-
Previous Article
Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains
- CPAA Home
- This Issue
-
Next Article
Shock polars for non-polytropic compressible potential flow
Global behavior for the classical solution 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. | 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 |
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.
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
Other articles
by authors
[Back to Top]