We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with previous works. For the magnetic field we mostly employ the classical boundary conditions where the field extends in the whole space. We establish critical values of hydrodynamic and magnetic Reynolds numbers below which arbitrarily large initial perturbations of the hydrodynamic flow decay. This involves generalization of the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic operator. For high Reynolds number turbulence we provide an estimate of critical magnetic Reynolds number below which arbitrarily large fluctuations of the magnetic field decay.

中文翻译：

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层流和湍流中的线性和非线性水磁稳定性

我们考虑导电流体的规定纯流体力学流动的任意大扰动的演化。我们研究流量扰动以及所产生的磁场是否随时间衰减或增长，并构成了一个发电机过程。为此，我们推导了一个广义的雷诺兹-奥尔（Reynolds-Orr）方程，该方程用于流体动力扰动的动能和磁能之和。流动被限制在有限的体积内，因此边界处速度的法线分量为零。与先前的作品相比，切向分量是任意的。对于磁场，我们主要采用经典边界条件，其中磁场在整个空间中延伸。我们建立了水动力和雷诺数的临界值，低于该临界值时，水动力流衰减的任意大的初始扰动。对于椭圆算子的最小特征值，这涉及到Rayleigh-Faber-Krahn不等式的推广。对于高雷诺数湍流，我们提供了临界磁雷诺数的估计值，低于该值时，磁场衰减的任意大波动。