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

The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.

Flows of Non-Smooth Vector Fields and Degenerate Elliptic Equations: With Applications to the Vlasov-Poisson and Semigeostrophic Systems (Publications of the Scuola Normale Superiore, 22)

Earth Sciences

Edizioni della Normale

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

Books by Colombo, Maria