Geodesic
FLOW WITH $A_{\infty }(\mathbb{R})$ DENSITY AND TRANSPORT EQUATION IN $\text{BMO}(\mathbb{R})$
Forum of Mathematics, Sigma, Tome 7 (2019)

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We show that, if $b\in L^{1}(0,T;L_{\operatorname{loc}}^{1}(\mathbb{R}))$ has a spatial derivative in the John–Nirenberg space $\operatorname{BMO}(\mathbb{R})$, then it generates a unique flow $\unicode[STIX]{x1D719}(t,\cdot )$ which has an $A_{\infty }(\mathbb{R})$ density for each time $t\in [0,T]$. Our condition on the map $b$ is not only optimal but also produces a sharp quantitative estimate for the density. As a killer application we achieve the well-posedness for a Cauchy problem of the transport equation in $\operatorname{BMO}(\mathbb{R})$. 
@article{10_1017_fms_2019_41,
     author = {RENJIN JIANG and KANGWEI LI and JIE XIAO},
     title = {FLOW {WITH} $A_{\infty }(\mathbb{R})$ {DENSITY} {AND} {TRANSPORT} {EQUATION} {IN} $\text{BMO}(\mathbb{R})$},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {7},
     year = {2019},
     doi = {10.1017/fms.2019.41},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.41/}
}
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RENJIN JIANG; KANGWEI LI; JIE XIAO. FLOW WITH $A_{\infty }(\mathbb{R})$ DENSITY AND TRANSPORT EQUATION IN $\text{BMO}(\mathbb{R})$. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2019.41

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