Nonlinear echoes and Landau damping with insufficient regularity
Nonlinear echoes and Landau damping with insufficient regularity
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DOI：
10.2140/tunis.2021.3.121
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发表时间：
2016-05
影响因子：
0.9
通讯作者：
J. Bedrossian
中科院分区：
文献类型：
--
作者：
J. Bedrossian
本文证明了Mouhot和Villani关于$\mathbb{T}_x \times \mathbb{R}_v$上Vlasov-Poisson方程平衡点附近的朗道阻尼定理在一般情况下不能推广到高Sobolev空间.这是通过在每个索伯列夫空间中显示，存在背景分布，使得人们可以构造任意小的扰动，这些扰动在密度中表现出任意多个孤立的非线性振荡。这些振荡在物理学界被称为等离子体回波。对于静电相互作用的情况下，我们证明了一系列的小背景分布和渐近较小的扰动在$H^S$显示类似的非线性回波。这表明，在静电的情况下，任何延伸的Mouhot和Villani的定理索伯列夫空间将不得不依赖于一些额外的非共振效应来自背景-不同的情况下，Gevrey-$\nu$与$\nu < 3$正则性，其结果是统一的小背景的大小。特别地，在Gevrey类中的Mouhot和Villani定理中得到的对小背景分布的一致依赖在Sobolev空间中是错误的。
We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations on $\mathbb{T}_x \times \mathbb{R}_v$ cannot, in general, be extended to high Sobolev spaces in the case of gravitational interactions. This is done by showing in every Sobolev space, there exists background distributions such that one can construct arbitrarily small perturbations that exhibit arbitrarily many isolated nonlinear oscillations in the density. These oscillations are known as plasma echoes in the physics community. For the case of electrostatic interactions, we demonstrate a sequence of small background distributions and asymptotically smaller perturbations in $H^s$ which display similar nonlinear echoes. This shows that in the electrostatic case, any extension of Mouhot and Villani's theorem to Sobolev spaces would have to depend crucially on some additional non-resonance effect coming from the background -- unlike the case of Gevrey-$\nu$ with $\nu < 3$ regularity, for which results are uniform in the size of small backgrounds. In particular, the uniform dependence on small background distributions obtained in Mouhot and Villani's theorem in Gevrey class is false in Sobolev spaces.