I’ve just released this book titled Stability of shear flows near a boundary, written jointly with E. Grenier, on the arXiv. 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. We had plan to complete a few additional chapters near the end of the book, but ultimately decided to release the full unpublished version that we had back in August 2020.
boundary layers
The inviscid limit problem for Navier-Stokes equations
A longstanding open problem is to establish the inviscid limit of classical solutions to the incompressible Navier-Stokes equations for smooth initial data on a domain with boundaries. The question is of great physical and mathematical interest, and it deeply links to the transition to turbulence in fluids that may possibly take place faster than expected due to the presence of a boundary. In this article, I shall give a quick overview of this subject, and then highlight some recent works with my former student, Trinh T. Nguyen, (currently a Van Vleck Assistant Professor at University of Wisconsin, Madison), whose main results establish the inviscid limit for smooth data that are only required to be analytic locally near the boundary. This may be the best possible type of positive results that one can hope for, given the known violent instabilities at the boundary, which I shall discuss below. Before getting on, this picture should already hint at the great delicacy in studying boundary layers (source internet):
Invalidity of Prandtl’s boundary layers
I’ve just submitted this paper with Grenier (ENS Lyon) which studies Prandtl’s boundary layer asymptotic expansions for incompressible fluids on the half-space in the inviscid limit. In 1904, Prandtl introduced his well known boundary layers in order to describe the transition from Navier-Stokes to Euler equations in the inviscid limit.
Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers
Emmanuel Grenier and I have just submitted this 84-page! long paper, also posted on arxiv (arXiv:1705.05323). This work is a continuation and completion of the program (initiated in Grenier-Toan1 and Grenier-Toan2) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a stationary boundary layer profile.
Sublayer of Prandtl boundary layers
The aim of this paper (arXiv:1705.04672), with E. Grenier, is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: 
Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms
I’ve just uploaded this paper, with E. Grenier, on the arXiv (arXiv:1703.00881), entitled Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms, aiming a better understanding of the classical Prandtl’s boundary layers. Indeed, one of the key difficulties in dealing with boundary layers is the creation of (unbounded) vorticity in the inviscid limit.
Green function for linearized Navier-Stokes around boundary layers: away from critical layers
I’ve just submitted this new paper with E. Grenier (ENS de Lyon) on arxiv (scheduled to announce next Tuesday 1:00GMT), in which we construct the Green function for the classical Orr-Sommerfeld equations and derive sharp semigroup bounds for linearized Navier-Stokes equations around a boundary layer profile. This is part of the long program to understand the stability of classical Prandtl’s layers appearing in the inviscid limit of incompressible Navier-Stokes flows.
Grenier’s nonlinear iterative scheme
In his paper [Grenier, CPAM 2000], Grenier introduced a nonlinear iterative scheme to prove the instability of Euler and Prandtl equations. Recently, the scheme is also proved to be decisive in the study of water waves: [Ming-Rousset-Tzvetkov, SIAM J. Math. Anal., 2015], and plasma physics: [Han-Kwan & Hauray, CMP 2015] or my recent paper with Han-Kwan (see also my previous blog discussions). I am certain that it can be useful in other contexts as well. In this blog post, I’d like to give a sketch of the scheme to prove instability.