Existence and uniqueness of Leray-Hopf weak solution for the inhomogeneous 2D Navier--Stokes equations without vacuum

We prove the existence and uniqueness of weak solutions of the inhomogeneous incompressible Navier--Stokes equations without vacuum using the relative energy method. We present a novel and direct proof of the existence of weak solutions based on approximation with more regular solutions. The analysis we employ to justify the strong convergence reveals how to conclude the stability and uniqueness of weak solutions. To the best of our knowledge, these stability estimates are completely new. Furthermore, for the first time, we establish energy conservation for weak solutions.