<p>In this paper, we prove a sharp and strong nonuniqueness for a class of weak solutions to the three-dimensional magnetohydrodynamic (MHD) system. More precisely, we show that any weak solution <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10201_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">\((v,b)\in L^p_tL^{\infty }_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mi>L</mi> <mi>t</mi> <mi>p</mi> </msubsup> <msubsup> <mi>L</mi> <mi>x</mi> <mi>∞</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> is nonunique in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10201_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^p_tL^{\infty }_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>t</mi> <mi>p</mi> </msubsup> <msubsup> <mi>L</mi> <mi>x</mi> <mi>∞</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10201_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\le p&lt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>&lt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, which reveals the strong nonuniqueness, and the sharpness in terms of the classical Ladyzhenskaya–Prodi–Serrin criteria at endpoint <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10201_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\((2, \infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, for any <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10201_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we can also construct infinitely many weak solutions of the ideal MHD system in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10201_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^1_tC^{1-\epsilon }_x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>t</mi> <mn>1</mn> </msubsup> <msubsup> <mi>C</mi> <mi>x</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>ϵ</mi> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. Our result shows the nonuniqueness for any weak solution (<i>v</i>,&#xa0;<i>b</i>) including nontrivial magnetic field <i>b</i>.</p>

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Sharp and Strong Nonuniqueness for the Magnetohydrodynamic Equations

  • Yao Nie,
  • Weikui Ye

摘要

In this paper, we prove a sharp and strong nonuniqueness for a class of weak solutions to the three-dimensional magnetohydrodynamic (MHD) system. More precisely, we show that any weak solution \((v,b)\in L^p_tL^{\infty }_x\) ( v , b ) L t p L x is nonunique in \(L^p_tL^{\infty }_x\) L t p L x with \(1\le p<2\) 1 p < 2 , which reveals the strong nonuniqueness, and the sharpness in terms of the classical Ladyzhenskaya–Prodi–Serrin criteria at endpoint \((2, \infty )\) ( 2 , ) . Moreover, for any \(\epsilon >0\) ϵ > 0 , we can also construct infinitely many weak solutions of the ideal MHD system in \(L^1_tC^{1-\epsilon }_x\) L t 1 C x 1 - ϵ . Our result shows the nonuniqueness for any weak solution (vb) including nontrivial magnetic field b.