Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik (Annals of Mathematics Studies)

An essential companion to M. Vishik's groundbreaking work in fluid mechanics The incompressible Euler equations are a system of partial differential equations introduced by Leonhard Euler more than 250 years ago to describe the motion of an inviscid incompressible fluid. These equations can be derived from the classical conservations laws of mass and momentum under some very idealized assumptions. While they look simple compared to many other equations of mathematical physics, several fundamental mathematical questions about them are still unanswered. One is under which assumptions it can be rigorously proved that they determine the evolution of the fluid once we know its initial state and the forces acting on it. This book addresses a well-known case of this question in two space dimensions. Following the pioneering ideas of M. Vishik, the authors explain in detail the optimality of a celebrated theorem of V. Yudovich from the 1960s, which states that, in the vorticity formulation, the solution is unique if the initial vorticity and the acting force are bounded. In particular, the authors show that Yudovich's theorem cannot be generalized to the L^p setting.

Science & Math

Brué, Elia

Colombo, Maria

Lellis, Camillo De

Albritton, Dallas

Giri, Vikram

Janisch, Maximilian

Kwon, Hyunju

Mathematics

Princeton University Press

Review “This book is particularly suited for students who desire to learn from a text which closely follows the organization of a course, as well as for researchers and professors looking for inspiration for their own lecturers on the topic. … The exposition is clear and mostly self-contained, with a nice list of examples that show the necessity of the assumptions of some classical results of the theory.” (Nicolò De Ponti, zbMATH 1485.49001, 2022)“This book is very well written and will be accessible to graduate students without background on optimal transport … . All in all, this textbook is recommended to graduate students and researchers who want to discover the fundamental theory of optimal transport and its ramifications to several areas of mathematics. It can also easily be used by professors who want to teach a graduate course on the topic.” (Hugo Lavenant, Mathematical Reviews, June, 2022) Product Description This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations. From the Back Cover This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations. About the Author Prof. Luigi Ambrosio is a Professor of Mathematical Analysis, a former student of the Scuola Normale Superiore and presently its Director. 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 and the Balzan Prize in 2019.Dr. Elia Brué is a postdoctoral member at the Institute for Advanced Studies in Princeton. He earned his PhD degree at the Scuola Normale Superiore in 2020. His research interests include geometric measure theory, optimal transport, non-smooth geometry and PDE. Dr. Daniele Semola is a postdoctoral research assistant at the Mathematical Institute of the University of Oxford. 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.

Lectures on Optimal Transport (UNITEXT, 130)

Ambrosio, Luigi

Semola, Daniele

Springer

Lectures on Optimal Transport (UNITEXT, 169)

See the best price to sell, buy, or rent books by Brué, Elia. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Brué, Elia