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

TL;DR

This paper proves the unique global solvability of nonhomogeneous incompressible asymmetric fluid equations in 2D and 3D, allowing initial vacuum and large data, using a Lagrangian approach for uniqueness.

Contribution

It establishes the first comprehensive proof of global existence and uniqueness for these equations with minimal initial regularity and vacuum conditions.

Findings

Global existence of solutions in 2D for large data

Local existence in 3D for large data, global for small data

Uniqueness under soft regularity assumptions

Abstract

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