• Previous Article
    Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance
  • DCDS-B Home
  • This Issue
  • Next Article
    A mathematical model for biodiversity diluting transmission of zika virus through competition mechanics
doi: 10.3934/dcdsb.2021021
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Singularity formation to the nonhomogeneous magneto-micropolar fluid equations

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.

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 SilvaL. 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. LüZ. 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. TanW. 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 SilvaL. 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. LüZ. 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. TanW. 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

[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]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations & Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019

[3]

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

[4]

Kazuo Yamazaki. Large deviation principle for the micropolar, magneto-micropolar fluid systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 913-938. doi: 10.3934/dcdsb.2018048

[5]

Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115

[6]

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 & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583

[7]

Kazuo Yamazaki. Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2193-2207. doi: 10.3934/dcds.2015.35.2193

[8]

Yong Zhou, Jishan Fan. Regularity criteria of strong solutions to a problem of magneto-elastic interactions. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1697-1704. doi: 10.3934/cpaa.2010.9.1697

[9]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[10]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[11]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[12]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164

[13]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090

[14]

Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062

[15]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811

[16]

Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165

[17]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[18]

Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4921-4941. doi: 10.3934/dcds.2021062

[19]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348

[20]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834

2020 Impact Factor: 1.327

Article outline

[Back to Top]