The present paper studies the nonhomogeneous incompressible asymmetric fluids in two and three dimensions. The main aim is to obtain the unique global solvability of the system with only bounded nonnegative initial density. More precisely, we construct the global existence of the solution with large data in 2-D and the existence of global in time for some smallness conditions in 3-D. In addition, the uniqueness of the solution is proved through a Lagrangian approach under some quite soft assumptions about its regularity. These conclusions can be viewed as a generalization of the one established by Braz e Silva et al. [J. Differ. Equations 269, 1319–1348 (2020)].
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