On the one-dimensional continuity equation with a nearly incompressible vector field

TL;DR

This paper proves that in one dimension, near incompressibility of the vector field guarantees both existence and uniqueness of weak solutions to the continuity equation, unlike the multi-dimensional case.

Contribution

It establishes the sufficiency of near incompressibility for existence and uniqueness of solutions in one dimension, extending understanding beyond the multi-dimensional scenario.

Findings

Existence of solutions in 1D under near incompressibility.

Uniqueness of solutions in 1D with near incompressibility.

Analysis of Lagrangian flow compactness properties.

Abstract

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) \times \mathbb R^d \to \mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system). It is well known that in the generic multi-dimensional case ($d\ge 1$) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of $b$ (e.g. Sobolev regularity) are needed in order to obtain uniqueness. We prove that in the one-dimensional case ($d=1$) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.