Instability and Nonuniqueness for the 2d Euler Equations, after M. Vishik


Princeton University Press
Camillo De Lellis, Elia Brué, Dallas Albritton, Maria Colombo, Vikram Giri, Maximilian Janisch, Hyunju Kwon
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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 in the sixties, 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.

Contributor Bio

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.

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