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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  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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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