Uniqueness of solution for incompressible inhomogeneous Navier-Stokes equations in dimension two

The global existence of solution for 2D inhomogeneous incompressible Navier-Stokes equations is established by Abidi et al. (2024), and the uniqueness of solution is also investigated under some additional conditions on initial density. The purpose of this paper is to obtain the uniqueness of the solution without any additional assumptions on the initial density in case of 2 <= p<4. The key strategy is to establish a new estimate of solution in Lagrangian coordinates.