-
Previous Article
Uniform stabilization of 1-D Schrödinger equation with internal difference-type control
- DCDS-B Home
- This Issue
-
Next Article
Limiting behavior of unstable manifolds for spdes in varying phase spaces
Singularity formation to the nonhomogeneous magneto-micropolar fluid equations
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
We consider the Cauchy problem of nonhomogeneous magneto-micropolar fluid equations with zero density at infinity in the entire space $ \mathbb{R}^2 $. We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity and magnetic field.
References:
[1] |
G. Ahmadi and M. Shahinpoor,
Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.
doi: 10.1016/0020-7225(74)90042-1. |
[2] |
B. Berkovski and V. Bashtovoy, Magnetic Fluids and Applications Handbook, Begell House, New York, 1996. |
[3] |
P. Braz e Silva, L. Friz and M. A. Rojas-Medar,
Exponential stability for magneto-micropolar fluids, Nonlinear Anal., 143 (2016), 211-223.
doi: 10.1016/j.na.2016.05.015. |
[4] |
H. Brézis and S. Wainger,
A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[5] |
B. Desjardins,
Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.
doi: 10.1007/s002050050025. |
[6] |
J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), Paper No. 7.
doi: 10.1007/s40818-019-0064-5. |
[7] |
M. Li and H. Shang,
Large time decay of solutions for the 3D magneto-micropolar equations, Nonlinear Anal. Real World Appl., 44 (2018), 479-496.
doi: 10.1016/j.nonrwa.2018.05.013. |
[8] |
Z. Liang,
Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differential Equations, 258 (2015), 2633-2654.
doi: 10.1016/j.jde.2014.12.015. |
[9] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.
![]() ![]() |
[10] |
G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Baston, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[11] |
B. Lü, Z. Xu and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.
doi: 10.1016/j.matpur.2016.10.009. |
[12] |
L. Ma,
On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity, Nonlinear Anal. Real World Appl., 40 (2018), 95-129.
doi: 10.1016/j.nonrwa.2017.08.014. |
[13] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[14] |
M. A. Rojas-Medar,
Magneto-micropolar fluid motion: on the convergence rate of the spectral Galerkin approximations, Z. Angew. Math. Mech., 77 (1997), 723-732.
doi: 10.1002/zamm.19970771003. |
[15] |
M. A. Rojas-Medar,
Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.
|
[16] |
H. Shang and J. Zhao,
Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion, Nonlinear Anal., 150 (2017), 194-209.
doi: 10.1016/j.na.2016.11.011. |
[17] |
Z. Tan, W. Wu and J. Zhou,
Global existence and decay estimate of solutions to magneto-micropolar fluid equations, J. Differential Equations, 266 (2019), 4137-4169.
doi: 10.1016/j.jde.2018.09.027. |
[18] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001.
doi: 10.1090/chel/343. |
[19] |
Y. Wang and K. Wang,
Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity, Nonlinear Anal. Real World Appl., 33 (2017), 348-362.
doi: 10.1016/j.nonrwa.2016.07.003. |
[20] |
Y.-Z. Wang and Y.-X. Wang,
Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity, Math. Methods Appl. Sci., 34 (2011), 2125-2135.
|
[21] |
K. Yamazaki,
Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207.
doi: 10.3934/dcds.2015.35.2193. |
[22] |
K. Yamazaki,
Large deviation principle for the micropolar, magneto-micropolar fluid systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 913-938.
doi: 10.3934/dcdsb.2018048. |
[23] |
J. Yuan,
Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130.
|
[24] |
P. Zhang and M. Zhu,
Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304-2314.
doi: 10.1016/j.camwa.2018.08.041. |
[25] |
X. Zhong, Global existence and exponential decay of strong solutions to nonhomogeneous magneto-micropolar fluid equations with large initial data and vacuum, J. Math. Fluid Mech., 22 (2020), Paper No. 35.
doi: 10.1007/s00021-020-00498-3. |
[26] |
X. Zhong,
A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4603-4615.
doi: 10.3934/dcdsb.2020115. |
[27] |
X. Zhong, Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density, Anal. Appl. (Singap.).
doi: 10.1142/S0219530519500167. |
show all references
References:
[1] |
G. Ahmadi and M. Shahinpoor,
Universal stability of magneto-micropolar fluid motions, Internat. J. Engrg. Sci., 12 (1974), 657-663.
doi: 10.1016/0020-7225(74)90042-1. |
[2] |
B. Berkovski and V. Bashtovoy, Magnetic Fluids and Applications Handbook, Begell House, New York, 1996. |
[3] |
P. Braz e Silva, L. Friz and M. A. Rojas-Medar,
Exponential stability for magneto-micropolar fluids, Nonlinear Anal., 143 (2016), 211-223.
