The Magnetized Vlasov-Ampère system and the Bernstein-Landau paradox
Frédérique Charles, Bruno Després, Alexandre Rege, Ricardo Weder
备注：
We have edited the paper to include new results and to add details that improve the presentation. Further, we have added the word magnetized to the title
We study the Bernstein-Landau paradox in the collisionless motion of an electrostatic plasma in the presence of a constant external magnetic field. The Bernstein-Landau paradox consists in that in the presence of the magnetic field, the electric field and the charge density fluctuation have an oscillatory behavior in time. This is radically different from Landau damping, in the case without magnetic field, where the electric field tends to zero for large times. We consider this problem from a new point of view. Instead of analyzing the linear magnetized Vlasov-Poisson system, as it is usually done, we study the linear magnetized Vlasov-Ampère system. We formulate the magnetized Vlasov-Ampère system as a Schrödinger equation with a selfadjoint magnetized Vlasov-Ampère operator in the Hilbert space of states with finite energy. The magnetized Vlasov-Ampère operator has a complete set of orthonormal eigenfunctions, that include the Bernstein modes. The expansion of the solution of the magnetized Vlasov-Ampère system in the eigenfunctions shows the oscillatory behavior in time. We prove the convergence of the expansion under optimal conditions, assuming only that the initial state has finite energy. This solves a problem that was recently posed in the literature. The Bernstein modes are not complete. To have a complete system it is necessary to add eigenfunctions that are associated with eigenvalues at all the integer multiples of the cyclotron frequency. These special plasma oscillations actually exist on their own, without the excitation of the other modes. In the limit when the magnetic fields goes to zero the spectrum of the magnetized Vlasov-Ampère operator changes drastically from pure point to absolutely continuous in the orthogonal complement to its kernel, due to a sharp change on its domain. This explains the Bernstein-Landau paradox.
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Departamento de Física Matemática（数学物理系）；Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas（应用数学与系统研究所）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）
Constant residual electrostatic electron plasma mode in Vlasov-Ampere system
在无碰撞V-P电子等离子体系统中，存在两种电场扰动模式：指数朗道阻尼电子等离子体波和初值敏感弹道模式。这里，V-P系统被稍微修改为Vlasov-Ampere（V-A）系统。揭示了一种新的常余模。在数学上，该模式来自初始电场扰动的拉普拉斯变换，并且在物理上表示初始扰动（例如，外部电场扰动）不会被衰减掉。因此，该残余模式比弹道模式更难被阻尼。[Physics of Plasmas 20，112108（2013）; doi：10.1063/1.4831761]
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