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.

在Vlasov-Poisson方程中，空间上均质的每个构型都提供了固定解。彭罗斯（Penrose）于1960年提出了使这种配置线性不稳定的标准。尽管此标准在量度值设置中有意义，但有关非线性不稳定性的现有结果始终假设速度变量具有一定规律性。在这里，由于问题的多相重构，我们可以证明Vlasov-Poisson方程的“几乎Lyapounov不稳定性”结果，以及动力学Euler方程和Vlasov-Benney方程的不适定结果（两个准中性极限） （Vlasov-Poisson方程），都围绕任何不稳定的度量。