On Landau damping
On Landau damping
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DOI：
10.1007/s11511-011-0068-9
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发表时间：
2011-09-01
期刊：
影响因子：
3.7
通讯作者：
Villani, Cedric
中科院分区：
文献类型：
--
作者：
Mouhot, Clement;Villani, Cedric
超越线性化研究是朗道阻尼器理论中一个长期存在的问题。本文建立了解析正则性下的指数朗道阻尼。根据运动变量和空间变量之间的规律性转移，而不是能量交换，重新解释了阻尼现象；相混合是驱动机制。分析涉及到新的解析范数族，通过与自由传输方程的解进行比较来衡量正则性；新的泛函不等式；非线性回波的控制；尖锐的“偏转”估计；以及牛顿近似方案。我们的结果适用于任何不比库仑或牛顿相互作用更奇异的势；极限情况包含了特定的技术努力。在尖锐的假设下，建立了非线性Vlasov方程齐次平衡点的稳定性。我们指出了与KAM理论的强烈相似之处，并讨论了物理含义。最后，我们将这些结果推广到一些Gevrey(非解析)分布函数。
Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp "deflection" estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.