On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

Mingyu Zhang; Mingyu Zhang

doi:10.3934/math.20241133

AIMS Mathematics

2024, Volume 9, Issue 9: 23313-23330. doi: 10.3934/math.20241133

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Research article

On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

Mingyu Zhang ^,

School of Mathematics and Statistics, Weifang University, Weifang 261061, China

Received: 17 June 2024 Revised: 18 July 2024 Accepted: 26 July 2024 Published: 01 August 2024
MSC : 35Q35, 76D03

In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space $ \mathbb{R}^3 $ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.
- nonhomogeneous micropolar fluids,
- density-dependent viscosity,
- global well-posedness
Citation: Mingyu Zhang. On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity[J]. AIMS Mathematics, 2024, 9(9): 23313-23330. doi: 10.3934/math.20241133

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Abstract

In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space $ \mathbb{R}^3 $ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.

References

[1]	H. Abidi, G. L. Gui, P. Zhang, On the decay and stability to global solutions of the 3D inhomogeneous Navier-Stokes equations, Commun. Pure Appl. Math., 64 (2011), 832–881. https://doi.org/10.1002/cpa.20351 doi: 10.1002/cpa.20351
[2]	H. Abidi, P. Zhang, On the global well-posedness of 2-D inhomogeneous incompressible Navier–Stokes system with variable viscous coefficient, J. Differ. Equations, 259 (2015), 3755–3802. https://doi.org/10.1016/j.jde.2015.05.002 doi: 10.1016/j.jde.2015.05.002
[3]	P. Braz e Silva, E. G. Santos, Global weak solutions for variable density asymmetric incompressible fluids, J. Math. Anal. Appl., 387 (2012), 953–969. https://doi.org/10.1016/j.jmaa.2011.10.015 doi: 10.1016/j.jmaa.2011.10.015
[4]	P. Braz e Silva, F. W. Cruz, M. Loayza, M. A. Rojas-Medar, Global unique solvability of nonhomogeneous asymmetric fluids: a Lagrangian approach, J. Differ. Equations, 269 (2020), 1319–1348. https://doi.org/10.1016/j.jde.2020.01.001 doi: 10.1016/j.jde.2020.01.001
[5]	B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135–158. https://doi.org/10.1007/s002050050025 doi: 10.1007/s002050050025
[6]	R. Danchin, Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, 9 (2004), 353–386. https://doi.org/10.57262/ade/1355867948 doi: 10.57262/ade/1355867948
[7]	R. Danchin, P. B. Mucha, A Lagrangian approach for the incompressible Navier-Stokes equations with variable density, Commun. Pure Appl. Math., 65 (2012), 1458–1480. https://doi.org/10.1002/cpa.21409 doi: 10.1002/cpa.21409
[8]	C. He, J. Li, B. Q. Lü, Global well-posedness and exponential stability of 3D Navier-Stokes equations with density-dependent viscosity and vacuum in unbounded domains, Arch. Rational Mech. Anal., 239 (2021), 1809–1835. https://doi.org/10.1007/s00205-020-01604-5 doi: 10.1007/s00205-020-01604-5
[9]	J. C. Huang, M. Paicu, P. Zhang, Global solutions to 2D inhomogeneous Navier-Stokes system with general velocity, J. Math. Pures Appl., 100 (2013), 806–831. https://doi.org/10.1016/j.matpur.2013.03.003 doi: 10.1016/j.matpur.2013.03.003
[10]	X. D. Huang, Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differ. Equations, 259 (2015), 1606–1627. https://doi.org/10.1016/j.jde.2015.03.008 doi: 10.1016/j.jde.2015.03.008
[11]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Providence: American Mathematical Society, 1968. https://doi.org/10.1090/mmono/023
[12]	L. X. Liu, X. Zhong, Global existence and expinential decay of strong solutions for 2D nonhomogeneous micropolar fluids with density-dependent viscosity, J. Math. Phys., 62 (2021), 061508. https://doi.org/10.1063/5.0055689 doi: 10.1063/5.0055689
[13]	B. Q. Lü, S. S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum, Nonlinear Anal. Real, 46 (2019), 58–81. https://doi.org/10.1016/j.nonrwa.2018.09.001 doi: 10.1016/j.nonrwa.2018.09.001
[14]	C. Y. Qian, Y. Qu, Global well-posedness for 3D incompressible inhomogeneous asymmetric fluids with density-dependent viscosity, J. Differ. Equations, 306 (2022), 333–402. https://doi.org/10.1016/j.jde.2021.10.045 doi: 10.1016/j.jde.2021.10.045
[15]	C. Y. Qian, H. Chen, T. Zhang, Global existence of weak solutions for 3D incompressible inhomogeneous asymmetric fluids, Math. Ann., 386 (2023), 1555–1593. https://doi.org/10.1007/s00208-022-02427-3 doi: 10.1007/s00208-022-02427-3
[16]	C. Y. Qian, B. B. He, T. Zhang, Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data, Sci. China Math., 67 (2024), 527–556. https://doi.org/10.1007/s11425-022-2099-1 doi: 10.1007/s11425-022-2099-1
[17]	J. W. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density-dependent viscosity coefficient, J. Differ. Equations, 259 (2015), 1722–1742. https://doi.org/10.1016/j.jde.2015.03.011 doi: 10.1016/j.jde.2015.03.011
[18]	P. X. Zhang, 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
[19]	L. Zhou, C. L. Tang, Global well-posedness to the 3D Cauchy problem of nonhomogeneous Navier-Stokes equations with density-dependent viscosity and large initial velocity, J. Math. Phys., 64 (2023), 111509. https://doi.org/10.1063/5.0144133 doi: 10.1063/5.0144133
[20]	X. Zhong, Global well-posedness for 3D nonhomogeneous micropolar fluids with density-dependent viscosity, Bull. Malays. Math. Sci. Soc., 46 (2023), 23. https://doi.org/10.1007/s40840-022-01399-6 doi: 10.1007/s40840-022-01399-6

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