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This book are notes prepared for the PhD courses that the author has been teaching during the last 10 years. The material available in the already existing literature (papers and essays) has been collected in this unique text, presenting the results with all the details for the reader's convenience, fixing a unified notation, and providing a consistent framework for the subject. These notes cover many of the arguments that usually can be found in high level essays, where the proofs are simply sketched, and in papers, which are not easily available and not always self-contained.
This book is intended for1. PhD students in Mathematics, Physics and Mechanical Engineering in order to learn the basic features of nonlinear scalar equations,2. researchers interested in nonlinear hyperbolic PDEs in order to learn the details behind some known and deep results on nonlinear scalar equations,3. teachers of courses on nonlinear PDEs.The readers are expected to know the basic measure theory and Sobolev spaces.
List of contents
Chapter 1 Introduction.- Chapter 2 Entropy Solutions.- Chapter 3 Riemann Problem.- Chapter 4 Functions with Bounded Variation.- Chapter 5 Wave Front Tracking.- Chapter 6 Vanishing Viscosity.- Chapter 7 Compensated Compactness.- Chapter 8 Periodic solutions.- Chapter 9 Oleinik Estimate.- Chapter 10 Lax-Oleinik Formula.
About the author
Dr. ¿Giuseppe Maria Coclite, Full Professor in Mathematical Analysis at the Department of Mechanics, Mathematics and Management of the Polytechnic University of Bari (Italy). His main research interests are Boundary Controllability for Systems of Conservation Laws, Traffic Models, Parabolic equations, Conservation laws with discontinuous flows, Nonlocal models in continuum mechanics, optimization in measure spaces, etc.
Report
"This book provides an overview on some of the basic and classical results for mathematical theory of hyperbolic scalar conservation laws in one space dimension. Sufficient technical details, examples and exercise problems are provided throughout the book which would make it comprehensible to both graduate students and researchers. " (Yogiraj Mantri, Mathematical Reviews, March, 2025)
