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

2024/04/19 by Timothée Crin-Barat, Crin-Barat, Timothée, Stefan Škondrić +3
Engineering · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Lattice Boltzmann Simulation Studies

paper · pdf · doi:10.48550/arxiv.2404.12858

openalex publication_date 2024/04/19 · openalex created_date 2024/04/23 · openalex updated_date 2026/07/28

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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