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On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation

Kihyun Kim, Frank Merle

TL;DR

This work delivers a complete classification of global dynamics for energy-critical, symmetry-reduced PDEs: the D-equivariant harmonic map heat flow on the plane and the radial energy-critical heat equation in high dimensions. Central to the approach is a novel energy-method framework that avoids maximum principles and leverages a monotonicity estimate near multi-bubble configurations, together with the construction of modified multi-bubble profiles to gain a robust L^2_t control. The authors prove that HMHF solutions with D>=3 exist globally and decompose into a fixed number J=|m−ℓ| of bubbles with universal scaling laws, while NLH radial solutions in N>=7 admit a full multi-bubble soliton-resolution-type description under radial symmetry. The results establish universal bubble-speed laws and signs, provide first rigorous bubble-tree dynamics in this symmetry class, and open pathways to extend the energy-method framework to other gradient flows and geometric evolution equations.

Abstract

We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices $D\geq3$; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^{1}$-bounded radial solutions to (NLH) in dimensions $N\geq7$, building upon soliton resolution for such solutions. To our knowledge, this provides the first rigorous classification of bubble tree dynamics within symmetry. We introduce a new approach based on the energy method that does not rely on maximum principle. The key ingredient of the proof is a monotonicity estimate near any bubble tree configurations, which in turn requires a delicate construction of modified multi-bubble profiles also.

On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation

TL;DR

This work delivers a complete classification of global dynamics for energy-critical, symmetry-reduced PDEs: the D-equivariant harmonic map heat flow on the plane and the radial energy-critical heat equation in high dimensions. Central to the approach is a novel energy-method framework that avoids maximum principles and leverages a monotonicity estimate near multi-bubble configurations, together with the construction of modified multi-bubble profiles to gain a robust L^2_t control. The authors prove that HMHF solutions with D>=3 exist globally and decompose into a fixed number J=|m−ℓ| of bubbles with universal scaling laws, while NLH radial solutions in N>=7 admit a full multi-bubble soliton-resolution-type description under radial symmetry. The results establish universal bubble-speed laws and signs, provide first rigorous bubble-tree dynamics in this symmetry class, and open pathways to extend the energy-method framework to other gradient flows and geometric evolution equations.

Abstract

We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices ; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of -bounded radial solutions to (NLH) in dimensions , building upon soliton resolution for such solutions. To our knowledge, this provides the first rigorous classification of bubble tree dynamics within symmetry. We introduce a new approach based on the energy method that does not rely on maximum principle. The key ingredient of the proof is a monotonicity estimate near any bubble tree configurations, which in turn requires a delicate construction of modified multi-bubble profiles also.
Paper Structure (27 sections, 23 theorems, 282 equations)

This paper contains 27 sections, 23 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. Energy-critical equivariant harmonic map heat flow
  3. Energy-critical radial nonlinear heat equation
  4. Discussions on the results
  5. Notation
  6. Basic notation
  7. Unified notation for \ref{['eq:NLH-rad']} and \ref{['eq:HMHF-equiv']}.
  8. Notation for multi-bubbles
  9. Sequences of universal constants in main theorems
  10. Strategy of the proof
  11. Preliminaries
  12. Statements of bubble decomposition and soliton resolution
  13. Basic pointwise estimates
  14. Formal inversion of $H$
  15. Coercivity estimate for $H_{\vec{\lambda}}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $D\geq3$. Let $\ell,m\in\mathbb{Z}$ and $v_{0}\in\mathcal{E}_{\ell,m}$ be an initial data. Then, the corresponding solution $v(t)$ to eq:HMHF-equiv is global and admits the decomposition for some real $L_{\infty}\in(0,\infty)$ and Here, the rates $\lambda_{j,L_{\infty}}^{\mathrm{ex}}(t)$ are given by and $\beta_{j}$s are the universal constants defined by eq:def-beta_j and eq:def-kappa-expl

Theorems & Definitions (49)

  • Theorem 1.1: Classification of $D$-equivariant solutions within high equivariance
  • Theorem 1.2: Classification of $\dot{H}^{1}$-bounded radial solutions in large dimensions
  • Remark 1.3
  • Theorem 1.4: Classification of radial bubble trees for $D>2$
  • Proposition 2.1: Bubble decomposition for equivariant solutions; see JendrejLawrie2023CVPDE
  • Lemma 2.3
  • Remark 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 39 more