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Finite-time singularity formation for the heat flow of the $H$-system
Analysis of PDEs · Submitted 24 Nov 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right.
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We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right. \end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$. The singularity at finite time forms as a scaled least energy $H$-bubble, denoted as $W$, exhibiting type II blow-up speed. One key observation is that the linearized operators around $W$ projected onto $W^\perp$ and in the $W$-direction are in fact decoupled. On $W^\perp$, the linearization is the linearized harmonic map heat flow, while in the $W$-direction, it is the linearized Liouville-type flow. Based on this, we also prove the non-degeneracy of the $H$-bubbles with any degree.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right. | contribution 0.87 | 0.94 | 0.81 | asserted 1.00 |
| s1 | \end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$. | background 0.90 | 0.16 | 0.03 | unclear 0.54 |
| s2 | The singularity at finite time forms as a scaled least energy $H$-bubble, denoted as $W$, exhibiting type II blow-up speed. | result 0.97 | 0.88 | 0.67 | asserted 1.00 |
| s3 | One key observation is that the linearized operators around $W$ projected onto $W^\perp$ and in the $W$-direction are in fact decoupled. | result 0.65 | 0.85 | 0.05 | asserted 1.00 |
| s4 | On $W^\perp$, the linearization is the linearized harmonic map heat flow, while in the $W$-direction, it is the linearized Liouville-type flow. | method 0.50 | 0.77 | 0.05 | asserted 1.00 |
| s5 | Based on this, we also prove the non-degeneracy of the $H$-bubbles with any degree. | result 0.83 | 0.94 | 0.11 | asserted 1.00 |
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Original abstract
We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right. \end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$. The singularity at finite time forms as a scaled least energy $H$-bubble, denoted as $W$, exhibiting type II blow-up speed. One key observation is that the linearized operators around $W$ projected onto $W^\perp$ and in the $W$-direction are in fact decoupled. On $W^\perp$, the linearization is the linearized harmonic map heat flow, while in the $W$-direction, it is the linearized Liouville-type flow. Based on this, we also prove the non-degeneracy of the $H$-bubbles with any degree.
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How the literature uses it
3 arXiv papers cite this in 7 sentences
Papers citing it · 6 sentences without a confident label
Cites it as background 1 paper
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As a geometrical problem, it constitutes a toy model that could be useful to provide some information for understanding other, more difficult geometrical issues in higher dimension, e.g., the $H$-bubble problem (see the very recent paper [ref] and the references therein).
Planar closed curves with prescribed curvature · Introduction · 0.81
What it cites · 7 sentences without a confident label
Uses the method or tool of 2 papers
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Here we follow the argument in [ref] and [ref] .
A stability theory beyond the co-rotational setting for critical Wave Maps blow up · Generalized eigenfunctions and density of spectral measure · 1.00
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The linear theory for the mode $0$ is stated as follows, and its proof is in a similar spirit as [ref] .
Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two · Duhamel's representation by DFT · 0.88
Cites as background 1 paper
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Non-degeneracy of harmonic maps proved in [ref] is a rather important property in the study of singularity formation for the harmonic map heat flow and the quantitative stability of harmonic maps, see [ref] and the references therein.
Quantitative stability of harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$ with higher degree · Introduction and main results · 0.87
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