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.

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