A uniqueness result for the decomposition of vector fields in Rd

• 成果类型: Article
• 署名作者: Bianchini, Stefano; Bonicatto, Paolo
• 署名单位: International School for Advanced Studies (SISSA)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00928-8
• 发表日期: 2020
• 页码: 255-393
• 关键词: conservation-laws; cauchy-problem; Chain rule; equation; SYSTEM

摘要

Given a vector field.(1, b). L1 loc(R+ x Rd, Rd+1) such that divt, x(.(1, b)) is a measure, we consider the problem of uniqueness of the representation. of.(1, b)Ld+1 as a superposition of characteristics. : (t ., t +.). Rd,.. (t) = b(t,. (t)). We give conditions in terms of a local structure of the representation. on suitable sets in order to prove that there is a partition of Rd+1 into disjoint trajectories Pa, a. A, such that the PDE divt, x u.(1, b). M(Rd+1), u. L 8 (R+ x Rd), can be disintegrated into a family of ODEs along P a with measure r.h.s. The decomposition P a is essentially unique. We finally show that b. L1t (BVx)loc satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible BV vector fields.