Uniqueness of solutions to the 2D compressible Navier-Stokes equations for bounded densities

We prove the uniqueness of global solutions to the two dimensional barotropic compressible Navier–Stokes equations within the regularity class introduced by Danchin and Mucha [14], for general $C^1$ pressure laws. A crucial assumption is that both solutions have bounded divergence, locally integrable in time, which ensures that their densities remain bounded without excluding vacuum or requiring spatial regularity of the density.