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.

## Archive for the ‘Instabilities’ Category

## 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 »

## 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.

## Grenier’s nonlinear iterative scheme

Posted in Instabilities on July 28, 2015| Leave a Comment »

(originally posted here on toannguyen.org)

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.

## Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

Posted in Instabilities, New papers, Plasma Physics on July 1, 2015| Leave a Comment »

(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit , with being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from in arbitrary negative Sobolev norms within time of order . Second, we deduce the invalidity of the quasineutral limit in in arbitrarily short time.

## The onset of instability in first-order systems

Posted in Instabilities, New papers on April 19, 2015| Leave a Comment »

(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

Nicolas Lerner, Ben Texier and I have just submitted to arxiv our long paper on “The onset of instability in first-order systems”, in which we prove the Hadamard’s instability for first-order quasilinear systems that lose its hyperbolicity in positive times. Precisely, we consider the Cauchy problem for the following first-order systems of partial differential equations: