<p>In this work, we study the finite time blow-up dynamics of Landau–Lifshitz flow <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">\(u(t, \cdot ):\mathbb {R}^2\rightarrow \mathbb S^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">→</mo> <msup> <mi mathvariant="double-struck">S</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We show that starting from any initial data with (anti-)holomorphic energy lower than <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(4\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>, if the Landau–Lifshitz flow blows up at finite times <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(T&lt;+\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>&lt;</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, then the blow-up rate is at least <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(T-t)^{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with some <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(p&gt;1/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> decided by the coupling constants, and the limit map <i>u</i>(<i>T</i>) is Hölder continuous. Moreover, for arbitrary time sequence <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(t_n\rightarrow T^-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <msup> <mi>T</mi> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2994_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(u(t_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> sub-converges to <i>u</i>(<i>T</i>) in the bubble-tree sense, with no necks.</p>

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Finite-time singularities of 2D Landau–Lifshitz flow for almost-holomorphic maps

  • Ze Li,
  • Chong Song

摘要

In this work, we study the finite time blow-up dynamics of Landau–Lifshitz flow \(u(t, \cdot ):\mathbb {R}^2\rightarrow \mathbb S^2\) u ( t , · ) : R 2 S 2 . We show that starting from any initial data with (anti-)holomorphic energy lower than \(4\pi \) 4 π , if the Landau–Lifshitz flow blows up at finite times \(T<+\infty \) T < + , then the blow-up rate is at least \(O(T-t)^{p}\) O ( T - t ) p with some \(p>1/2\) p > 1 / 2 decided by the coupling constants, and the limit map u(T) is Hölder continuous. Moreover, for arbitrary time sequence \(t_n\rightarrow T^-\) t n T - , \(u(t_n)\) u ( t n ) sub-converges to u(T) in the bubble-tree sense, with no necks.