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A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
doi: 10.1512/iumj.1967.16.16001. |
[8] |
A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0555-5. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[17] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
doi: 10.1512/iumj.1967.16.16001. |
[8] |
A. C. Eringen, Microcontinuum Field Theories. I. Foundations and Solids, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0555-5. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[17] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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