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Quantitative Estimates for Regular Lagrangian Flows with BV Vector Fields
Author(s) -
Nguyen QuocHung
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21992
Subject(s) - vector field , mathematics , lagrangian , field (mathematics) , vector potential , mathematical analysis , combinatorics , pure mathematics , mathematical physics , physics , geometry , magnetic field , quantum mechanics
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.