Transport Equations and Multi-D Hyperbolic Conservation Laws (Lecture Notes of the Unione Matematica Italiana, Band 5) - Softcover

Über die Autorin bzw. den Autor

Prof. Luigi Ambrosio is a Professor of Mathematical Analysis, a former student of the Scuola Normale Superiore and presently its Rector. His research interests include calculus of variations, geometric measure theory, optimal transport and analysis in metric spaces. For his scientific achievements, he has been awarded several prizes, in particular the Fermat prize in 2003, the Balzan Prize in 2019, the Riemann Prize in 2023 and the Nemmers Prize in 2024. Dr. Elia Bruè is an Associate Professor at Bocconi University, Milan, Italy. He earned his PhD from the Scuola Normale Superiore in 2020. His research interests lie in the fields of Geometric Analysis and Partial Differential Equations, with a focus on Ricci curvature, metric geometry, incompressible fluid mechanics, and passive scalars with rough velocity fields. Dr. Daniele Semola is an Assistant Professor at the University of Vienna. He was a student in Mathematics at the Scuola Normale Superiore, where he earned his PhD degree in 2020. His research interests lie at the interface between geometric analysis and analysis on metric spaces, mainly with a focus on lower curvature bounds.

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The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements thanks to ideas and techniques related to the structure and fine properties of functions of bounded variation. 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. Geometric and measure theoretic tools play a key role to obtain some fundamental advances: the well-posedness theory of linear transport equations with irregular coefficients, and the study of the BV-like structure of  bounded entropy solutions to multi-dimensional scalar conservation laws.

The volume contains surveys of recent deep results, provides an overview of further developments and related open problems, and will capture the interest of members both of the hyperbolic and the elliptic community willing to explore the intriguing interplays that link their worlds. Readers should have basic knowledge of PDE and measure theory.