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 = (B¹, …, B^d) ∈ L¹(ℝ₊; L¹(ℝ^d) + L^∞(ℝ^d)) satisfying b_j ∈ L¹(ℝ₊, BV(ℝ^d)) and div(B) ∈ L¹(ℝ₊; L^∞(ℝ^d)) for d, m ≥ 2, where are singular kernels in ℝ^d. Moreover, we also show that there exist an autonomous vector‐field B ∈ L¹(ℝ²) + L^∞(ℝ²) and singular kernels , singular Radon measures μ_ijk in ℝ² satisfying 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.

中文翻译：

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具有BV向量场的规则Lagrangian流的定量估计

本文专门研究与非光滑向量场相关的流动。我们证明定期拉格朗日的适定性相关联的矢量场流乙 =（乙¹，...， 乙^d）∈ 大号¹（ℝ ₊ ; 大号¹（ℝ ^d）+ 大号^∞（ℝ ^d））满足 b _Ĵ  ∈ 大号¹（ℝ ₊，  BV（ℝ ^d））和DIV（乙）∈ 大号¹（ℝ ₊; 大号^∞（ℝ ^d））为d，米 ≥2 ，其中是在单数的内核ℝ ^d。此外，我们还表明，存在一个自治矢量场乙 ∈ 大号¹（ℝ ²）+ 大号^∞（ℝ ²）和奇异内核，奇异氡措施μ _IJK在ℝ ²满足在分布式感一些米 ≥2和对于k，i  = 1，2因此与向量场B相关的规则拉格朗日流不是唯一的。版权©2021 Wiley Periodicals LLC。