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arXiv · 2404.12858

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

Timothée Crin-Barat·Stefan Škondrić·Alessandro Violini
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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.

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Timothée Crin-Barat, Stefan Škondrić, Alessandro Violini. 2024-04-19. Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations. https://arxiv.org/abs/2404.12858

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