Flow with $A_\infty(\mathbb R)$ density and transport equation in $\mathrm{BMO}(\mathbb R)$

TL;DR

This paper proves the existence and uniqueness of flows with $A_ abla( )$ density under BMO conditions on the velocity field's derivative, and applies this to establish well-posedness of the transport equation in BMO.

Contribution

It introduces optimal conditions on velocity fields with BMO derivatives to ensure unique flows with $A_ abla( )$ density and applies these results to the transport equation.

Findings

Existence of unique flows with $A_ abla( )$ density under BMO derivative conditions.

Sharp quantitative estimates for the density of the flow.

Well-posedness of the transport equation in BMO space.

Abstract

We show that, if $b\in L^1(0,T;L^1_{\mathrm{loc}}(\mathbb{R}))$ has spatial derivative in the John-Nirenberg space $\mathrm{BMO}(\mathbb{R})$, then it generalizes a unique flow $\phi(t,\cdot)$ which has an $A_\infty(\mathbb R)$ density for each time $t\in [0,T]$. Our condition on the map $b$ is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in $\mathrm{BMO}(\mathbb R)$.