Notes on Hyperbolic Systems of Conservation Laws and Transport Equations

Contents 1. Introduction 2 1.1. The Keyfitz and Kranzer system 2 1.2. Bressan's compactness conjecture 3 1.3. Ambrosio's renormalization Theorem 4 1.4. Well–posedness for the Keyfitz and Kranzer system 5 1.5. Renormalization conjecture for nearly incompressible BV fields 6 1.6. Plan of the paper 7 2. Preliminaries 8 2.1. Notation 8 2.2. Measure theory 9 2.3. Approximate continuity and approximate jumps 10 2.4. BV functions 11 2.5. Caccioppoli sets and Coarea formula 12 2.6. The Volpert Chain rule 12 2.7. Alberti's Rank–one Theorem 13 3. DiPerna–Lions theory for nearly incompressible flows 13 3.1. Lagrangian flows 13 3.2. Nearly incompressible fields and fields with the renormalization property 17 3.3. Existence and uniqueness of solutions to transport equations 20 3.4. Stability of solutions to transport equations 24 3.5. Existence, uniqueness, and stability of regular Lagrangian flows 26 4. Commutator estimates and Ambrosio's Renormalization Theorem 30 4.