J. Math. Phys., 60 (2019), 021507, 14 pp. doi: 10.1063/1.5017652.  Google Scholar

[23]

L. Huang and C. X. Kong, Global behavior for compressible viscous micropolar fluid with spherical symmetry, J. Math. Anal. Appl., 443 (2016), 1158-1178.  doi: 10.1016/j.jmaa.2016.05.056.  Google Scholar

[24]

L. Huang and R. X. Lian, Exponential stability of spherically symmetric solutions for compressible viscous micropolar fluid, J. Math. Phys., 56 (2015), 071503, 12 pp. doi: 10.1063/1.4926426.  Google Scholar

[25]

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

[26]

L. Huang, Z. Sun and X. Yang, Large time behavior of spherically symmetrical micropolar fluid in unbounded domain, Preprint, (2019). Google Scholar

[27]

L. Huang, X.-G. Yang, Y. J. Lu and T. wang, Global attractor for a nonlinear one-dimensional compressible viscous micropolar fluid model, Z. Angew. Math. Phys., 70 (2019), 20 pp. doi: 10.1007/s00033-019-1083-5.  Google Scholar

[28]

T. Kato, Strong $L^P$-solutions of the Navier-Stokes equations in $R^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.  Google Scholar

[29]

J. Li and Z. L. Liang, Some uniform estimates and large-time behavior for one dimensional compressible Navier-Stokes in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.  Google Scholar

[30]

Z. Liang and F. H. Lin, Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[31]

Y. K. LiaoT. Wang and H. J. Zhao, Global spherical symmetric flows for a viscous radiative and reactive gas in an exterior domain, J. Differential Equations, 266 (2019), 6459-6506.  doi: 10.1016/j.jde.2018.11.008.  Google Scholar

[32]

G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[33]

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

[34]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A global existence theorem, Glas. Mat. Ser. III, 33 (1998), 199-208.   Google Scholar

[35]

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

[36]

N. Mujaković, Nonhomogenerous 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

[37]

N. Mujaković, Nonhomogenerous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution, Bound. Value Probl., 2008 (2008), 189748, 15 pp. doi: 10.1155/2008/189748.  Google Scholar

[38]

N. Mujaković, Nonhomogenerous 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

[39]

I. PapautskyJ. BrazzleT. Ameel and A. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory, Sens. and Actuators A: Phys., 73 (1999), 101-108.  doi: 10.1016/S0924-4247(98)00261-1.  Google Scholar

[40]

W. H. WangT. G. Qin and Q. Y. Bie, Global well-posedness and analyticity results to 3-D generalized, Nonlinear Anal. RWA, 59 (2016), 65-70.   Google Scholar

[41]

Y. Z. Wang and K. Y. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. RWA, 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[42]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

show all references

References:
[1]

S. Antontscv, A. Kazhikhov and V. Monakhov, Boundary Problems in Mechanics of Nonhomogeneous Fluids, Amsterdam, New York, 1990. Google Scholar

[2]

A. BorrelliG. Giantesio and M. C. Patria, An exact solution for the 3D MHD stagnation-point flow of a micropolar fluid, Commun. Nonlinear Sci. Numer. Simul., 20 (2015), 121-135.  doi: 10.1016/j.cnsns.2014.04.011.  Google Scholar

[3]

J. ChenC. L. Liang and J. D. Lee, Numerical simulation for unsteady compressible micropolar fluid flow, Comput. Fluids, 66 (2012), 1-9.  doi: 10.1016/j.compfluid.2012.05.015.  Google Scholar

[4]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  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]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A local existence theorem, Bound. Value Probl., 2012 (2012), 25 pp. doi: 10.1186/1687-2770-2012-69.  Google Scholar

[7]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Uniqueness of a generalized solution, Bound. Value Probl., 2014 (2014), 17 pp. doi: 10.1186/s13661-014-0226-z.  Google Scholar

[8]

I. Dražić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A global existence theorem, Bound. Value Probl., 2015 (2015), 21 pp. doi: 10.1186/s13661-015-0357-x.  Google Scholar

[9]

I. Dražić and N. Mujaković, 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. DražićL. Simčić and N. Mujaković, 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]

I. DražićN. Mujaković and N. Črnjarić-Žic, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Derivation of the model and a numerical solution, Math. Comput. Simul., 140 (2017), 107-124.  doi: 10.1016/j.matcom.2017.03.006.  Google Scholar

[12]

I. Dražić, 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: A global existence theorem, Math. Meth. Appl. Sci., 40 (2017), 4785-4801.   Google Scholar

[13]

I. DražićL. Simčić and N. Mujaković, Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Uniqueness of a generalized solution, Math. Meth. Appl. Sci., 40 (2017), 2686-2701.  doi: 10.1002/mma.4191.  Google Scholar

[14]

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

[15]

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. RWA, 42 (2018), 71-92.  doi: 10.1016/j.nonrwa.2017.12.006.  Google Scholar

[16]

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

[17]

