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December  2021, 26(12): 6339-6357. doi: 10.3934/dcdsb.2021021

Singularity formation to the nonhomogeneous magneto-micropolar fluid equations

Xin Zhong

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  August 2020 Revised  November 2020 Published  December 2021 Early access  January 2021

Fund Project: This research was partially supported by National Natural Science Foundation of China (No. 11901474) and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2020082)

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.

Keywords: nonhomogeneous magneto-micropolar fluid equations, strong solutions, Cauchy problem, singularity formation.
Mathematics Subject Classification: 35Q35, 35B65.
Citation: 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
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.

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