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<p>We consider finite-time and <i>k</i>-equivariant solutions to the harmonic map heat flow from <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(B^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>B</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(S^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> under general time-dependent boundary data and prove that the bubble tree decomposition contains only one bubble. The method relies on the Maximum and Comparison Principle. We also exhibit solutions blowing up in infinite time for any <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(k \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On blow-up trees for the harmonic map heat flow from \(B^2\) to \(S^2\)

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月23日

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  • Dylan Samuelian

摘要

We consider finite-time and k-equivariant solutions to the harmonic map heat flow from \(B^2\) B 2 to \(S^2\) S 2 under general time-dependent boundary data and prove that the bubble tree decomposition contains only one bubble. The method relies on the Maximum and Comparison Principle. We also exhibit solutions blowing up in infinite time for any \(k \ge 1\) k ≥ 1 .