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Quantitative Estimates for Regular Lagrangian Flows with BV Vector Fields
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DOI：10.1002/cpa.21992.png)
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摘要
En 中文
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, horizontal ellipsis , B-d) is an element of L-1(Double-struck capital R+; L-1(Double-struck capital R-d) + L-infinity(Double-struck capital R-d)) satisfying Bi= n-ary sumation j=1mKji*bj, b(j) is an element of L-1(Double-struck capital R+, BV(Double-struck capital R-d)) and div(B) is an element of L-1(Double-struck capital R+; L-infinity(Double-struck capital R-d)) for d, m >= 2, where Kjii,j are singular kernels in Double-struck capital R-d. Moreover, we also show that there exist an autonomous vector-field B is an element of L-1(Double-struck capital R-2) + L-infinity(Double-struck capital R-2) and singular kernels Kjii,j, singular Radon measures mu(ijk) in Double-struck capital R-2 satisfying partial differential xkBi= n-ary sumation j=1mKji mu 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. (c) 2021 Wiley Periodicals LLC.
Keyword:
ORDINARY DIFFERENTIAL-EQUATIONS
WEAK TYPE 1
TRANSPORT-EQUATIONS
RENORMALIZED SOLUTIONS
CONTINUITY EQUATIONS
WELL-POSEDNESS
CAUCHY-PROBLEM
UNIQUENESS
SYSTEM
BOUNDS
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