<p>In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt [<CitationRef CitationID="CR20">20</CitationRef>] and describes a low resistivity viscous plasma in which the pressure and the inertia are much smaller than the magnetic pressure. In the limit of resistivity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\log (\varepsilon ^{-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon ^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> characterized by the influence of the resistivity. We show that in this second time scale, as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$b_\varepsilon (t,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> can be approximated as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> by the solution of a PDE whose solutions exhibit blow up for some choices of initial data.</p>

Effective Dynamics and Blow Up in a Model of Magnetic Relaxation

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2026年6月1日

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Dimitri Cobb,
Daniel Sánchez-Simón del Pino,
Juan J. L. Velázquez

摘要

In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt [20] and describes a low resistivity viscous plasma in which the pressure and the inertia are much smaller than the magnetic pressure. In the limit of resistivity $\varepsilon \rightarrow 0$ $ε \to 0$ , we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order $\log (\varepsilon ^{-1})$ $log (ε^{- 1})$ in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order $\varepsilon ^{-1}$ $ε^{- 1}$ characterized by the influence of the resistivity. We show that in this second time scale, as $\varepsilon \rightarrow 0$ $ε \to 0$ , the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field $b_\varepsilon (t,x)$ $b_{ε} (t, x)$ can be approximated as $\varepsilon \rightarrow 0$ $ε \to 0$ by the solution of a PDE whose solutions exhibit blow up for some choices of initial data.