On vanishing diffusivity selection for the advection equation

2024/11/19 by Giulia Mescolini, Mescolini, Giulia, Jules Pitcho +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods

paper · pdf · doi:10.48550/arxiv.2411.12910

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 L¹_loc((0, T ]; BV (\mathbbT^d;ℝ^d))∩ L²((0, T ) ×\mathbbT^d;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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