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## L-infinity instability of Prandtl layers

In 1904, Prandtl introduced his famous boundary layer theory in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in $L^\infty$ in the inviscid limit. In this post, I briefly announce my recent work with E. Grenier (ENS Lyon) on the Prandtl’s boundary layer theory, where we prove

• the Prandtl’s Ansatz is false for shear profiles that are unstable to Rayleigh equations;
• the Prandtl’s asymptotic expansion is invalid for shear profiles that are monotone and stable to Rayleigh equations.

## Graduate Student Seminar: Topics in Fluid Dynamics

Today, I give a Graduate Student Seminar lecture whose goal is to introduce to the first and second year graduate students at Penn State a few topics of research in Fluid Dynamics. There are many recent exciting developments in the field, which I only have time to present a few (many students haven’t taken any PDE course!). You may find the slides of my lecture here (up to many details delivered on the board!). You may also enjoy my similar lecture on Kinetic Theory of Gases, also aiming at first and second year students.

## 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: ${\nu \rightarrow 0}$.  In his CPAM2000 paper, Grenier proved that there exists no Prandtl’s asymptotic expansion involving one Prandtl’s boundary layer with thickness of order ${\sqrt\nu}$, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order ${\nu^{3/4}}$. 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 ${L^\infty}$.  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

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.

## Graduate student seminar: Kinetic theory of gases

Last week, I gave a graduate student seminar, whose purpose is to introduce to first and second year graduate students (at Penn State) an active and beautiful topics of research, and suggest a few possible ideas for students’ presentation later in the semester. Here are slides of my talk, which focuses on Kinetic Theory of Gases, a topics that I will teach as a graduate topics course, next fall (2017).

## The Maxwell-Boltzmann approximation for ion kinetic modeling

In this short paper with C. Bardos, F. Golse, and R. Sentis, we are aimed to justify the Maxwell-Boltzmann approximation from kinetic models, widely used in the literature for electrons density distribution; namely, the relation

$\displaystyle n_e =e^{q_{e} \phi / \theta }$

in which ${n_e, q_e, \theta,\phi}$ denote the electrons density, the elementary charge, the electron temperature, and the electric potential.