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Continuous in time bubble decomposition for the harmonic map heat flow

Jacek Jendrej, Andrew Lawrie, Wilhelm Schlag

TL;DR

This work provides a time-continuous description of bubble formation for the harmonic map heat flow $u:\mathbb{R}^2\to\mathbb{S}^2$, showing that every time sequence admits a subsequence along which bubbling occurs and the solution converges to a multi-bubble configuration. The authors develop a minimal collision-energy framework and localized bubbling tools to extract bubbles at shrinking scales without relying on Palais–Smale sequences, and they prove continuous-in-time convergence to a fixed finite collection of harmonic maps. The results unify and extend the classical bubble-branching picture by proving convergence to multi-bubble configurations in continuous time and for both finite-time blow-up and global-in-time solutions, with energy quantization and neck-control properties. This advances our understanding of dynamic bubble resolution and provides a robust framework for analyzing time-evolving singularities in low-dimensional geometric flows. The techniques rely on precise scale/center analysis for harmonic maps, localized energy estimates, and compactness arguments to navigate interactions among bubbles.

Abstract

We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.

Continuous in time bubble decomposition for the harmonic map heat flow

TL;DR

This work provides a time-continuous description of bubble formation for the harmonic map heat flow , showing that every time sequence admits a subsequence along which bubbling occurs and the solution converges to a multi-bubble configuration. The authors develop a minimal collision-energy framework and localized bubbling tools to extract bubbles at shrinking scales without relying on Palais–Smale sequences, and they prove continuous-in-time convergence to a fixed finite collection of harmonic maps. The results unify and extend the classical bubble-branching picture by proving convergence to multi-bubble configurations in continuous time and for both finite-time blow-up and global-in-time solutions, with energy quantization and neck-control properties. This advances our understanding of dynamic bubble resolution and provides a robust framework for analyzing time-evolving singularities in low-dimensional geometric flows. The techniques rely on precise scale/center analysis for harmonic maps, localized energy estimates, and compactness arguments to navigate interactions among bubbles.

Abstract

We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.
Paper Structure (18 sections, 18 theorems, 238 equations)

This paper contains 18 sections, 18 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Statement of the results
  4. Summary of the proof
  5. Notational conventions
  6. Preliminaries
  7. Properties of harmonic maps
  8. The scale and center of a harmonic map
  9. Properties of the harmonic map heat flow
  10. Well-posedness
  11. Local $L^\infty$ estimates for the heat flow
  12. Concentration properties of the heat flow
  13. Localized sequential bubbling
  14. Proofs of the main results
  15. The minimal collision energy
  16. ...and 3 more sections

Key Result

Theorem 1

Let $u(t)$ be the unique solution to eq:hmhf associated to initial data $u_0 \in \mathcal{E}$. Let $T_+ = T_+(u_0) \in (0, \infty]$ denote the maximal time of existence. (Finite time blow-up) Suppose $T_+ < \infty$. There exist a finite energy map $u^*: \mathbb{R}^2 \to \mathbb{S}^2$, an integer $L and where $\omega_{j}^{(\ell)}(\infty) := \lim_{\left\lvert{x}\right\rvert \to \infty} \omega_{j}^

Theorems & Definitions (49)

  • Theorem 1: Bubble decomposition along any time sequence
  • Remark 1.1
  • Remark 1.2
  • Definition 1.3: Scale of a harmonic map
  • Definition 1.4: Center of a harmonic map
  • Definition 1.5: Multi-bubble configuration
  • Definition 1.6: Localized distance to a multi-bubble configuration
  • Definition 1.7: Localized multi-bubble proximity function
  • Theorem 2: Convergence to multi-bubbles in continuous time
  • Remark 1.8
  • ...and 39 more