American Institute of Mathematical Sciences

Advanced
May  2022, 21(5): 1595-1620. doi: 10.3934/cpaa.2022033

Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry

Lan Huang 1, , Zhiying Sun 1, , Xin-Guang Yang 2,, and  Alain Miranville 2,3,

1. 

College of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
2. 

Department of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
3. 

Laboratoire de Mathématiques et Applications, UMR CNRS 7348-SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie-Téléport 2 86962, Chasseneuil Futuroscope Cedex, France

* Corresponding Author

Received  September 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: This paper was partially supported by the NSFC (No. 11501199), the Young Key Teachers Project in Higher Vocational Colleges of Henan Province (No. 2020GZGG109), Young Backbone Teachers in Henan Province (No. 2018GGJS039), Incubation Fund Project of Henan Normal University (No. 2020PL17), Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003)

This paper is concerned with the global solutions of the 3D compressible micropolar fluid model in the domain to a subset of $ R^3 $ bounded with two coaxial cylinders that present the solid thermo-insulated walls, which is in a thermodynamical sense perfect and polytropic. Compared with the classical Navier-Stokes equations, the angular velocity $ w $ in this model brings benefit that is the damping term -$ uw $ can provide extra regularity of $ w $. At the same time, the term $ uw^2 $ is bad, it increases the nonlinearity of our system. Moreover, the regularity and exponential stability in $ H^4 $ also are proved.

Keywords: Exponential stability, cylindrical symmetry, regularity, micropolar fluid.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35Q35.
Citation: Lan Huang, Zhiying Sun, Xin-Guang Yang, Alain Miranville. Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1595-1620. doi: 10.3934/cpaa.2022033
References:
[1]

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

[2]

A. Bašić-Šiško and I. Dražić, Global solution to a one-dimensional model of viscous and heat-conducting micropolar real gas flow, J. Math. Anal. Appl., (2021), 124690. 

[3]

A. Bašić-Šiško and I. Dražić, Uniqueness of generalized solution to micropolar viscous real gas flow with homogeneous boundary conditions, Math. Meth. Appl. Sci., 44 (2021), 4330-4341.  doi: 10.1002/mma.7032.

[4]

A. Borrelli, G. Giantesio and M. 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.

[5]

G. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609-634.  doi: 10.1137/0523031.

[6] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. 
[7]

G. Chen, D. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Commun. Partial Differ. Equ., 25 (2000), 2233-2257.  doi: 10.1080/03605300008821583.

[8]

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

[9]

H. Cui and H. 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.

[10]

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.

[11]

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.

[12]

I. Dražić, L. Simčić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution, J. Math. Anal. Appl., 435 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.

[13]

I. Dražić, N. 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.

[14]

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. 

[15]

I. Dražić, L. 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. 

[16]

I. Dražić, A. Bašić-Šiško and L. Simčić, One-dimensional model and numerical solution to the viscous and heat-conducting micropolar real gas flow with homogeneous boundary conditions, 2020, Preprint.

[17]

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.

[18]

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

[19]

Z. Feng and C. Zhu, Global classical large solution to compressible viscous micropolar and heat-conduting fluids with vacuum, Discret. Contin. Dynam. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.

[20] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[21]

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

[22]

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

[23]

L. Huang and R. 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.

[24]

L. Huang and C. 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.

[25]

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.

[26]

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.

[27]

L. Huang, Z. Sun and X. Yang, Large time behavior of spherically symmetrical micropolar fluid in unbounded domain, Appl. Math. Optim., 84 (2021), S1607-S1638.  doi: 10.1007/s00245-021-09806-3.

[28]

A. Kazhikhov and V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.

[29]

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.

[30]

M. Lewicka and P. Mucha, On temporal asymptotics for the pth power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969.  doi: 10.1016/j.na.2003.12.001.

[31]

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

[32]

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

[33]

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.

[34]

D. Maltese, M. Michálek, B. Mucha Piotr, A. Novotný, M. Pokorný and E. Zatorska, Existence of weak solutions for compressible Navier-Stokes equations with entropy transport, J. Differ. Equ., 261 (2016), 4448-4485.  doi: 10.1016/j.jde.2016.06.029.

[35]

I. Papautsky, J. Brazzle, T. Ameel and A. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory, Sens. and Actuators A: Phys., 73 (1999), 101-108. 

[36]

Z. Sun, L. Huang and X. Yang, Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry, Electron. Res. Arch., 28 (2020), 861-878.  doi: 10.3934/era.2020045.

[37]

L. Wan and T. Wang, Asymptotic behavior for the one-dimensional pth power Newtonian fluid in unbounded domains, Math. Meth. Appl. Sci., 39 (2016), 1020-1025.  doi: 10.1002/mma.3539.

[38]

L. Wan and L. Zhang, Global solutions to the micropolar compressible flow with constant coefficients and vacuum, Nonlinear Anal. RWA, 51 (2020), 102990, 14 pp. doi: 10.1016/j.nonrwa.2019.102990.

