<p>In this paper, we are concerned with energy equality of weak solutions to the electron magnetohydrodynamic equations. In the spirit of Cheskidov and Luo’s and Berselli’s works, it is shown that weak solutions in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2532_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{2/\alpha ,\infty }(0,T;C^{\alpha }(\mathbb {R}^{3}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mi>α</mi> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>C</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2532_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0\le \alpha &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> satisfy energy equality in this system. In addition, we prove that magnetic helicity identity holds if weak solutions <i>H</i> satisfy <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2532_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="206" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\in L^{ {2}/{(\alpha +1)},\infty }(0,T;C^{\alpha }(\mathbb {R}^{3}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>∞</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>C</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2532_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(0&lt;\alpha &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>α</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. This indicates that the conservation of magnetic helicity is easier than that of energy and agrees with the Taylor conjecture in magnetohydrodynamic turbulence.</p>

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Energy identity of weak solutions to the electron magnetohydrodynamic equations in Hölder spaces

  • Lingxian Meng,
  • Ruomeng Sun,
  • Yanqing Wang

摘要

In this paper, we are concerned with energy equality of weak solutions to the electron magnetohydrodynamic equations. In the spirit of Cheskidov and Luo’s and Berselli’s works, it is shown that weak solutions in \(L^{2/\alpha ,\infty }(0,T;C^{\alpha }(\mathbb {R}^{3}))\) L 2 / α , ( 0 , T ; C α ( R 3 ) ) with \(0\le \alpha <1\) 0 α < 1 satisfy energy equality in this system. In addition, we prove that magnetic helicity identity holds if weak solutions H satisfy \(H\in L^{ {2}/{(\alpha +1)},\infty }(0,T;C^{\alpha }(\mathbb {R}^{3}))\) H L 2 / ( α + 1 ) , ( 0 , T ; C α ( R 3 ) ) with \(0<\alpha <1\) 0 < α < 1 . This indicates that the conservation of magnetic helicity is easier than that of energy and agrees with the Taylor conjecture in magnetohydrodynamic turbulence.