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

Abstract

We show that, if $b\in L^1(0,T;L_{\mathrm{loc}}^1(\mathbb{R}))$ has spatial derivative in the John-Nirenberg space $\mathrm{BMO}(\mathbb{R})$, then it generalizes a unique flow $\varphi (t,\cdot )$ 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 $\mathrm{BMO}(\mathbb{R})$.

26 pages, comments are very welcome