Nonlinear instability in Vlasov type equations around rough velocity profiles
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 489-547.
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In the Vlasov-Poisson equation, every configuration which is homogeneous in space provides a stationary solution. Penrose gave in 1960 a criterion for such a configuration to be linearly unstable. While this criterion makes sense in a measure-valued setting, the existing results concerning nonlinear instability always suppose some regularity with respect to the velocity variable. Here, thanks to a multiphasic reformulation of the problem, we can prove an “almost Lyapounov instability” result for the Vlasov-Poisson equation, and an ill-posedness result for the kinetic Euler equation and the Vlasov-Benney equation (two quasineutral limits of the Vlasov-Poisson equation), both around any unstable measure.

DOI : 10.1016/j.anihpc.2019.12.002
Mots-clés : Vlasov–Poisson, Nonlinear instability, Penrose condition, Measure-valued solutions 
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Baradat, Aymeric. Nonlinear instability in Vlasov type equations around rough velocity profiles. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 489-547. doi : 10.1016/j.anihpc.2019.12.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.12.002/

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