Instability and Non-uniqueness for the 2D Euler Equations, after M. Vishik: (AMS-219)

The incompressible Euler equations are a set of partial differential equations introduced by Leonhard Euler to describe the motion of an inviscid incompressible fluid. This book discusses a well-known case of this question in two space dimensions, showing that V. Yudovich's theorem cannot be generalized to the Lp setting.

Format: Hardback
Length: 144 pages
Publication date: 13 February 2024
Publisher: Princeton University Press

The incompressible Euler equations, a groundbreaking system of partial differential equations introduced by Leonhard Euler over 250 years ago, serve as an indispensable companion to M. Vishiks seminal work in fluid mechanics. These equations, derived from classical conservations laws of mass and momentum under idealized assumptions, describe the motion of an inviscid incompressible fluid. Despite their apparent simplicity compared to other equations in mathematical physics, several fundamental mathematical questions surrounding these equations remain unanswered. One central concern revolves around the conditions under which it can be rigorously proven that the Euler equations determine the evolution of the fluid once we know its initial state and the forces acting upon it.

In this book, the authors delve into a well-known case of this question in two spatial dimensions, building upon the pioneering ideas of M. Vishik. Through detailed explanations, they elucidate the optimality of a celebrated theorem by V. Yudovich from the 1960s. This theorem asserts that, in the vorticity formulation, the solution is unique if the initial vorticity and the acting force are bounded. Importantly, the authors demonstrate that Yudovichs theorem cannot be generalized to the Lp setting, highlighting the limitations and complexities of the Euler equations in various contexts.

The incompressible Euler equations continue to captivate mathematicians and physicists, as they represent a fundamental framework for understanding the behavior of fluids in a wide range of applications. This book provides valuable insights into the mathematical challenges and complexities associated with these equations, shedding light on their historical development and ongoing research endeavors. By exploring the interplay between theory and application, it contributes to our understanding of the intricate dynamics of fluids and their role in shaping the world around us.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691257525