On vanishing diffusivity selection for the advection equation

TL;DR

This paper proves the existence and uniqueness of vanishing diffusivity solutions for the advection equation with certain singular divergence-free vector fields, addressing cases with multiple solutions and establishing conditions for well-posedness.

Contribution

It establishes the existence and uniqueness of vanishing diffusivity solutions for the advection equation with specific singular vector fields, including previously problematic cases.

Findings

Unique vanishing diffusivity solutions exist for the studied class of vector fields.

The class includes Depauw's vector field, for which multiple solutions previously existed.

The results clarify conditions under which the advection equation is well-posed.

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.