Related papers: The adjoint Rayleigh and Orr-Sommerfeld equations:…
The classical Orr-Sommerfeld equations are the resolvent equations of the linearized Navier Stokes equations around a stationary shear layer profile in the half plane. In this paper, we derive pointwise bounds on the Green function of the…
This is a continuation and completion of the program (initiated in \cite{GrN1,GrN2}) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a generic stationary boundary…
We study the dynamics of the two dimensional Navier-Stokes equations linearized around a shear flow on a (non-square) torus which possesses exactly two non-degenerate critical points. We obtain linear inviscid damping and vorticity…
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…
In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear…
For shear flows in a 2D channel, we define resonances near regular values of the shear profile for the Rayleigh equation under an analyticity assumption. This is done via complex deformation of the interval on which Rayleigh equation is…
We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on $\mathbb{T}\times\mathbb{R}$. The main task is to understand the associated Rayleigh and Orr-Sommerfeld equations,…
The Rayleigh equation, which is the linearized Euler equations near a shear flow in vorticity formulation, is a key ingredient in the study of the long time behavior of solutions of linearized Euler equations, in the study of the linear…
Ordinary Differential Equations are derived for the adjoint Euler equations firstly using the method of characteristics in 2D. For this system of partial-differential equations, the characteristic curves appear to be the streamtraces and…
We consider the flow of an { ideal} fluid in a 2D-bounded domain, admitting flows through the boundary of this domain. The flow is described by Euler equations with \textit{non-homogeneous } Navier slip boundary conditions. These conditions…
In this note we present the current status of the derivation of a viscous Serre-Green-Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom…
Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two and three dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a…
The aim of this paper is to give a detailed presentation of long wave instabilities of shear layers for Navier Stokes equations, and in particular to give a simple and easy to read presentation of the study of Orr Sommerfeld equation and to…
In this paper, we are interested in the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile $\rho_0(x_2)$ in a…
For homogeneous and isotropic linearly elastic solids and for incompressible fluids under low-Reynolds-number conditions the fundamental solutions of the associated continuum equations were derived a long time ago for bulk systems. That is,…
The hydrodynamic stability analysis of gravity-driven, soluble surfactant-laden fluid streaming down a slippery, slanted plane in the presence of external shear force is being explored in this article. The Navier-Stokes equations are…
In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by…
This paper is the continuation of a program, initiated in Grenier-Nguyen [8,9], to derive pointwise estimates on the Green function of Orr Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely…
We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of ${\mathbb{R}}^2$ satisfies the Euler equation if the solutions' local enstrophies are…
We prove that the Navier-Stokes equation is well-posed in function spaces on $\mathbb{R}^d$, $d\ge 2$, that contain vector fields of order $O(|x|^\kappa)$ as $|x|\to\infty$ with $\kappa<1/2$. The corresponding solutions depend continuously…