Quantitative Estimates for Regular Lagrangian Flows with BV Vector Fields

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 B  = ( B 1 , …,  B d ) ∈  L 1 (ℝ + ;  L 1 (ℝ d ) +  L ∞ (ℝ d )) satisfying B i = ∑ j = 1 m K j i * b j ,b j  ∈  L 1 (ℝ + ,  BV (ℝ d )) and div ( B ) ∈  L 1 (ℝ + ;  L ∞ (ℝ d )) for d , m  ≥ 2 , whereK j ii , jare singular kernels in ℝ d . Moreover, we also show that there exist an autonomous vector‐field B  ∈  L 1 (ℝ 2 ) +  L ∞ (ℝ 2 ) and singular kernelsK j ii , j , singular Radon measures μ ijk in ℝ 2 satisfying ∂ x kB i = ∑ j = 1 m K j i ⋆ μ ijk in distributional sense for some m  ≥ 2 and for k , i  = 1, 2 such that regular Lagrangian flows associated to vector field B are not unique. © 2021 Wiley Periodicals LLC.