Scuola Normale Superiore di Pisa

Contents Introduction 5 Chapter 1. The theory in the smooth framework 11 1.1. The ordinary differential equation 11 1.2. Existence and uniqueness in the classical setting 11 1.3. The classical flow of a vector field 16 1.4. The transport equation and the continuity equation 19 Part 1. The Eulerian viewpoint 23 Chapter 2. Renormalized solutions and well-posedness of the PDE 25 2.1. A strategy to obtain uniqueness 25 2.2. Weak solutions and existence 26 2.3. Renormalized solutions 28 2.4. The DiPerna–Lions regularization scheme 31 2.5. Vector fields with Sobolev regularity 32 2.6. Vector fields with bounded variation 33 2.7. Other various renormalization results 40 2.8. Nearly incompressible vector fields 41 Chapter 3. An abstract characterization of the renormalization property 45 3.1. Forward-backward formulation 45 3.2. One-way formulation 51 3.3. Nearly incompressible vector fields 55 Chapter 4. Well-posedness in the two-dimensional case 59 4.1. Bounded planar divergence-free vector fields 59 4.2. Splitting on the level sets of the Hamiltonian 61 4.3. The weak Sard property 63 4.4. Structure of the level sets 64 4.5. Uniqueness on the level sets 65 Chapter 5. Counterexamples to the well-posedness and related constructions 69 5.1. Depauw's counterexample 69 5.2. Oscillatory solutions to transport equations 70 5.3. Two counterexamples by DiPerna and Lions 73 5.4. Lack of propagation of regularity 78 3 4 CONTENTS Part 2. The Lagrangian viewpoint 79 Chapter 6. The connection between PDE and ODE 81 6.1. Pointwise uniqueness and measure valued solutions 81 6.2. The superposition principle 82 6.3. The regular Lagrangian flow 90 6.4. Ambrosio's theory of regular Lagrangian flows 92 6.5. DiPerna–Lions' notion of flow 96 6.6. The density of a regular Lagrangian flow 97 6.7. Bressan's compactness conjecture 100 Chapter 7. A priori estimates for regular Lagrangian flows 103 7.1. A purely Lagrangian approach 103 7.2. A priori estimates for bounded vector fields and corollaries 105 7.3. Estimates for more general vector fields and corollaries 114 7.4. A direct proof of compactness 121 7.5. Lipexp p –regularity for transport equations with W 1,p coefficients 125 7.6. An application to a conjecture on mixing flows 127 Appendix: Background material and technical results 131 A.1. Measure theory 131 A.2. Lipschitz functions, extension theorems and approximate differentiability 132 A.3. Functions with bounded variation 133 A.4. Maximal functions 135 Bibliography 137 Introduction The main topic of this thesis is the study of the ordinary differential equation (0.1) ˙ γ(t) = b(t, γ(t)) γ(0) …