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

TL;DR

This paper proves weak-strong uniqueness for the inhomogeneous Navier-Stokes equations in 2D and 3D using a relative energy approach and stability estimates, ensuring weak solutions match strong solutions when initial data agree.

Contribution

It introduces a novel relative energy method and $W^{-1,p}$-type stability estimates to establish weak-strong uniqueness for inhomogeneous Navier-Stokes equations with densities far from vacuum.

Findings

Weak solutions remain distant from vacuum over time.

Weak and strong solutions coincide if initial data agree.

New stability estimates for density are established.

Abstract

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.