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

2024/04/19 by Crin-Barat, Timothée, Škondrić, Stefan, Violini, Alessandro
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.12858

Abstract

We present a weak-strong uniqueness result for the inhomogeneous Navier-Stokes (INS) equations in 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.

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