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In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.

Heuristically, what is the role of one of the key assumptions of the paper: that $\mathrm{div}\, a$ is absolutely continuous with respect to the Lebesgue measure?

Note. A related question is asked in the post BV function with absolutely continuous divergence

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