Abstract
In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids with the initial (angular) velocity being located in sub-critical Sobolev spaces \(H^{s}(\mathbb{R}^2)\) (0 < s< 1) and the initial density being bounded from above and below by some positive constants. The global unique solvability of the 2D incompressible inhomogeneous asymmetric fluids with the initial data in the critical Besov space \((u_{0},w_{0})\in\dot{B}_{2,1}^{0}(\mathbb{R}^{2})\) and \(\rho^{-1}-1\in\dot{B}_{2/\varepsilon,1}^{\varepsilon}(\mathbb{R}^{2})\) is established. In particular, the uniqueness of the solution is also obtained without any more regularity assumptions on the initial density which is an improvement on the recent result of Abidi and Gui (2021) for the 2-D inhomogeneous incompressible Navier-Stokes system.
Similar content being viewed by others
References
Abidi H, Gui G L. Global well-posedness for the 2-D inhomogeneous incompressible Navier-Stokes system with large initial data in critical spaces. Arch Ration Mech Anal, 2021, 242: 1533–1570
Abidi H, Gui G L, Zhang P. On the wellposedness of three-dimensional inhomogeneous Navier-Stokes equations in the critical spaces. Arch Ration Mech Anal, 2012, 204: 189–230
Abidi H, Gui G L, Zhang P. Well-posedness of 3-D inhomogeneous Navier-Stokes equations with highly oscillatory initial velocity field. J Math Pures Appl (9), 2013, 100: 166–203
Abidi H, Paicu M. Existence globale pour un fluide inhomogène. Ann Inst Fourier (Grenoble), 2007, 57: 883–917
Bahouri H, Chemin J-Y, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343. Heidelberg: Springer, 2011
Boldrini J L, Durán M, Rojas-Medar M A. Existence and uniqueness of strong solution for the incompressible micropolar fluid equations in domains of \(\mathbb{R}^{3}\). Ann Univ Ferrara Sez VII Sci Mat, 2010, 56: 37–51
Boldrini J L, Rojas-Medar M A, Fernández-Cara E. Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids. J Math Pures Appl (9), 2003, 82: 1499–1525
Braz e Silva P, Cruz F W, Loayza M, et al. Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach. J Differential Equations, 2020, 269: 1319–1348
Braz e Silva P, Santos E G. Global weak solutions for variable density asymmetric incompressible fluids. J Math Anal Appl, 2012, 387: 953–969
Conca C, Gormaz R, Ortega E, et al. Existence and uniqueness of a strong solution for nonhomogeneous micropolar fluids. Stud Math Appl, 2002, 31: 213–241
Condiff D W, Dahler J S. Fluid mechanical aspects of antisymmetric stress. Phys Fluids, 1964, 7: 842–854
Danchin R. Density-dependent incompressible viscous fluids in critical spaces. Proc Roy Soc Edinburgh Sect A, 2003, 133: 1311–1334
Danchin R. Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv Differential Equations, 2004, 9: 353–386
Danchin R. The inviscid limit for density-dependent incompressible fluids. Ann Fac Sci Toulouse Math (6), 2006, 15:637–688
Danchin R. A Lagrangian approach for the compressible Navier-Stokes equations. Ann Inst Fourier (Grenoble), 2014, 64: 753–791
Danchin R, Mucha P B. A Lagrangian approach for the incompressible Navier-Stokes equations with variable density. Comm Pure Appl Math, 2012, 65: 1458–1480
Danchin R, Mucha P B. Incompressible flows with piecewise constant density. Arch Ration Mech Anal, 2013, 207: 991–1023
Danchin R, Mucha P B. The incompressible Navier-Stokes equations in vacuum. Comm Pure Appl Math, 2019, 72: 1351–1385
Danchin R, Zhang P. Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density. J Funct Anal, 2014, 267: 2371–2436
Farwig R, Qian C Y, Zhang P. Incompressible inhomogeneous fluids in bounded domains of \(\mathbb{R}^{3}\) with bounded density. J Funct Anal, 2020, 278: 108394
Ferrari C. On lubrication with structured fluids. Appl Anal, 1983, 15: 127–146
Grafakos L. Classical and Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250. New York: Springer, 2009
Huang J C, Paicu M, Zhang P. Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity. Arch Ration Mech Anal, 2013, 209: 631–682
Lukaszewicz G. On non-stationary flows of incompressible asymmetric fluids. Math Methods Appl Sci, 1990, 13: 219–232
Lukaszewicz G. Micropolar Fluids: Theory and Applications. Modeling and Simulation in Science, Engineering and Technology. Boston: Birkhauser, 1999
Paicu M, Zhang P, Zhang Z F. Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density. Comm Partial Differential Equations, 2013, 38: 1208–1234
Prakash J, Sinha P. Lubrication theory for micropolar fluids and its application to a journal bearing. Internat J Engrg Sci, 1975, 13: 217–323
Qian C Y, Qu Y. Global well-posedness for 3D incompressible inhomogeneous asymmetric fluids with density-dependent viscosity. J Differential Equations, 2022, 306: 333–402
Zhang P. Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system. Adv Math, 2020, 363: 107007
Acknowledgements
Chenyin Qian was supported by Natural Science Foundation of Zhejiang Province (Grant No. LY20A010017). Ting Zhang was supported by National Natural Science Foundation of China (Grant No. 11931010) and Natural Science Foundation of Zhejiang Province (Grant No. LDQ23A010001).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Qian, C., He, B. & Zhang, T. Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data. Sci. China Math. 67, 527–556 (2024). https://doi.org/10.1007/s11425-022-2099-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-022-2099-1