Effective Dynamics and Blow Up in a Model of Magnetic Relaxation

TL;DR

This paper analyzes a one-dimensional reduced MHD model for magnetic relaxation, uncovering a two-time-scale structure in the vanishing-resistivity limit: a fast, viscosity-dominated regime at times $t = O\left(\log(\varepsilon^{-1})\right)$ and a slow, resistivity-driven regime at $t = O(\varepsilon^{-1})$. On the slow scale the magnetic field modulus becomes time-dependent and its phase obeys a limiting PDE that can blow up for certain initial data; the authors derive the limit system in polar form and prove convergence of the time-rescaled full system $b_\varepsilon$ to the limit dynamics on intervals where the limit is well posed. They show that the limit PDE may experience finite-time blow-up, and provide a rigorous bridge showing $b_\varepsilon$ converges to the limit solution before blow-up, supplemented by numerical experiments illustrating both blow-up and global behavior. The results shed light on the asymptotics of magnetic relaxation and highlight how the full system can regularize or fail to regularize the limit dynamics, offering a framework to interpret long-time MHD relaxation phenomena.

Abstract

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.

Figures (6)

• Figure 1: Illustration of the viscosity-driven time scale. The magnetic field is represented as a parametrized curve $x \mapsto (x, b_1(x), b_2(x))$. The initial magnetic field (the thin blue line) relaxes to a state where $x \mapsto |b|$ is constant (the thick black line lies at the surface of a cylinder).
• Figure 2: An illustration of two different values of the number of turns $N \in \mathbb{Z}$. As in Figure \ref{['fig:Relaxation']}, the magnetic field is represented as a parametrized curve $x \mapsto (0, b_1(x), b_2(x))$. The thick blue line represents a situation where $N = 0$ and the magnetic field is "topologically trivial" with respect to the axis $(1, 0, 0)$, and the thin black line represents the converse situation, where the magnetic field circles around the axis $N \neq 0$ (here $N = 1$).
• Figure 3: The first panel displays the initial datum $\theta_0(x)$ from \ref{['eq:NumericlInitial']}. The second and third panels are the solution evaluated at time $T_1 = 5.625 \cdot 10^{-4}$ and $T_2 = 6.619 \cdot 10^{-4}$ just prior and after singularity formation.
• Figure 4: Initial datum $\lambda \theta_0(x)$, which gives rise to a global solution.
• Figure 5: Numerical solution at times $T_1 = 2.5 \cdot 10^{-4}$, $T_2 = 5 \cdot 10^{-3}$ and $T_3 = 10^{-2}$. No formation of singularity is observed.

Theorems & Definitions (72)

• Theorem 1.1: see Theorems \ref{['t:existencelimit']}, \ref{['t:smallOscillations']} and \ref{['prop:blowup']}
• Theorem 1.2: See Theorem \ref{['teor:main']}
• Definition 1.3
• Definition 1.4
• Remark 1.5
• Theorem 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof