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arXiv · 2411.12910

On vanishing diffusivity selection for the advection equation

Giulia Mescolini·Jules Pitcho·Massimo Sorella
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Abstract

We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in $L^1_{loc}((0, T ]; BV (\mathbb{T}^d;\mathbb{R}^d))\cap L^2((0, T ) \times\mathbb{T}^d;\mathbb{R}^d)$, there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw, for which there are infinitely many distinct bounded solutions to the advection equation.

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Giulia Mescolini, Jules Pitcho, Massimo Sorella. 2024-11-19. On vanishing diffusivity selection for the advection equation. https://arxiv.org/abs/2411.12910

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