Global well-posedness and large-time behavior for anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation

Ying Wang; Ying Wang

doi:10.3934/era.2025194

Electronic Research Archive

2025, Volume 33, Issue 7: 4284-4306. doi: 10.3934/era.2025194

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Global well-posedness and large-time behavior for anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation

Ying Wang ^{1,2
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1.
School of Mathematics and Information Science, Guangxi University, Nanning 530004, China
2.
School of Physical Science and Technology, Guangxi University, Nanning 530004, China

Received: 11 May 2025 Revised: 02 July 2025 Accepted: 08 July 2025 Published: 22 July 2025

In this paper, we consider the Cauchy problem for the two-dimensional anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation, where the density at infinity is in the state $ \widetilde{\rho} > 0 $. We obtain the global well-posedness of strong solutions and the large-time behavior of the velocity field under the condition that $ \Vert \sqrt{\rho_0} \mathbf{u}_0 \Vert_{L^2} $ is sufficiently small. It is worth noting that our results include the case of vacuum.
- inhomogeneous Navier-Stokes equations,
- global strong solutions,
- partial dissipation,
- decay rates,
- vacuum
Citation: Ying Wang. Global well-posedness and large-time behavior for anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation[J]. Electronic Research Archive, 2025, 33(7): 4284-4306. doi: 10.3934/era.2025194

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Abstract

In this paper, we consider the Cauchy problem for the two-dimensional anisotropic inhomogeneous Navier-Stokes equations with mixed partial dissipation, where the density at infinity is in the state $ \widetilde{\rho} > 0 $. We obtain the global well-posedness of strong solutions and the large-time behavior of the velocity field under the condition that $ \Vert \sqrt{\rho_0} \mathbf{u}_0 \Vert_{L^2} $ is sufficiently small. It is worth noting that our results include the case of vacuum.

