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Evolutionary de Rham-Hodge method
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. Google Scholar |
[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. Google Scholar |
[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. |
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