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

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 ${\mathbf{B}}^i={\sum }_{j=1}^m{\mathbf{K}}_j^i\ast b_j,$ $b_j\in L^1({\mathbb{R}}_{+},BV({\mathbb{R}}^d))$ and $\mathrm{div}(\mathbf{B})\in L^1({\mathbb{R}}_{+};L^{\infty }({\mathbb{R}}^d))$ for $d,m\ge 2$, 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 $m\ge 2$ and for $k,i=1,2$ such that regular Lagrangian flows associated to vector field $\mathbf{B}$ are not unique.

51 pages

Cited by in corpus (1)

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