Nader Masmoudi's Books

Intermittent Convex Integration for the 3D Euler Equations

(AMS-217)

By: Tristan Buckmaster,Nader Masmoudi,Matthew Novack,Vlad Vicol

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Intermittent Convex Integration for the 3D Euler Equations

(AMS-217)

By: Tristan Buckmaster,Nader Masmoudi,Matthew Novack,Vlad Vicol

A new threshold for the existence of weak solutions to the 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 L^2 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.

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Page Count

256

Publisher

Princeton University Press

Published Date

ISBN 10

0691249547

ISBN 13

9780691249544

Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

By: Jacob Bedrossian,Pierre Germain,Nader Masmoudi

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Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

By: Jacob Bedrossian,Pierre Germain,Nader Masmoudi

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Page Count

148

Publisher

American Mathematical Society

Published Date

ISBN 10

1470472252

ISBN 13

9781470472252

Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

By: Jacob Bedrossian,Pierre Germain,Nader Masmoudi

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Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

By: Jacob Bedrossian,Pierre Germain,Nader Masmoudi

The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $epsilon leq c_0mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t rightarrow infty $. For times $t gtrsim mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of ``2.5 dimensional'' streamwise-independent solutions referred to as streaks.

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