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The present paper deals with the nonhomogeneous incompressible asymmetricfluids equations in dimension $d= 2,3$. The aim is to prove the unique globalsolvability of the system with only bounded nonnegative initial density and$H^{1}$ initial velocities. We first construct the global existence of thesolution with large data in 2-D. Next, we establish the existence of local intime solution for arbitrary large data and global in time for some smallnessconditions in 3-D. Finally, the uniqueness of the solution is proved underquite soft assumptions about its regularity through a Lagrangian approach. Inparticular, the initial vacuum is allowed.

The unique global solvability of the nonhomogeneous incompressible asymmetric fluids with vacuum