[39]

T. Wang, One dimensional p-th power Newtonian fluid with temperature-dependent thermal conductivity, Commun. Pure Appl. Anal., 15 (2016), 477-494.  doi: 10.3934/cpaa.2016.15.477.

show all references

References:
[1]

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

[2]

A. Bašić-Šiško and I. Dražić, Global solution to a one-dimensional model of viscous and heat-conducting micropolar real gas flow, J. Math. Anal. Appl., (2021), 124690. 

[3]

A. Bašić-Šiško and I. Dražić, Uniqueness of generalized solution to micropolar viscous real gas flow with homogeneous boundary conditions, Math. Meth. Appl. Sci., 44 (2021), 4330-4341.  doi: 10.1002/mma.7032.

[4]

A. Borrelli, G. Giantesio and M. 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.

[5]

G. Chen, Global solutions to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal., 23 (1992), 609-634.  doi: 10.1137/0523031.

[6] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. 
[7]

G. Chen, D. Hoff and K. Trivisa, Global solutions of the compressible Navier-Stokes equations with large discontinuous initial data, Commun. Partial Differ. Equ., 25 (2000), 2233-2257.  doi: 10.1080/03605300008821583.

[8]

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

[9]

H. Cui and H. 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.

[10]

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.

[11]

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.

[12]

I. Dražić, L. Simčić and N. Mujaković, 3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution, J. Math. Anal. Appl., 435 (2016), 162-183.  doi: 10.1016/j.jmaa.2016.01.071.

[13]

I. Dražić, N. 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.

[14]

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. 

[15]

I. Dražić, L. 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. 

[16]

I. Dražić, A. Bašić-Šiško and L. Simčić, One-dimensional model and numerical solution to the viscous and heat-conducting micropolar real gas flow with homogeneous boundary conditions, 2020, Preprint.

[17]

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.

[18]

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

[19]

Z. Feng and C. Zhu, Global classical large solution to compressible viscous micropolar and heat-conduting fluids with vacuum, Discret. Contin. Dynam. Syst., 39 (2019), 3069-3097.  doi: 10.3934/dcds.2019127.

[20] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[21]

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

[22]

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

[23]

L. Huang and R. 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.

[24]

L. Huang and C. 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.

[25]

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.

[26]

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.

[27]

L. Huang, Z. Sun and X. Yang, Large time behavior of spherically symmetrical micropolar fluid in unbounded domain, Appl. Math. Optim., 84 (2021), S1607-S1638.  doi: 10.1007/s00245-021-09806-3.

[28]

A. Kazhikhov and V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.

[29]

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.

[30]

M. Lewicka and P. Mucha, On temporal asymptotics for the pth power viscous reactive gas, Nonlinear Anal., 57 (2004), 951-969.  doi: 10.1016/j.na.2003.12.001.

[31]

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

[32]

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

[33]

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.

[34]

D. Maltese, M. Michálek, B. Mucha Piotr, A. Novotný, M. Pokorný and E. Zatorska, Existence of weak solutions for compressible Navier-Stokes equations with entropy transport, J. Differ. Equ., 261 (2016), 4448-4485.  doi: 10.1016/j.jde.2016.06.029.

[35]

I. Papautsky, J. Brazzle, T. Ameel and A. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory, Sens. and Actuators A: Phys., 73 (1999), 101-108. 

[36]

Z. Sun, L. Huang and X. Yang, Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry, Electron. Res. Arch., 28 (2020), 861-878.  doi: 10.3934/era.2020045.

[37]

L. Wan and T. Wang, Asymptotic behavior for the one-dimensional pth power Newtonian fluid in unbounded domains, Math. Meth. Appl. Sci., 39 (2016), 1020-1025.  doi: 10.1002/mma.3539.

[38]

L. Wan and L. Zhang, Global solutions to the micropolar compressible flow with constant coefficients and vacuum, Nonlinear Anal. RWA, 51 (2020), 102990, 14 pp. doi: 10.1016/j.nonrwa.2019.102990.

[39]

T. Wang, One dimensional p-th power Newtonian fluid with temperature-dependent thermal conductivity, Commun. Pure Appl. Anal., 15 (2016), 477-494.  doi: 10.3934/cpaa.2016.15.477.

[1]

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
[2]

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
[3]

Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure and Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185
[4]

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

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 and Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583
[6]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[7]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045
[8]

Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048
[9]

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
[10]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations and Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016
[11]

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 and Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635
[12]

Haibo Cui, Junpei Gao, Lei Yao. Asymptotic behavior of the one-dimensional compressible micropolar fluid model. Electronic Research Archive, 2021, 29 (2) : 2063-2075. doi: 10.3934/era.2020105
[13]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations and Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019
[14]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021
[15]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083
[16]

Bo-Qing Dong, Jiahong Wu, Xiaojing Xu, Zhuan Ye. Global regularity for the 2D micropolar equations with fractional dissipation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4133-4162. doi: 10.3934/dcds.2018180
[17]

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
[18]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189
[19]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
[20]

Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (128)
  • HTML views (98)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy