Alberto Bressan - Böcker

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 monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by the author together with several collaborators. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader's visual intuition with over 70 figures. A set of problems, with varying difficulty, is given at the end of each chapter. These exercises are designed to verify and expand a student's understanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference for the state of the art, in the field of hyperbolic systems of conservation laws. The last chapter contains a large, up to date list of references, preceded by extensive bibliographical notes.

This volume contains the proceedings of the Summer Program on Nonlinear Conservation Laws and Applications held at the IMA on July 13--31, 2009. Hyperbolic conservation laws is a classical subject, which has experienced vigorous growth in recent years. The present collection provides a timely survey of the state of the art in this exciting field, and a comprehensive outlook on open problems. Contributions of more theoretical nature cover the following topics: global existence and uniqueness theory of one-dimensional systems, multidimensional conservation laws in several space variables and approximations of their solutions, mathematical analysis of fluid motion, stability and dynamics of viscous shock waves, singular limits for viscous systems, basic principles in the modeling of turbulent mixing, transonic flows past an obstacle and a fluid dynamic approach for isometric embedding in geometry, models of nonlinear elasticity, the Monge problem, and transport equations with rough coefficients. In addition, there are a number of papers devoted to applications. These include: models of blood flow, self-gravitating compressible fluids, granular flow, charge transport in fluids, and the modeling and control of traffic flow on networks.

This volume contains the proceedings of the Summer Program on Nonlinear Conservation Laws and Applications held at the IMA on July 13--31, 2009. Hyperbolic conservation laws is a classical subject, which has experienced vigorous growth in recent years. The present collection provides a timely survey of the state of the art in this exciting field, and a comprehensive outlook on open problems. Contributions of more theoretical nature cover the following topics: global existence and uniqueness theory of one-dimensional systems, multidimensional conservation laws in several space variables and approximations of their solutions, mathematical analysis of fluid motion, stability and dynamics of viscous shock waves, singular limits for viscous systems, basic principles in the modeling of turbulent mixing, transonic flows past an obstacle and a fluid dynamic approach for isometric embedding in geometry, models of nonlinear elasticity, the Monge problem, and transport equations with rough coefficients. In addition, there are a number of papers devoted to applications. These include: models of blood flow, self-gravitating compressible fluids, granular flow, charge transport in fluids, and the modeling and control of traffic flow on networks.

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.

This volume includes lectures presented at the CIME School on “PDEs, Control and Deep Learning”, held in Cetraro (Italy) from July 22 to 26, 2024. It provides a comprehensive and up-to-date view of the diverse and rapidly evolving field of nonlinear partial differential equations (PDEs), with an emphasis on modeling, analysis, control, and deep learning aspects. The theory of PDEs interacts closely with almost all areas of physics and many branches of mathematics. As explicit solutions of PDEs are rarely available (except in the simplest cases), numerical approximations play a central role in their study. Machine learning, particularly through artificial neural networks, introduces powerful methods for function approximation through layered structures of interconnected units (neurons) that combine linear transformations and nonlinear activations. Deep learning (the use of neural networks with many hidden layers) has proven to be remarkably effective in a wide range of applications. At the same time, recent advances in PDEs and control theory are beginning to inform machine learning, providing new theoretical perspectives.The book will be a valuable resource for PhD students and researchers seeking to deepen their understanding of partial differential equations, control, and their connections to modern machine learning.

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.

In recent years flows in networks have attracted the interest of many researchers from different areas, e.g. 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.