Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929320878931968 |
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| author | Crin-Barat, Timothée Škondrić, Stefan Violini, Alessandro |
| author_facet | Crin-Barat, Timothée Škondrić, Stefan Violini, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations Crin-Barat, Timothée Škondrić, Stefan Violini, Alessandro Analysis of PDEs 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. |
| title | Relative Energy Method For Weak-Strong Uniqueness Of The Inhomogeneous Navier-Stokes Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.12858 |