*Preprint*

**Inserted:** 13 apr 2018

**Last Updated:** 3 may 2018

**Pages:** 50

**Year:** 2018

**Abstract:**

In this paper, we solve a main open problem mentioned in \cite{AmbCrip}. Specifically, we prove the well posedness of regular Lagrangian flows to vector fields $\mathbf{B}=(\mathbf{B}^1,...,\mathbf{B}^d)\in L^1((0,T);L^1\cap L^\infty(\mathbb{R}^d))$ satisfying $ \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i*b_j,$ $b_j\in L^1((0,T),BV(\mathbb{R}^d))$ and $\operatorname{div}(\mathbf{B})\in L^1((0,T);L^\infty(\mathbb{R}^d))$ for $d\geq 2$, where $(\mathbf{K}_j^i)_{i,j}$ are singular kernels in $\mathbb{R}^d$.

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