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doi: 10.3934/dcdsb.2021021
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.   Google Scholar
[10]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[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.  Google Scholar

[15]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.   Google Scholar
[10]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Baston, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[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.  Google Scholar

[15]

M. A. Rojas-Medar, Magneto-micropolar fluid motion: Existence and uniqueness of strong solution, Math. Nachr., 188 (1997), 301-319.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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