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Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations
Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, China |
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.
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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