Relative energy method for weak–strong uniqueness of the inhomogeneous Navier–Stokes equations far from vacuum

We present a weak–strong uniqueness result for the inhomogeneous Navier–Stokes equations in \(\mathbb {R}^d\) ${\mathbb{R}}^d$ ( \(d=2,3\) $d=2,3$ ) for bounded initial densities that are far from vacuum. Given a strong solution, i.e. a solution satisfying the equation as an identity in \(L^2\) $L^2$ , and a Leray–Hopf weak solution, we establish that they coincide if the initial data agree. Our proof strategy is based on the relative energy method and new \(W^{-1,p}\) $W^{-1,p}$ -type stability estimates for the density. A key point lies in proving that every Leray–Hopf weak solution originating from initial densities far from vacuum remains distant from vacuum at all times.