Intermittent Convex Integration for the 3D Euler Equations: (AMS-217)

Mathematicians use convex integration techniques to construct weak solutions to the 3D Euler equations with non-negligible regularity and intermittency,which match the measured intermittent nature of turbulent flows. This book develops convex integration techniques at the local level to handle the inhomogeneities inherent in intermittent solutions.

Format: Paperback / softback
Length: 256 pages
Publication date: 11 July 2023
Publisher: Princeton University Press

Mathematicians explore the intricate dynamics of turbulent fluids by starting with experimental observations and translating them into mathematical properties for solutions of the fundamental fluid partial differential equations (PDEs). They then construct solutions that exhibit turbulent characteristics, contributing to our understanding of these complex systems. This book is part of a program that has integrated convex integration techniques into hydrodynamics. Convex integration methods have been employed to generate solutions with precise regularity, essential for resolving the Onsager conjecture for the 3D Euler equations and constructing dissipative weak solutions for the Navier-Stokes equations.

In this groundbreaking work, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions possess a spatial regularity index in L^2, allowing it to be closely approached to 1/2, which is a threshold achieved by all known convex integration methods. This remarkable property aligns with the measured intermittent nature of turbulent flows. The construction of such solutions necessitates specialized technology tailored to the inherent inhomogeneities present in intermittent solutions.

The central technical contribution of this book lies in the development of convex integration techniques at the local level. This localization procedure operates as an ad hoc wavelet decomposition of the solution, capturing information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates. By employing these techniques, mathematicians can effectively analyze and understand the complex behavior of turbulent fluids, paving the way for further advancements in this field.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691249544