Extension of transport and regularity theory from K*BV to K*BD

Investigate whether the chain-rule, renormalization, quantitative stability, and well-posedness theory developed for vector fields of the form K*BV extends to the class K*BD when the input vector field belongs to BD.

Background

The paper discusses applications of BV-like vector fields to transport equations and regular Lagrangian flows. In particular, prior work established quantitative stability and well-posedness for vector fields of the form B^i = \sum_{j=1}^m K_j^i * b_j with b_j in L^1((0,T);BV(ℝ^n)), under suitable assumptions on the kernels, growth, and divergence.

Replacing BV inputs by BD inputs would broaden the admissible class from fields whose full distributional derivatives are finite measures to fields whose symmetric derivatives are finite measures. The paper explains that strong approximation of arbitrary BD inputs by BV fields would have provided a possible route to this extension, but the main theorem proves that such strong approximation fails in general. Thus the extension from K*BV to K*BD remains an explicitly posed question not resolved by the paper.