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 $\epsilon \leq c_0\mathbf {{Re}}^-1$ 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.

Dynamics Near the Subcritical Transition of the 3d Couette Flow: Below Threshold Case (Memoirs of the American Mathematical Society)

Science & Math

Bedrossian, Jacob

Germain, Pierre

Masmoudi, Nader

Mathematics

Amer Mathematical Society

The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations (The Graduate Studies in Mathematics, 225)

The aim of this book is to provide beginning graduate students who completed the first two semesters of graduate-level analysis and PDE courses with a first exposure to the mathematical analysis of the incompressible Euler and Navier-Stokes equations. The book gives a concise introduction to the fundamental results in the well-posedness theory of these PDEs, leaving aside some of the technical challenges presented by bounded domains or by intricate functional spaces. Chapters 1 and 2 cover the fundamentals of the Euler theory: derivation, Eulerian and Lagrangian perspectives, vorticity, special solutions, existence theory for smooth solutions, and blowup criteria. Chapters 3, 4, and 5 cover the fundamentals of the Navier-Stokes theory: derivation, special solutions, existence theory for strong solutions, Leray theory of weak solutions, weak-strong uniqueness, existence theory of mild solutions, and Prodi-Serrin regularity criteria. Chapter 6 provides a short guide to the must-read topics, including active research directions, for an advanced graduate student working in incompressible fluids. It may be used as a roadmap for a topics course in a subsequent semester. The appendix recalls basic results from real, harmonic, and functional analysis. Each chapter concludes with exercises, making the text suitable for a one-semester graduate course. Prerequisites to this book are the first two semesters of graduate-level analysis and PDE courses.

The Mathematical Analysis of the Incompressible Euler and Navier-stokes Equations: An Introduction (Graduate Studies in Mathematics, 225)

Vicol, Vlad

Dynamics Near the Subcritical Transition of the 3d Couette Flow: Above Threshold Case (Memoirs of the American Mathematical Society, 279)

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