Quantitative estimates for regular Lagrangian flows with $BV$ vector fields

created by nguyen on 13 Apr 2018
modified on 03 May 2018

[BibTeX]

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 $\mathbb{R}^d$.