American Institute of Mathematical Sciences

Advanced
December  2020, 25(12): 4603-4615. doi: 10.3934/dcdsb.2020115

A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum

Xin Zhong

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

Received  August 2019 Published  December 2020 Early access  March 2020

Fund Project: This research is supported by National Natural Science Foundation of China (No. 11901474) and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2019130)

We deal with the Cauchy problem of nonhomogeneous 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.

Keywords: Nonhomogeneous micropolar fluid equations, strong solutions, Cauchy problem, blow-up criterion.
Mathematics Subject Classification: 35Q35, 35B65.
Citation: 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
References:
[1]

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

[2]

M. Chen, X. Xu and J. Zhang, The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect, Z. Angew. Math. Phys., 65 (2014), 687-710.  doi: 10.1007/s00033-013-0345-x.  Google Scholar

[3]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar

[4]

B.-Q. Dong, J. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.  Google Scholar

[5]

B.-Q. Dong, J. Wu, X. Xu and Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst., 38 (2018), 4133-4162.  doi: 10.3934/dcds.2018180.  Google Scholar

[6]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[7]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[8]

A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0555-5.  Google Scholar

[9]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[10]

R. H. Guterres, J. R. Nunes and C. F. Perusato, On the large time decay of global solutions for the micropolar dynamics in $L^2(\Bbb{R}^n)$, Nonlinear Anal. Real World Appl., 45 (2019), 789-798.  doi: 10.1016/j.nonrwa.2018.08.002.  Google Scholar

[11]

Q. Jiu, J. Liu, J. Wu and H. Yu, On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation, Z. Angew. Math. Phys., 68 (2017), 24 pp. doi: 10.1007/s00033-017-0855-z.  Google Scholar

[12]

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), 37 pp. doi: 10.1007/s40818-019-0064-5.  Google Scholar

[13]

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

[14]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I. Incompressible Models, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[15]

J. Liu and S. Wang, Initial-boundary value problem for 2D micropolar equations without angular viscosity, Commun. Math. Sci., 16 (2018), 2147-2165.  doi: 10.4310/CMS.2018.v16.n8.a5.  Google Scholar

[16]

G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[17]

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

[18]

B. Nowakowski, Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal. Real World Appl., 14 (2013), 635-660.  doi: 10.1016/j.nonrwa.2012.07.023.  Google Scholar

[19]

Z. Ye, Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6725-6743.  doi: 10.3934/dcdsb.2019164.  Google Scholar

[20]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.  Google Scholar

[21]

X. Zhong, Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density, Analysis and Applications, (2019), 1–29. doi: 10.1142/S0219530519500167.  Google Scholar

[22]

W. Zhu, Sharp well-posedness and ill-posedness for the 3-D micropolar fluid system in Fourier-Besov spaces, Nonlinear Anal. Real World Appl., 46 (2019), 335-351.  doi: 10.1016/j.nonrwa.2018.09.022.  Google Scholar

show all references

References:
[1]

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

[2]

M. Chen, X. Xu and J. Zhang, The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect, Z. Angew. Math. Phys., 65 (2014), 687-710.  doi: 10.1007/s00033-013-0345-x.  Google Scholar

[3]

Q. Chen and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations, 252 (2012), 2698-2724.  doi: 10.1016/j.jde.2011.09.035.  Google Scholar

[4]

B.-Q. Dong, J. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523.  doi: 10.1016/j.jde.2016.11.029.  Google Scholar

[5]

B.-Q. Dong, J. Wu, X. Xu and Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst., 38 (2018), 4133-4162.  doi: 10.3934/dcds.2018180.  Google Scholar

[6]

B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249 (2010), 200-213.  doi: 10.1016/j.jde.2010.03.016.  Google Scholar

[7]

A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.  doi: 10.1512/iumj.1967.16.16001.  Google Scholar

[8]

A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0555-5.  Google Scholar

[9]

G. P. Galdi and S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. Sci., 15 (1977), 105-108.  doi: 10.1016/0020-7225(77)90025-8.  Google Scholar

[10]

R. H. Guterres, J. R. Nunes and C. F. Perusato, On the large time decay of global solutions for the micropolar dynamics in $L^2(\Bbb{R}^n)$, Nonlinear Anal. Real World Appl., 45 (2019), 789-798.  doi: 10.1016/j.nonrwa.2018.08.002.  Google Scholar

[11]

Q. Jiu, J. Liu, J. Wu and H. Yu, On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation, Z. Angew. Math. Phys., 68 (2017), 24 pp. doi: 10.1007/s00033-017-0855-z.  Google Scholar

[12]

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), 37 pp. doi: 10.1007/s40818-019-0064-5.  Google Scholar

[13]

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

[14]

P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I. Incompressible Models, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[15]

J. Liu and S. Wang, Initial-boundary value problem for 2D micropolar equations without angular viscosity, Commun. Math. Sci., 16 (2018), 2147-2165.  doi: 10.4310/CMS.2018.v16.n8.a5.  Google Scholar

[16]

G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-0641-5.  Google Scholar

[17]

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

[18]

B. Nowakowski, Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal. Real World Appl., 14 (2013), 635-660.  doi: 10.1016/j.nonrwa.2012.07.023.  Google Scholar

[19]

Z. Ye, Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6725-6743.  doi: 10.3934/dcdsb.2019164.  Google Scholar

[20]

P. Zhang and M. Zhu, Global regularity of 3D nonhomogeneous incompressible micropolar fluids, Acta Appl. Math., 161 (2019), 13-34.  doi: 10.1007/s10440-018-0202-1.  Google Scholar

[21]

X. Zhong, Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density, Analysis and Applications, (2019), 1–29. doi: 10.1142/S0219530519500167.  Google Scholar

[22]

W. Zhu, Sharp well-posedness and ill-posedness for the 3-D micropolar fluid system in Fourier-Besov spaces, Nonlinear Anal. Real World Appl., 46 (2019), 335-351.  doi: 10.1016/j.nonrwa.2018.09.022.  Google Scholar

[1]

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
[2]

Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923
[3]

Xin Zhong. Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021296
[4]

Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167
[5]

Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139
[6]

Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370
[7]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[8]

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
[9]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6339-6357. doi: 10.3934/dcdsb.2021021
[10]

Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327
[11]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[12]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271
[13]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
[14]

Xin Zhong. A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3249-3264. doi: 10.3934/dcdsb.2018318
[15]

Jeongho Kim, Weiyuan Zou. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain. Kinetic & Related Models, 2020, 13 (3) : 623-651. doi: 10.3934/krm.2020021
[16]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333
[17]

Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140
[18]

Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021093
[19]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[20]

Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (259)
  • HTML views (334)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy