Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909447228489728 |
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| author | Huysmans, Lucas Titi, Edriss S. |
| author_facet | Huysmans, Lucas Titi, Edriss S. |
| contents | We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in\mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in\mathbb{R}^2$. This is described by the initial value problem to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$, or with positive viscosity/diffusivity $ν>0$, to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -νΔf = 0$. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_00809 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation Huysmans, Lucas Titi, Edriss S. Analysis of PDEs 76F25 (Primary) 35A02, 35D30, 35Q35, 60J60, 76R10, 76R50 (Secondary) We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in\mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in\mathbb{R}^2$. This is described by the initial value problem to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$, or with positive viscosity/diffusivity $ν>0$, to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -νΔf = 0$. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy. |
| title | Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation |
| topic | Analysis of PDEs 76F25 (Primary) 35A02, 35D30, 35Q35, 60J60, 76R10, 76R50 (Secondary) |
| url | https://arxiv.org/abs/2307.00809 |