Z. F. Feng and C. J. Zhu, Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum, Discrete Contin. Dyn. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.  Google Scholar

[18]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.  Google Scholar

[19]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Ration. Mech. Anal., 16 (1964), 269-315.  doi: 10.1007/BF00276188.  Google Scholar

[20]

B. L. Guo and P. C. Zhu, Asymptotic behavior of the solution to the system for a viscous reactive gas, J. Differential Equstions, 155 (1999), 177-202.  doi: 10.1006/jdeq.1998.3578.  Google Scholar

[21]

L. Huang and I. Dražić, Large-time behavior of solutions to the 3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry, Math. Meth. Appl. Sci., 41 (2018), 7888-7905.  doi: 10.1002/mma.5250.  Google Scholar

[22]

L. Huang and I. Drazic, Exponential stability for the compressible micropolar fluid with cylinder symmetry in $R^3$, J. Math. Phys., 60 (2019), 021507, 14 pp. doi: 10.1063/1.5017652.  Google Scholar

[23]

L. Huang and C. X. Kong, Global behavior for compressible viscous micropolar fluid with spherical symmetry, J. Math. Anal. Appl., 443 (2016), 1158-1178.  doi: 10.1016/j.jmaa.2016.05.056.  Google Scholar

[24]

L. Huang and R. X. Lian, Exponential stability of spherically symmetric solutions for compressible viscous micropolar fluid, J. Math. Phys., 56 (2015), 071503, 12 pp. doi: 10.1063/1.4926426.  Google Scholar

[25]

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

[26]

L. Huang, Z. Sun and X. Yang, Large time behavior of spherically symmetrical micropolar fluid in unbounded domain, Preprint, (2019). Google Scholar

[27]

L. Huang, X.-G. Yang, Y. J. Lu and T. wang, Global attractor for a nonlinear one-dimensional compressible viscous micropolar fluid model, Z. Angew. Math. Phys., 70 (2019), 20 pp. doi: 10.1007/s00033-019-1083-5.  Google Scholar

[28]

T. Kato, Strong $L^P$-solutions of the Navier-Stokes equations in $R^m$ with applications to weak solutions, Math. Z., 187 (1984), 471-480.  doi: 10.1007/BF01174182.  Google Scholar

[29]

J. Li and Z. L. Liang, Some uniform estimates and large-time behavior for one dimensional compressible Navier-Stokes in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.  Google Scholar

[30]

Z. Liang and F. H. Lin, Global mild solutions of Navier-Stokes equations, Commun. Pure Appl. Math., 64 (2011), 1297-1304.  doi: 10.1002/cpa.20361.  Google Scholar

[31]

Y. K. LiaoT. Wang and H. J. Zhao, Global spherical symmetric flows for a viscous radiative and reactive gas in an exterior domain, J. Differential Equations, 266 (2019), 6459-6506.  doi: 10.1016/j.jde.2018.11.008.  Google Scholar

[32]

G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[33]

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

[34]

N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: A global existence theorem, Glas. Mat. Ser. III, 33 (1998), 199-208.   Google Scholar

[35]

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

[36]

N. Mujaković, Nonhomogenerous 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

[37]

N. Mujaković, Nonhomogenerous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution, Bound. Value Probl., 2008 (2008), 189748, 15 pp. doi: 10.1155/2008/189748.  Google Scholar

[38]

N. Mujaković, Nonhomogenerous 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

[39]

I. PapautskyJ. BrazzleT. Ameel and A. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory, Sens. and Actuators A: Phys., 73 (1999), 101-108.  doi: 10.1016/S0924-4247(98)00261-1.  Google Scholar

[40]

W. H. WangT. G. Qin and Q. Y. Bie, Global well-posedness and analyticity results to 3-D generalized, Nonlinear Anal. RWA, 59 (2016), 65-70.   Google Scholar

[41]

Y. Z. Wang and K. Y. Wang, Global well-posedness of the three dimensional magnetohydrodynamics equations, Nonlinear Anal. RWA, 17 (2014), 245-251.  doi: 10.1016/j.nonrwa.2013.12.002.  Google Scholar

[42]

Z. Ye, Global well-posedness and decay results to 3D generalized viscous magnetohydrodynamic equations, Ann. Mat. Pura Appl., 195 (2016), 1111-1121.  doi: 10.1007/s10231-015-0507-x.  Google Scholar

[1]

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

[2]

Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193

[3]

Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583

[4]

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

[5]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138

[6]

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

[7]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[8]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations & Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019

[9]

Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure & Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635

[10]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations & Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016

[11]

Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180

[12]

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

[13]

Bo-Qing Dong, Zhi-Min Chen. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 765-784. doi: 10.3934/dcds.2009.23.765

[14]

Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020115

[15]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020189

[16]

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, 2020  doi: 10.3934/dcdsb.2020210

[17]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164

[18]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886

[19]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219

[20]

Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425

2018 Impact Factor: 0.263

Metrics

  • PDF downloads (17)
  • HTML views (51)
  • Cited by (0)

Other articles
by authors

[Back to Top]