An $L^{p}$--approach to the well-posedness of transport equations associated to a regular field

Abstract

We investigate transport equations associated to a Lipschitz field F on some subspace of R N endowed with some general space measure µ in some L p -spaces 1 < p < ∞, extending the results obtained in two previous contributions [2,3] in the L 1 -context. We notably prove the well-posedness of boundary-value transport problems with a large variety of boundary conditions. New explicit formula for the transport semigroup are in particular given. CONTENTS 1. Introduction 2. Preliminary results 2.1. Integration along characteristic curves 2.2. The maximal transport operator and trace results 2.3. Fundamental representation formula: mild formulation 2.4. Additional Properties 3. Well-posedness for initial and boundary-value problems 3.1. Absorption semigroup 3.2. Some useful operators. 3.3. Generalized Cessenat's theorems 3.4. Boundary-value problem 3.5. Additional properties of the traces 4. Generation properties for unbounded boundary operators 4.1. The transport operator associated to bounded boundary operators 5. Explicit transport semigroup for bounded boundary operators 5.1. Boundary Dyson-Phillips iterated operators 5.2. Generation Theorem Appendix A. Proof of Theorem 2.6 Appendix B. Additional properties of T max, p . Appendix C. On the family (U k (t)) t 0