# Untangling of trajectories for non-smooth vector fields and Bressan's Compactness Conjecture

created by bonicatto on 18 Jul 2019

[BibTeX]

Proceedings

Inserted: 18 jul 2019
Last Updated: 18 jul 2019

Year: 2019
Notes:

Proceeding of the "XVII International Conference on Hyperbolic Problems: Theory, Numerics, Applications."

Abstract:

Given $d \ge 1$, $T > 0$ and a vector field $b: [0,T]×\mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u+b \cdot \nabla u = 0$ where $u: [0,T]\times \mathbb R^d \to \mathbb R$ is a scalar function. In the classical setting, the method of characteristics provides an explicit formula for the solution of the PDE, in terms of the flow of $b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost. We present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We finally show that, if $b$ is locally of class BV in the space variable, the decomposition satisfies this structural assumption, yielding a positive answer to the (weak) Bressan’s Compactness Conjecture.