Quantitative Estimates for Regular Lagrangian Flows with BV Vector Fields

• Published in: Communications on pure and applied mathematics Vol. 74; no. 6; pp. 1129 - 1192
• Main Author: Nguyen, Quoc‐Hung
• Format: Journal Article
• Language: English
• Published: Melbourne John Wiley & Sons Australia, Ltd 01.06.2021; John Wiley and Sons, Limited
• Subjects: Fields (mathematics); Kernels
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21992

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, …, Bd) ∈ L1(ℝ+; L1(ℝd) + L∞(ℝd)) satisfying Bi=∑j=1mKji*bj, bj ∈ L1(ℝ+, BV(ℝd)) and div(B) ∈ L1(ℝ+; L∞(ℝd)) for d, m ≥ 2, where Kjii,j are singular kernels in ℝd. Moreover, we also show that there exist an autonomous vector‐field B ∈ L1(ℝ2) + L∞(ℝ2) and singular kernels Kjii,j, singular Radon measures μijk in ℝ2 satisfying ∂xkBi=∑j=1mKji⋆μijk 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.