Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data

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 Hs(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s}(\mathbb{R}^2)$$\end{document} (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 (u0,w0)∈B˙2,10(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_{0},w_{0})\in\dot{B}_{2,1}^{0}(\mathbb{R}^{2})$$\end{document} and ρ−1−1∈B˙2/ε,1ε(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho^{-1}-1\in\dot{B}_{2/\varepsilon,1}^{\varepsilon}(\mathbb{R}^{2})$$\end{document} 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.