Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)2239110.22363/2413-3639-2017-63-3-418-436New ResultsНовые результатыResearch ArticleLagrangian Representations for Linear and Nonlinear TransportЛагранжевы представления для линейного и нелинейного переносаBianchiniStefanoБьянкиниСbianchin@sissa.itBonicattoPaoloБоникаттоПpaolo.bonicatto@sissa.itMarconiElioМаркониЭelio.marconi@sissa.itS.I.S.S.A15122017633Differential and Functional Differential EquationsДифференциальные и функционально-дифференциальные уравнения41843606122019Copyright ©; 2019, Contemporary Mathematics. Fundamental DirectionsCopyright ©; 2019, Современная математика. Фундаментальные направления2019Contemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направленияhttps://journals.rudn.ru/CMFD/article/view/22391In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.Представлен подход, объединяющий два класса дифференциальных уравнений в частных производных первого порядка: вводится понятие лагранжева представления для уравнения неразрывности и скалярных законов сохранения. С одной стороны, это дает единственность слабых решений уравнений переноса, определяемых двумерными почти несжимаемыми векторными полями ограниченной вариации. 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