doi: 10.1016/j.na.2016.05.015. |
[4] |
H. Brézis and S. Wainger,
A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.
doi: 10.1080/03605308008820154. |
[5] |
B. Desjardins,
Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.
doi: 10.1007/s002050050025. |
[6] |
J. Li and Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE, 5 (2019), Paper No. 7.
doi: 10.1007/s40818-019-0064-5. |
[7] |
M. Li and H. Shang,
Large time decay of solutions for the 3D magneto-micropolar equations, Nonlinear Anal. Real World Appl., 44 (2018), 479-496.
doi: 10.1016/j.nonrwa.2018.05.013. |
[8] |
Z. Liang,
Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differential Equations, 258 (2015), 2633-2654.
doi: 10.1016/j.jde.2014.12.015. |
[9] |
P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.
![]() ![]() |
[10] |
G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Baston, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[11] |
B. Lü, Z. Xu and X. Zhong,
Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.
doi: 10.1016/j.matpur.2016.10.009. |
[12] |
L. Ma,
On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity, Nonlinear Anal. Real World Appl., 40 (2018), 95-129.
doi: 10.1016/j.nonrwa.2017.08.014. |
[13] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[14] |
M. A. Rojas-Medar,
Magneto-micropolar fluid motion: on the convergence rate of the spectral Galerkin approximations, Z. Angew. Math. Mech., 77 (1997), 723-732.
doi: 10.1002/zamm.19970771003. |
[15] |
M. A. Rojas-Medar,
Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.
|
[16] |
H. Shang and J. Zhao,
Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion, Nonlinear Anal., 150 (2017), 194-209.
doi: 10.1016/j.na.2016.11.011. |
[17] |
Z. Tan, W. Wu and J. Zhou,
Global existence and decay estimate of solutions to magneto-micropolar fluid equations, J. Differential Equations, 266 (2019), 4137-4169.
doi: 10.1016/j.jde.2018.09.027. |
[18] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001.
doi: 10.1090/chel/343. |
[19] |
Y. Wang and K. Wang,
Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity, Nonlinear Anal. Real World Appl., 33 (2017), 348-362.
doi: 10.1016/j.nonrwa.2016.07.003. |
[20] |
Y.-Z. Wang and Y.-X. Wang,
Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity, Math. Methods Appl. Sci., 34 (2011), 2125-2135.
|
[21] |
K. Yamazaki,
Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst., 35 (2015), 2193-2207.
doi: 10.3934/dcds.2015.35.2193. |
[22] |
K. Yamazaki,
Large deviation principle for the micropolar, magneto-micropolar fluid systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 913-938.
doi: 10.3934/dcdsb.2018048. |
[23] |
J. Yuan,
Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci., 31 (2008), 1113-1130.
|
[24] |
P. Zhang and M. Zhu,
Global regularity of 3D nonhomogeneous incompressible magneto-micropolar system with the density-dependent viscosity, Comput. Math. Appl., 76 (2018), 2304-2314.
doi: 10.1016/j.camwa.2018.08.041. |
[25] |
X. Zhong, Global existence and exponential decay of strong solutions to nonhomogeneous magneto-micropolar fluid equations with large initial data and vacuum, J. Math. Fluid Mech., 22 (2020), Paper No. 35.
doi: 10.1007/s00021-020-00498-3. |
[26] |
X. Zhong,
A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4603-4615.
doi: 10.3934/dcdsb.2020115. |
[27] |
X. Zhong, Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density, Anal. Appl. (Singap.).
doi: 10.1142/S0219530519500167. |
[1] |
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 |
[2] |
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 |
[3] |
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 |
[4] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
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 |
[8] |
Xin Zhong. Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021296 |
[9] |
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 |
[10] |
Yong Zhou, Jishan Fan. Regularity criteria of strong solutions to a problem of magneto-elastic interactions. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1697-1704. doi: 10.3934/cpaa.2010.9.1697 |
[11] |
Sen Ming, Han Yang, Xiongmei Fan. Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1773-1792. doi: 10.3934/cpaa.2022046 |
[12] |
Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6017-6026. doi: 10.3934/dcdsb.2020377 |
[13] |
Yang Liu, Nan Zhou, Renying Guo. Global solvability to the 3D incompressible magneto-micropolar system with vacuum. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022061 |
[14] |
Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643 |
[15] |
Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163 |
[16] |
Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164 |
[17] |
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 |
[18] |
Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811 |
[19] |
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 |
[20] |
Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]