Stability of shear flows near a boundary

TL;DR

This book provides a comprehensive overview of recent advances in the stability analysis of shear flows and boundary layers in incompressible Navier-Stokes equations with small viscosity, covering linear and nonlinear aspects.

Contribution

It offers a detailed synthesis of classical and modern methods for analyzing boundary layer stability, including new constructions of Green functions and resolvent operators.

Findings

Analysis of Rayleigh and Orr-Sommerfeld equations

Construction of Green functions for Orr-Sommerfeld equations

Nonlinear instability results for shear flows

Abstract

This book is devoted to the study of the linear and nonlinear stability of shear flows and boundary layers for Navier Stokes equations for incompressible fluids with Dirichlet boundary conditions in the case of small viscosity. The aim of this book is to provide a comprehensive presentation to recent advances on boundary layers stability. It targets graduate students and researchers in mathematical fluid dynamics and only assumes that the readers have a basic knowledge on ordinary differential equations and complex analysis. No prerequisites are required in fluid mechanics, excepted a basic knowledge on Navier Stokes and Euler equations, including Leray's theorem. This book consists of three parts. Part I is devoted to the presentation of classical results and methods: Green functions techniques, resolvent techniques, analytic functions. Part II focuses on the linear analysis, first of Rayleigh equations, then of Orr Sommerfeld equations. This enables the construction of Green functions for Orr Sommerfeld, and then the construction of the resolvent of linearized Navier Stokes equations. Part III details the construction of approximate solutions for the complete nonlinear problem and nonlinear instability results.