Global solutions for inhomogeneous micropolar equations with density-dependent coefficients and large data

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we consider the Dirichlet problem of three-dimensional inhomogeneous incompressible micropolar equations with density-dependent viscosity. Under the assumption that the coefficients are power functions of the density, we establish the global existence of strong solutions as long as the initial density is linearly equivalent to a large constant state. There is no restriction on the size of the initial velocity and micro-rotational velocity. As a byproduct, we prove the exponential decay for the solution.

Keywords

inhomogeneous incompressible micropolar equations / density-dependent coefficients / global strong solutions / large initial data / exponential decay

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

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

[2] P. Kalita, G. Łukaszewicz, Micropolar meets Newtonian in 3D. The Rayleigh-Bénard problem for large Prandtl numbers, Nonlinearity, 33 (2020), 5686-5732. https://doi.org/10.1088/1361-6544/ab9729

[3] P. Kalita, J. A. Langa, G. Łukaszewicz, Micropolar meets Newtonian. The Rayleigh-Bénard problem, Phys. D, 392 (2019), 57-80. https://doi.org/10.1016/j.physd.2018.12.004

[4] G. Łukaszewicz, Micropolar Fluids: Theory and Applications , Birkhäuser Boston, MA, 1999. https://doi.org/10.1007/978-1-4612-0641-5

[5] P. Zhang, M. 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

[6] C. 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

[7] C. 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

[8] 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

[9] L. Zhou, C. L. Tang, Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity, Dyn. Partial Differ. Equations, 21 (2024), 77-96. https://doi.org/10.4310/DPDE.2024.v21.n1.a4

[10] L. Liu, X. Zhong, Global existence and exponential 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

[11] C. Qian, 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

[12] X. Zhong, A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4603-4615. https://doi.org/10.3934/dcdsb.2020115

[13] X. Zhong, Strong solutions to the Cauchy problem of two-dimensional nonhomogeneous micropolar fluid equations with nonnegative density, Dyn. Partial Differ. Equations, 18 (2021), 49-69. 10.4310/DPDE.2021.v18.n1.a4

[14] X. Zhong, Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 5805-5820. https://doi.org/10.3934/dcdsb.2021296

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

[16] P. Germain, Strong solutions and weak-strong uniqueness for the nonhomogeneous Navier-Stokes system, J. Anal. Math., 105 (2008), 169-196. https://doi.org/10.1007/s11854-008-0034-4

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

[18] D. Chen, Z. Zhang, W. Zhao, Fujita-Kato theorem for the 3-D inhomogeneous Navier-Stokes equations, J. Differ. Equations, 261 (2016), 738-761. https://doi.org/10.1016/j.jde.2016.03.024

[19] Z. Guo, Y. Wang, C. Xie, Global strong solutions to the inhomogeneous incompressible Navier-Stokes system in the exterior of a cylinder, SIAM J. Math. Anal., 53 (2021), 6804-6821. https://doi.org/10.1137/20M1384439

[20] X. Huang, Y. Wang, Global strong solution with vacuum to the two dimensional density-dependent Navier-Stokes system, SIAM J. Math. Anal., 46 (2014), 1771-1788. https://doi.org/10.1137/120894865

[21] Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differ. Equations, 258 (2015), 2633-2654. https://doi.org/10.1016/j.jde.2014.12.015

[22] X. 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

[23] J. 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

[24] H. Abidi, G. Gui, 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. https://doi.org/10.1002/cpa.20351

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

[26] R. Danchin, S. Wang, Global unique solutions for the inhomogeneous Navier-Stokes equations with only bounded density, in critical regularity spaces, Comm. Math. Phys., 399 (2023), 1647-1688. https://doi.org/10.1007/s00220-022-04592-7

[27] M. Paicu, P. Zhang, Striated regularity of 2-D inhomogeneous incompressible Navier-Stokes system with variable viscosity, Comm. Math. Phys., 376 (2020), 385-439. https://doi.org/10.1007/s00220-019-03446-z

[28] P. Zhang, Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system, Adv. Math., 363 (2020), 107007. https://doi.org/10.1016/j.aim.2020.107007

[29] X. Huang, J. Li, R. Zhang, Global large strong solution of the 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, preprint, arXiv:2408.00333. https://doi.org/10.48550/arXiv.2408.00333

[30] Y. Cho, H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Anal., 59 (2004), 465-489. https://doi.org/10.1016/j.na.2004.07.020

[31] G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations , World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.

[32] O. A. Ladyzhenskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type , American Mathematical Society, 1968.