E-Book, Englisch, Band 5, 131 Seiten
Ambrosio / Crippa / De Lellis Transport Equations and Multi-D Hyperbolic Conservation Laws
1. Auflage 2008
ISBN: 978-3-540-76781-7
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band 5, 131 Seiten
Reihe: Lecture Notes of the Unione Matematica Italiana
ISBN: 978-3-540-76781-7
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark
The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements. This volume provides an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to hyperbolic conservation laws. The captivating volume contains surveys of recent deep results and provides an overview of further developments and related open problems. Readers should have basic knowledge of PDE and measure theory.
Autoren/Hrsg.
Weitere Infos & Material
1;Preface;6
2;Contents;11
3;Authors;13
4;Part I;15
4.1;Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated toWeakly Differentiable Vector Fields;16
4.1.1;1 Introduction;16
4.1.2;2 The Continuity Equation;18
4.1.3;3 The Continuity Equation Within the Cauchy–Lipschitz Framework;20
4.1.4;4 (ODE) Uniqueness Vs. (PDE) Uniqueness;24
4.1.5;5 The Flow Associated to Sobolev or BV Vector Fields;32
4.1.6;6 Measure-Theoretic Differentials;45
4.1.7;7 Differentiability of the Flow in the W1,1 Case;51
4.1.8;8 Differentiability and Compactness of the Flow in the W1,p Case;53
4.1.9;9 Bibliographical Notes and Open Problems;65
4.1.10;References;67
5;Part II;71
5.1;A Note on Alberti’s Rank-One Theorem;72
5.1.1;1 Introduction;72
5.1.2;2 Dimensional Reduction;74
5.1.3;3 A Blow-Up Argument Leading to a Partial Result;76
5.1.4;4 The Fundamental Lemma;77
5.1.5;5 Proof of Theorem 1.1 in the Planar Case;79
5.1.6;References;85
6;Part III;86
6.1;Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws;87
6.1.1;1 Introduction;87
6.1.2;2 Background Material;89
6.1.3;3 Entropy Solutions with BV-Regularity;94
6.1.4;4 Structure of Entropy Solutions;97
6.1.5;5 Kinetic Formulation, Blow-Ups and Split States;101
6.1.6;6 Classification of Split States;108
6.1.7;7 Proof of the Main Theorem;116
6.1.8;8 Proofs of the Regularity Theorems;122
6.1.9;References;137
7;Index;139
