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.

## Archive for the ‘fluid dynamics’ Category

## Invalidity of Prandtl’s boundary layers

Posted in boundary layers, fluid dynamics, Instabilities, New papers on June 4, 2017| Leave a Comment »

## Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers

Posted in boundary layers, fluid dynamics, Instabilities on May 15, 2017| Leave a Comment »

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

Posted in boundary layers, fluid dynamics, Instabilities, Uncategorized on May 14, 2017| Leave a Comment »

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: . In his CPAM2000 paper, Grenier proved that there exists no Prandtl’s asymptotic expansion involving one Prandtl’s boundary layer with thickness of order , which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order . In this paper, we point out how the stability of the classical Prandtl’s layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in . That is, either the Prandtl’s layer or the boundary sublayer is nonlinearly unstable in the sup norm.

## Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms

Posted in boundary layers, fluid dynamics, Uncategorized on March 2, 2017| Leave a Comment »

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

Posted in boundary layers, fluid dynamics, New papers on February 25, 2017| Leave a Comment »

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.

## Prandtl’s layer expansions for steady Navier-Stokes

Posted in fluid dynamics, New papers on January 31, 2017| Leave a Comment »

In 1904, Prandtl conjectured that slightly viscous flows can be decomposed into the inviscid flows away from the boundary and a so-called Prandtl’s layer near the boundary. While various instabilities indicate the failure of the conjecture for unsteady flows (for instance, see Grenier 2000), recently with Y. Guo, we are able to prove that the conjecture holds for certain steady Navier-Stokes flows; see our paper which is to appear on Annals of PDEs.

## Inviscid limit for Navier-Stokes in a rough domain

Posted in fluid dynamics, New papers on January 28, 2017| Leave a Comment »

In this paper with Gérard-Varet, Lacave, and Rousset, we prove the inviscid limit of Navier-Stokes flows in domains with a rough or oscillating boundary. Precisely, we study the 2D incompressible Navier-Stokes flows with small viscosity , posed on the following rough domain: