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<p>In this work, we investigate the difference estimate for a class of Euler–Maxwell systems and those of magnetohydrodynamics (in short, MHD) systems in three dimensions. We decompose the Euler–Maxwell system into three parts, namely the MHD system, <i>auxiliary linear system</i> and <i>error part system</i>. As a result, we obtain the convergence of the fluid velocity <i>u</i>, electric field <i>E</i> and magnetic field <i>B</i> from the Euler–Maxwell system toward the MHD system in mixed-norm spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p_tL^2_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>t</mi> <mi>p</mi> </msubsup> <msubsup> <mi>L</mi> <mi>x</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> as the speed of light <i>c</i> approaches infinity for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\in [1,\infty ]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We also derive non-convergence results of electric current <i>j</i> or <i>cE</i>, and these results are classified by a certain threshold for <i>p</i>. Finally, we investigate how the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-energy flow of Euler–Maxwell system evolves as <i>c</i> tends to infinity, leading to the vanishing of Ampère’s term in the Euler–Maxwell system.</p>

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Convergence and non-convergence phenomena in Euler–Maxwell to MHD transitions

  • 期刊论文

  • 发表日期：

    2026年2月13日

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  • Dong-ha Kim,
  • Junha Kim,
  • Jihoon Lee

摘要

In this work, we investigate the difference estimate for a class of Euler–Maxwell systems and those of magnetohydrodynamics (in short, MHD) systems in three dimensions. We decompose the Euler–Maxwell system into three parts, namely the MHD system, auxiliary linear system and error part system. As a result, we obtain the convergence of the fluid velocity u, electric field E and magnetic field B from the Euler–Maxwell system toward the MHD system in mixed-norm spaces \(L^p_tL^2_x\) L t p L x 2 as the speed of light c approaches infinity for \(p\in [1,\infty ]\) p ∈ [ 1 , ∞ ] . We also derive non-convergence results of electric current j or cE, and these results are classified by a certain threshold for p. Finally, we investigate how the \(L^2\) L 2 -energy flow of Euler–Maxwell system evolves as c tends to infinity, leading to the vanishing of Ampère’s term in the Euler–Maxwell system.