标题： 有效动力学和磁弛豫模型中的爆破 显示英文标题

标题： Effective Dynamics and Blow Up in a Model of Magnetic Relaxation

摘要： 在本文中，我们研究了一个用于磁弛豫的一维模型。该模型由Moffatt提出，描述了一个低电阻率的粘性等离子体，在这种情况下，压力和惯性远小于磁压。在电阻率趋于$\varepsilon\rightarrow 0$的极限下，我们证明了磁场演化的两个时间尺度：一个是在时间尺度为$\log(\varepsilon^{-1})$的快速时间尺度，在此期间电阻率不起作用，能量仅通过粘性耗散；另一个是时间尺度为$\varepsilon^{-1}$的慢速时间尺度，其特征是电阻率的影响。我们表明，在第二个时间尺度下，当$\varepsilon\rightarrow 0$时，磁场的模趋近于仅依赖于时间的函数。我们还证明，在此条件下，磁场$b_\varepsilon(t,x)$可以通过一个解会出现爆破的偏微分方程的解来近似为$\varepsilon \rightarrow 0$。 显示英文摘要

摘要： In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt and describes a low resistivity viscous plasma, in which the pressure and the inercia are much smaller than the magnetic pressure. In the limit of resistivity $\varepsilon\rightarrow 0$, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order $\log(\varepsilon^{-1})$ in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order $\varepsilon^{-1}$ characterized by the influence of the resistivity. We show that in this second time scale, as $\varepsilon\rightarrow 0$, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field $b_\varepsilon(t,x)$ can be approximated as $\varepsilon \rightarrow 0$ by the solution of a PDE whose solutions exhibit blow up for some choices of initial data.