This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This volume is the first to present a comprehensive account of these new, fundamental advances. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. A set of problems, with varying difficulty is given at the end of each chapter to verify and expand understanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference to the state of the art in the field of hyperbolic systems of conservation laws.

Hyperbolic Systems of Conservation Laws: The One-Dimensional Cuachy Problem (Oxford Lecture Series in Mathematics and Its Applications)

Science & Math

Bressan, Alberto

Biological Sciences

Oxford University Press

This book is intended for graduate students and researchers interested in the mathematical physics and PDE.

Well-Posedness of the Cauchy Problem for Nxn Systems of Conservation Laws (Memoirs of the American Mathematical Society)

Crasta, Graziano

Piccoli, Benedetto

Mathematics

Amer Mathematical Society

This textbook is addressed to graduate students in mathematics or other disciplines who wish to understand the essential concepts of functional analysis and their applications to partial differential equations. The book is intentionally concise, presenting all the fundamental concepts and results but omitting the more specialized topics. Enough of the theory of Sobolev spaces and semigroups of linear operators is included as needed to develop significant applications to elliptic, parabolic, and hyperbolic PDEs. Throughout the book, care has been taken to explain the connections between theorems in functional analysis and familiar results of finite-dimensional linear algebra. The main concepts and ideas used in the proofs are illustrated with a large number of figures. A rich collection of homework problems is included at the end of most chapters. The book is suitable as a text for a one-semester graduate course.

Lecture Notes on Functional Analysis: With Applications to Linear Partial Differential Equations (Graduate Studies in Mathematics, 143)

This volume includes four lecture courses by Bressan, Serre, Zumbrun and Williams and a Tutorial by Bressan on the Center Manifold Theorem. Bressan introduces the vanishing viscosity approach and clearly explains the building blocks of the theory. Serre focuses on existence and stability for discrete shock profiles. The lectures by Williams and Zumbrun deal with the stability of multidimensional fronts.

Hyperbolic Systems of Balance Laws: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14-21, 2003 (Lecture Notes in Mathematics, 1911)

Williams, Mark

Serre, Denis

Zumbrun, Kevin

Springer

In recent years flows in networks have attracted the interest of many researchers from different areas, e.g. applied mathematicians, engineers, physicists, economists. The main reason for this ubiquity is the wide and diverse range of applications, such as vehicular traffic, supply chains, blood flow, irrigation channels, data networks and others. This book presents an extensive set of notes by world leaders on the main mathematical techniques used to address such problems, together with investigations into specific applications. The main focus is on partial differential equations in networks, but ordinary differential equations and optimal transport are also included. Moreover, the modeling is completed by analysis, numerics, control and optimization of flows in networks. The book will be a valuable resource for every researcher or student interested in the subject.

Modelling and Optimisation of Flows on Networks: Cetraro, Italy 2009, Editors: Benedetto Piccoli, Michel Rascle (Lecture Notes in Mathematics, 2062)

Computers & Technology

Ambrosio, Luigi

Helbing, Dirk

Zuazua, Enrique

Klar, Axel

Computer Science

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