*Submitted Paper*

**Inserted:** 23 oct 2020

**Last Updated:** 23 oct 2020

**Year:** 2020

**Abstract:**

We study the regularity of the flow $\boldsymbol{X}(t,y)$ which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution $\rho \in L^\infty(\mathbb R^{d+1})$ of the transport PDE $\partial_t \rho + \mathrm{div}(\rho\boldsymbol{b})=0,$ with $\boldsymbol{b} \in L^1_t BV_x$. We prove that $\boldsymbol{X}$ is differentiable in measure in the sense of Ambrosio-Maly, i.e. $\frac{\boldsymbol{X}(t,y+rz) - \boldsymbol{X}(t,y)}{r} \underset{r \to 0}{\to} W(t,y) z \quad \text{in measure}, $ where derivative $W(t,y)$ is a BV function satisfying the ODE $\frac{d}{d t} W(t, y) = \frac{(D \boldsymbol{b})_y(d t)}{J(t-,y)} W(t-, y), $ where $(D\boldsymbol{b})_y(d t)$ is the disintegration of the measure $\int D \boldsymbol{b}(t,\cdot) d t$ with respect to the partition given by the trajectories $\boldsymbol{X}(t, y)$ and the Jacobian $J(t,y)$ solves $\frac{d}{d t} J(t,y) = (\operatorname{div} \boldsymbol{b})_y(d t) = \mathrm{Tr} (D\boldsymbol{b})_y(d t). $ The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto (2019), on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a $BV$ vector field.

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