Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations

We present a weak-strong uniqueness result for the inhomogeneous Navier-Stokes (INS) equations in ${\mathbb{R}}^d$ ($d=2,3$) for bounded initial densities that are far from vacuum. Given a strong solution within the class employed in Paicu, Zhang and Zhang (2013) and Chen, Zhang and Zhao (2016), and a Leray-Hopf weak solution, we establish that they coincide if the initial data agree. The strategy of our proof is based on the relative energy method and new $W^{-1,p}$-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.