论文标题
磁化的Vlasov-ampère系统和伯恩斯坦 - 兰道悖论
The Magnetized Vlasov-Ampère system and the Bernstein-Landau paradox
论文作者
论文摘要
我们研究了恒定外部磁场存在静电等离子体的无碰撞运动中的伯恩斯坦 - 兰道悖论。 Bernstein-Landau悖论是在磁场存在的情况下，电场和电荷密度波动的及时振荡行为。在没有磁场的情况下，这与Landau阻尼完全不同，在没有磁场的情况下，电场倾向于零。我们从新的角度考虑了这个问题。我们研究线性磁化的vlasov-ampère系统，而不是分析线性磁化的Vlasov-Poisson系统。我们将磁化的vlasov-ampère系统作为schrödinger方程，并具有有限能量的州希尔伯特（Hilbert）空间中的自我偶像磁化的vlasov-ampère操作员。磁化的vlasov-ampère操作员具有一组完整的正统本征函数，其中包括伯恩斯坦模式。本征函数中磁化vlasov-ampère系统的溶液的扩展显示了及时的振荡行为。我们证明在最佳条件下扩展的收敛性仅假设初始状态具有有限的能量。这解决了最近在文献中提出的问题。伯恩斯坦模式不完整。要拥有一个完整的系统，有必要添加与环体频率所有整数倍数相关的特征功能。这些特殊的等离子体振荡实际上是独自存在的，而没有其他模式的激发。在磁场将磁化vlasov-ampère操作员的频谱变为零时，由于其域上的急剧变化，从纯点到绝对连续的磁点变化。这解释了伯恩斯坦 - 兰道悖论。
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.
