低维情形下1-等变调和映射热流与径向能量临界非线性热方程的无气泡树性质
No bubble trees for the $1$-equivariant harmonic map heat flow and the radial energy-critical nonlinear heat equation in low dimensions
AI总结：
该研究证明低维情形下1-等变调和映射热流与径向能量临界非线性热方程的解至多含一个气泡，通过调制分析排除气泡树，方法不依赖最大值原理且适用于任意气泡符号。
AI中文摘要：
我们研究从$\boldsymbol{\text{R}}^2$到$\boldsymbol{\text{S}}^2$的1-等变调和映射热流（HMHF），以及维数$d=3,4,5$下的径向能量临界非线性热方程（NLH）。我们证明，HMHF的每个有限能量解和NLH的每个$\boldsymbol{\text{\text{H}}}^1$有界解至多有一个气泡：每个有限时间爆破恰好有一个气泡，而每个全局解要么没有气泡，要么在无穷远处有一个气泡。我们从孤子分解出发，通过调制分析排除气泡树，从两个最内层尺度的相对动力学导出矛盾。该能量方法不依赖最大值原理，且特别地，对气泡符号无限制适用。
英文摘要：
We consider the $1$-equivariant harmonic map heat flow (HMHF) from $\mathbb R^2$ to $\mathbb S^2$ and the radial energy-critical nonlinear heat equation (NLH) in dimensions $d=3,4,5$. We prove that every finite-energy solution of (HMHF) and every $\dot H^1$-bounded solution of (NLH) has at most one bubble: every finite-time blow-up has exactly one bubble, whereas every global solution has either no bubble or one bubble at infinite time. Starting from the soliton resolution, we exclude bubble trees by modulation analysis, deriving a contradiction from the relative dynamics of the two innermost scales. This energy method does not rely on maximum principle and, in particular, applies without restriction on the bubble signs.
