-
Previous Article
Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise
- DCDS-B Home
- This Issue
-
Next Article
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals
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. |
[1] |
Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 |
[2] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
[3] |
Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021021 |
[4] |
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 |
[5] |
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 |
[6] |
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, 2020 doi: 10.3934/dcdsb.2020348 |
[7] |
Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 |
[8] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
[9] |
Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 |
[10] |
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 |
[11] |
Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 |
[12] |
Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 |
[13] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
[14] |
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 |
[15] |
Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 |
[16] |
Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171 |
[17] |
Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189 |
[18] |
Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 |
[19] |
Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 |
[20] |
Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]