Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik (Hardcover)

Camillo De Lellis

ISBN 10: 0691257523 / ISBN 13: 9780691257525

Published by Princeton University Press, New Jersey, 2024

About this Item

Hardcover. An essential companion to M. Vishiks groundbreaking work in fluid mechanicsThe 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 Yudovichs theorem cannot be generalized to the L^p setting. Shipping may be from our UK warehouse or from our Australian or US warehouses, depending on stock availability.

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Bibliographic Details

• Title: Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik (Hardcover)
• Publisher: Princeton University Press, New Jersey
• Publication Date: 2024
• Language: English
• ISBN 10: 0691257523
• ISBN 13: 9780691257525
• Binding: Hardcover
• Condition: new

About this title

Synopsis

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.

About the Author

Dallas Albritton is a mathematician and NSF postdoctoral fellow at Princeton University. Elia Brué is a mathematician at Bocconi University in Milan. Maria Colombo is a mathematician and professor at the Swiss Federal Institute of Technology in Lausanne. Camillo De Lellis is a mathematician at the Institute for Advanced Study in Princeton. Vikram Giri is a mathematician at Princeton. Maximilian Janisch is a PhD student in mathematics at the University of Zurich. Hyunju Kwon is a Hermann Weyl Instructor at ETH Zurich.