Intermittent convex integration for the 3D Euler equations / Tristan Buckmaster, Nader Masmoudi, Matthew Novack, Vlad Vicol.

Format:

Book

Author/Creator:

Buckmaster, Tristan, Author.

Masmoudi, Nader, 1974- Author.

Novack, Matthew, 1991- Author.

Vicol, Vlad, 1983- Author.

Series:

Annals of mathematics studies ; no. 217.

Annals of mathematics studies ; number 217

Language:

English

Subjects (All):

Differential equations, Nonlinear--Numerical solutions.

Differential equations, Nonlinear.

Induction (Mathematics).

Convex functions.

Fluid dynamics--Mathematics.

Fluid dynamics.

Physical Description:

vi, 246 pages : illustrations ; 24 cm.

Place of Publication:

Princeton : Princeton University Press, 2023.

Summary:

"To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L̂2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates"-- Provided by publisher.

"A new threshold for the existence of weak solutions to incompressible Euler equations. To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L̂2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates"-- Provided by publisher.

Contents:

Outline of the convex integration scheme

Inductive assumptions

Building blocks

Mollification

Cutoffs

From q to q + 1: breaking down the main inductive estimates

Proving the main inductive estimates

Parameters.

Notes:

Includes bibliographical references (pages [239]-243) and index.

Other Format:

Online version: Buckmaster, Tristan. Intermittent convex integration for the 3D Euler equations

ISBN:

9780691249551

0691249555

9780691249544

0691249547

OCLC:

1376982175