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December  2019, 24(12): 6725-6743. doi: 10.3934/dcdsb.2019164

Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations

Zhuan Ye ,

Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China

* Corresponding author: Zhuan Ye

Received  July 2018 Revised  February 2019 Published  December 2019 Early access  July 2019

Fund Project: The author is supported by the National Natural Science Foundation of China (No. 11701232) and the Natural Science Foundation of Jiangsu Province (No. BK20170224).

This paper addresses the Cauchy problem of the three-dimensional inhomogeneous incompressible micropolar equations. We prove the global existence and exponential decay-in-time of strong solution with vacuum over the whole space $ \mathbb{R}^{3} $ provided that the initial data are sufficiently small. The initial vacuum is allowed.

Keywords: Inhomogeneous micropolar equations, vacuum, global strong solution.
Mathematics Subject Classification: Primary: 35Q35, 35B65, 76N10; Secondary: 76D05.
Citation: Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164
References:
[1]

H. Abidi, G. Gui and P. Zhang, On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Ration. Mech. Anal., 204 (2012), 189-230.  doi: 10.1007/s00205-011-0473-4.

[2]

H. Abidi, G. L. Gui and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 832-881.  doi: 10.1002/cpa.20351.

[3]

S. N. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990.

[4]

Q. L. Chen and C. X. 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.

[5]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.

[6]

W. Craig, X. D. Huang and Y. Wang, Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations, J. Math. Fluid Mech., 15 (2013), 747-758.  doi: 10.1007/s00021-013-0133-6.

[7]

R. Danchin and P. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013), 991-1023.  doi: 10.1007/s00205-012-0586-4.

[8]

R. Danchin and P. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409.

[9]

B. -Q. Dong, J. N. Li and J. H. 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.

[10]

B. -Q. Dong, J. H. Wu, X. J. 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.

[11]

B. -Q. Dong and Z. F. 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.

[12]

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

[13]

L. C. F. Ferreira and J. C. Precioso, Existence of solutions for the 3D-micropolar fluid system with initial data in Besov-Morrey spaces, Z. Angew. Math. Phys., 64 (2013), 1699-1710.  doi: 10.1007/s00033-013-0310-8.

[14]

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.

[15]

C. He, J. Li and B. Lü, On the Cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, arXiv: 1709.05608v1.

[16]

X. D. Huang and Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 259 (2015), 1606-1627.  doi: 10.1016/j.jde.2015.03.008.

[17]

A. V. Kažhikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Dokl. Akad. Nauk., 216 (1974), 1008-1010. 

[18]

J. U. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007.

[19]

O. Ladyzhenskaya and V. Solonnikov, Unique solvability of an initial and boundary value problem for viscous incompressible non-homogeneous fluids, J. Soviet Math., 9 (1978), 697-749. 

[20]

J. K. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.

[21]

P. -L. Lions, Mathematical Topics in Fluid mMechanics. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, vol. 1. Clarendon Press/Oxford University Press, New York, 1996.

[22]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.

[23]

G. Łukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat., 12 (1988), 83-97. 

[24]

G. Łukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat., 13 (1989), 105-120. 

[25]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.  doi: 10.1016/j.jfa.2012.01.022.

[26]

M. Paicu, P. Zhang and Z. F. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.  doi: 10.1080/03605302.2013.780079.

[27]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.

[28]

D. Wang and Z. Ye, Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum, arXiv: 1806.04464v1.

[29]

L. T. Xue, Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777.  doi: 10.1002/mma.1491.

[30]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid systemin a bounded domain, Math. Methods Appl. Sci., 28 (2005), 1507-1526.  doi: 10.1002/mma.617.

[31]

B. Q. Yuan, On the regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025-2036.  doi: 10.1090/S0002-9939-10-10232-9.

[32]

J. W. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient, J. Differential Equations, 259 (2015), 1722-1742.  doi: 10.1016/j.jde.2015.03.011.

[33]

P. X. Zhang, C. Zhao and J. W. Zhang, Global regularity of the three-dimensional equations for nonhomogeneous incompressible fluids, Nonlinear Anal., 110 (2014), 61-76.  doi: 10.1016/j.na.2014.07.014.