References

[1]	B. Cushman-Roisin, J. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, $2^{nd}$ edition, Academic Press, 2011.
[2]	J. Pedlosky, Geophysical Fluid Dynamics, $2^{nd}$ edition, Springer, 1987. https://doi.org/10.1007/978-1-4612-4650-3
[3]	R. Temam, M. Ziane, Some mathematical problems in geophysical fluid dynamics, in Handbook of Mathematical Fluid Dynamic, 3 (2005), 535–658. https://doi.org/10.1016/S1874-5792(05)80009-6
[4]	A. V. Kažihov, Solvability of the initial-boundary value problem for the equations of the motion of an inhomogeneous viscous incompressible fluid, Dokl. Akad. Nauk SSSR, 216 (1974), 1008–1010.
[5]	S. N. Antontsev, A. V. Kazhikhov, V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Elsevier, 1989.
[6]	O. A. Ladyzhenskaya, V. A. Solonnikov, Unique solvability of an initial and boundary value problem for viscous incompressible nonhomogeneous fluids, J. Sov. Math., 9 (1978), 697–749. https://doi.org/10.1007/BF01085325 doi: 10.1007/BF01085325
[7]	R. Salvi, The equations of viscous incompressible non-homogeneous fluids: On the existence and regularity, J. Aust. Math. Soc. Ser. B Appl. Math., 33 (1991), 94–110. https://doi.org/10.1017/S0334270000008651 doi: 10.1017/S0334270000008651
[8]	J. Simon, Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093–1117. https://doi.org/10.1137/0521061 doi: 10.1137/0521061
[9]	H. J. Choe, H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Commun. Partial Differ. Equations, 28 (2003), 1183–1201. https://doi.org/10.1081/PDE-120021191 doi: 10.1081/PDE-120021191
[10]	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 doi: 10.1016/j.jde.2014.12.015
[11]	B. Lü, X. Shi, X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617. https://doi.org/10.1088/1361-6544/aab31f doi: 10.1088/1361-6544/aab31f
[12]	X. Huang, Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differ. Equations, 254 (2013), 511–527. https://doi.org/10.1016/j.jde.2012.08.029 doi: 10.1016/j.jde.2012.08.029
[13]	Y. Liu, The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations, J. Differ. Equations, 416 (2025), 1260–1284. https://doi.org/10.1016/j.jde.2024.10.026 doi: 10.1016/j.jde.2024.10.026
[14]	C. He, J. Li, B. Lü, Global well-posedness and exponential stability of 3D Navier Stokes equations with density-dependent viscosity and vacuum in unbounded domains, Arch. Ration. Mech. Anal., 239 (2021), 1809–1835. https://doi.org/10.1007/s00205-020-01604-5 doi: 10.1007/s00205-020-01604-5
[15]	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 doi: 10.1007/s00021-013-0133-6
[16]	H. Abidi, G. Gui, P. Zhang, On the well-posedness of three dimensional inhomogeneous Navier-Stokes equations in the critical spaces, Arch. Ration. Mech. Anal., 204 (2012), 189–230. https://doi.org/10.1007/s00205-011-0473-4 doi: 10.1007/s00205-011-0473-4
[17]	J. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Fluids with anisotropic viscosity, Math. Model. Numer. Anal., 34 (2000), 315–335. https://doi.org/10.1051/m2an:2000143 doi: 10.1051/m2an:2000143
[18]	M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 21 (2005), 179–235. https://doi.org/10.4171/RMI/420 doi: 10.4171/RMI/420
[19]	J. Chemin, P. Zhang, On the global wellposedness to the 3D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529–566. https://doi.org/10.1007/s00220-007-0236-0 doi: 10.1007/s00220-007-0236-0
[20]	T. Zhang, D. Fang, Global wellposed problem for the 3D incompressible anisotropic Navier-Stokes equations, J. Math. Pures Appl., 90 (2008), 413–449. https://doi.org/10.1016/j.matpur.2008.06.008 doi: 10.1016/j.matpur.2008.06.008
[21]	M. Fujii, Y. Li, Large time behavior for solutions to the anisotropic Navier-Stokes equations in a 3D half-space, Int. Math. Res. Not., 2025 (2025). https://doi.org/10.1093/imrn/rnae265
[22]	Y. Liu, Global well-posedness to the Cauchy problem of 2D density-dependent micropolar equations with large initial data and vacuum, J. Math. Anal. Appl., 491 (2020), 124294. https://doi.org/10.1016/j.jmaa.2020.124294 doi: 10.1016/j.jmaa.2020.124294
[23]	Y. Liu, P. Zhang, Global solutions of 3-D Navier-Stokes system with small unidirectional derivative, Arch. Ration. Mech. Anal., 235 (2020), 1405–1444. https://doi.org/10.1007/s00205-019-01447-9 doi: 10.1007/s00205-019-01447-9
[24]	D. Regmi, J. Wu, Global regularity for the 2D magneto-micropolar equations with partial dissipation, J. Math. Study, 49 (2016), 169–194. https://doi.org/10.4208/jms.v49n2.16.05 doi: 10.4208/jms.v49n2.16.05
[25]	H. Shang, D. Zhou, Optimal decay for the 2D anisotropic Navier-Stokes equations with mixed partial dissipation, Appl. Math. Lett., 144 (2023), 108696. https://doi.org/10.1016/j.aml.2023.108696 doi: 10.1016/j.aml.2023.108696
[26]	B. Dong, J. Wu, X. Xu, N. Zhu, Stability and exponential decay for the 2D anisotropic Navier-Stokes equations with horizontal dissipation, J. Math. Fluid Mech., 23 (2021), 100. https://doi.org/10.1007/s00021-021-00617-8 doi: 10.1007/s00021-021-00617-8
[27]	T. Hao, Global solutions of 3-D inhomogeneous Navier-Stokes system with large viscosity in one variable, Commun. Math. Res., 38 (2022), 62–80. https://doi.org/10.4208/cmr.2021-0040 doi: 10.4208/cmr.2021-0040
[28]	M. Paicu, N. Zhu, Global well-posedness of 3D inhomogeneous Navier-Stokes system with small unidirectional derivative, Nonlinearity, 36 (2023), 2403. https://doi.org/10.1088/1361-6544/acc37a doi: 10.1088/1361-6544/acc37a
[29]	M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Commun. Partial Differ. Equations, 11 (1986), 733–763. https://doi.org/10.1080/03605308608820443 doi: 10.1080/03605308608820443
[30]	L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat., 13 (1959), 115–162.
[31]	X. Zhong, Global well-posedness to the Cauchy problem of two dimensional nonhomogeneous heat conducting Navier-Stokes equations, J. Geom. Anal., 32 (2022), 200. https://doi.org/10.1007/s12220-022-00947-7 doi: 10.1007/s12220-022-00947-7

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