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

TL;DR

This paper establishes the well-posedness of regular Lagrangian flows for certain non-smooth BV vector fields with singular kernels, and demonstrates non-uniqueness of flows in specific cases.

Contribution

It proves existence and uniqueness of flows for BV vector fields with singular kernels and constructs examples showing non-uniqueness.

Findings

Well-posedness of flows for BV vector fields with singular kernels.

Existence of non-unique flows for certain vector fields in 2D.

Representation of derivatives of vector fields via singular measures and kernels.

Abstract

This paper is devoted to the study of flows associated to non-smooth vector fields. We prove the well-posedness of regular Lagrangian flows associated to vector fields $\mathbf{B}=(\mathbf{B}^1,...,\mathbf{B}^d)\in L^1(\mathbb{R}_+;L^1(\mathbb{R}^d)+L^\infty(\mathbb{R}^d))$ satisfying $ \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i*b_j,$ $b_j\in L^1(\mathbb{R}_+,BV(\mathbb{R}^d))$ and $\operatorname{div}(\mathbf{B})\in L^1(\mathbb{R}_+;L^\infty(\mathbb{R}^d))$ for $d,m\geq 2$, where $(\mathbf{K}_j^i)_{i,j}$ are singular kernels in $\mathbb{R}^d$. Moreover, we also show that there exist an autonomous vector-field $\mathbf{B}\in L^1(\mathbb{R}^2)+L^\infty(\mathbb{R}^2)$ and singular kernels $(\mathbf{K}_j^i)_{i,j}$, singular Radon measures $\mu_{ijk}$ in $\mathbb{R}^2$ satisfying $\partial_{x_k} \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i\star\mu_{ijk}$ in distributional sense for some $m\geq 2$ and for $k,i=1,2$ such that regular Lagrangian flows associated to vector field $\mathbf{B}$ are not unique.