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## Grenier’s nonlinear iterative scheme

(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.

## Ill-posedness of the hydrostatic Euler and singular Vlasov equations

(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 (also, available on arxiv): “Ill-posedness of the hydrostatic Euler and singular Vlasov equations”, dedicated to Claude Bardos on the occasion of his 75th birthday, as a token of friendship and admiration.

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 ${\varepsilon \rightarrow 0}$, with ${\varepsilon}$ being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution ${\mu}$ of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from ${\mu}$ in arbitrary negative Sobolev norms within time of order ${|\log \varepsilon|}$. Second, we deduce the invalidity of the quasineutral limit in ${L^2}$ in arbitrarily short time.