Published Paper

Inserted: 23 oct 2020
Last Updated: 19 nov 2022

Journal: Archive for Ratonal Mechanics and Analysis
Year: 2022
Doi: https://doi.org/10.1007/s00205-022-01820-1

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 continuity equation \[ \partial_t \rho + \operatorname{div}(\rho \boldsymbol{b}) = 0, \] with $\boldsymbol{b} \in L^1_t \mathrm{BV}_x$. We prove that $\boldsymbol{X}$ is differentiable in measure in the sense of Ambrosio-Mal\'y, 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 \[ \frac{\mathrm d}{\mathrm d t} W(t, y) = \frac{(D \boldsymbol{b})_y(\mathrm d t)}{J(t-,y)} W(t-, y), \] where $(D\boldsymbol{b})_y(\mathrm d t)$ is the disintegration of the measure $\int D \boldsymbol{b}(t,\cdot) \, \mathrm d t$ with respect to the partition given by the trajectories $\boldsymbol{X}(t, y)$ and the Jacobian $J(t,y)$ solves \[ \frac{\mathrm d}{\mathrm d t} J(t,y) = (\operatorname{div} \boldsymbol{b})_y(\mathrm d t) = \mathrm{Tr} (D\boldsymbol{b})_y(\mathrm d t). \] 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 $\mathrm{BV}$ vector field.

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