[34]

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

show all references

References:
[1]

H. Abidi, G. Gui and P. Zhang, On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Ration. Mech. Anal., 204 (2012), 189-230.  doi: 10.1007/s00205-011-0473-4.

[2]

H. Abidi, G. L. Gui and P. Zhang, On the decay and stability of global solutions to the 3D inhomogeneous Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 832-881.  doi: 10.1002/cpa.20351.

[3]

S. N. Antontesv, A. V. Kazhikov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990.

[4]

Q. L. Chen and C. X. 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.

[5]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.

[6]

W. Craig, X. D. Huang and Y. Wang, Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations, J. Math. Fluid Mech., 15 (2013), 747-758.  doi: 10.1007/s00021-013-0133-6.

[7]

R. Danchin and P. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013), 991-1023.  doi: 10.1007/s00205-012-0586-4.

[8]

R. Danchin and P. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Comm. Pure Appl. Math., 65 (2012), 1458-1480.  doi: 10.1002/cpa.21409.

[9]

B. -Q. Dong, J. N. Li and J. H. 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.

[10]

B. -Q. Dong, J. H. Wu, X. J. 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.

[11]

B. -Q. Dong and Z. F. 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.

[12]

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

[13]

L. C. F. Ferreira and J. C. Precioso, Existence of solutions for the 3D-micropolar fluid system with initial data in Besov-Morrey spaces, Z. Angew. Math. Phys., 64 (2013), 1699-1710.  doi: 10.1007/s00033-013-0310-8.

[14]

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.

[15]

C. He, J. Li and B. Lü, On the Cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, arXiv: 1709.05608v1.

[16]

X. D. Huang and Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 259 (2015), 1606-1627.  doi: 10.1016/j.jde.2015.03.008.

[17]

A. V. Kažhikov, Resolution of boundary value problems for nonhomogeneous viscous fluids, Dokl. Akad. Nauk., 216 (1974), 1008-1010. 

[18]

J. U. Kim, Weak solutions of an initial boundary value problem for an incompressible viscous fluid with nonnegative density, SIAM J. Math. Anal., 18 (1987), 89-96.  doi: 10.1137/0518007.

[19]

O. Ladyzhenskaya and V. Solonnikov, Unique solvability of an initial and boundary value problem for viscous incompressible non-homogeneous fluids, J. Soviet Math., 9 (1978), 697-749. 

[20]

J. K. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.

[21]

P. -L. Lions, Mathematical Topics in Fluid mMechanics. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications, vol. 1. Clarendon Press/Oxford University Press, New York, 1996.

[22]

G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-0641-5.

[23]

G. Łukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat., 12 (1988), 83-97. 

[24]

G. Łukaszewicz, On the existence, uniqueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat., 13 (1989), 105-120. 

[25]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.  doi: 10.1016/j.jfa.2012.01.022.

[26]

M. Paicu, P. Zhang and Z. F. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.  doi: 10.1080/03605302.2013.780079.

[27]

J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.  doi: 10.1137/0521061.

[28]

D. Wang and Z. Ye, Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum, arXiv: 1806.04464v1.

[29]

L. T. Xue, Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34 (2011), 1760-1777.  doi: 10.1002/mma.1491.

[30]

N. Yamaguchi, Existence of global strong solution to the micropolar fluid systemin a bounded domain, Math. Methods Appl. Sci., 28 (2005), 1507-1526.  doi: 10.1002/mma.617.

[31]

B. Q. Yuan, On the regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. Amer. Math. Soc., 138 (2010), 2025-2036.  doi: 10.1090/S0002-9939-10-10232-9.

[32]

J. W. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient, J. Differential Equations, 259 (2015), 1722-1742.  doi: 10.1016/j.jde.2015.03.011.

[33]

P. X. Zhang, C. Zhao and J. W. Zhang, Global regularity of the three-dimensional equations for nonhomogeneous incompressible fluids, Nonlinear Anal., 110 (2014), 61-76.  doi: 10.1016/j.na.2014.07.014.

[34]

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

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