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Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow

Yezhou Yi

TL;DR

This work establishes the existence and $(l-1)$-codimension stability of type II blow-up for the $SO(d)$-equivariant Yang-Mills heat flow in dimensions $d>10$. The authors develop a robust modulation analysis around a universal blow-up profile $Q$, constructing a finite-parameter family $Q_b$ and proving precise coercivity and energy estimates for the residual $q$ that governs nonlinear corrections. They prove that blow-up occurs at a quantized rate $\\lambda(t)\\sim c(u_0)(T-t)^{\\frac{l}{\\gamma}}$, with the ground state $Q$ giving the leading concentration profile and $q$ vanishing in Sobolev norms as $t\to T$, while the initial data can be chosen to realize any prescribed finite set of unstable directions of codimension $l-1$. The analysis hinges on a detailed spectral study of the linearized operator $\\mathscr{L}$, the construction of an approximate solution with controlled errors, and energy-virial type estimates that close a bootstrap argument, yielding a rigorous framework for high-dimensional, energy-supercritical blow-up phenomena in geometric flows.

Abstract

We consider the $SO(d)$-equivariant Yang-Mills heat flow \begin{equation*} \partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{equation*} in dimensions $d>10.$ We construct a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of a universal profile \begin{equation*} u(t,r)\sim Q\left(\frac{r}{λ(t)}\right), \end{equation*}where $Q$ is a stationary state of the equation and the blow-up rates are quantized by \begin{equation*} λ(t)\sim c_{u}(T-t)^{\frac{l}γ},\,\,\,l\,\,\,\text{is any positive integer},\,\,\,γ=γ(d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{equation*} Moreover, such solutions are in fact $(l-1)$-codimension stable under pertubation of the initial data.

Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow

TL;DR

This work establishes the existence and -codimension stability of type II blow-up for the -equivariant Yang-Mills heat flow in dimensions . The authors develop a robust modulation analysis around a universal blow-up profile , constructing a finite-parameter family and proving precise coercivity and energy estimates for the residual that governs nonlinear corrections. They prove that blow-up occurs at a quantized rate , with the ground state giving the leading concentration profile and vanishing in Sobolev norms as , while the initial data can be chosen to realize any prescribed finite set of unstable directions of codimension . The analysis hinges on a detailed spectral study of the linearized operator , the construction of an approximate solution with controlled errors, and energy-virial type estimates that close a bootstrap argument, yielding a rigorous framework for high-dimensional, energy-supercritical blow-up phenomena in geometric flows.

Abstract

We consider the -equivariant Yang-Mills heat flow \begin{equation*} \partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{equation*} in dimensions We construct a family of solutions which blow up in finite time via concentration of a universal profile \begin{equation*} u(t,r)\sim Q\left(\frac{r}{λ(t)}\right), \end{equation*}where is a stationary state of the equation and the blow-up rates are quantized by \begin{equation*} λ(t)\sim c_{u}(T-t)^{\frac{l}γ},\,\,\,l\,\,\,\text{is any positive integer},\,\,\,γ=γ(d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{equation*} Moreover, such solutions are in fact -codimension stable under pertubation of the initial data.
Paper Structure (17 sections, 20 theorems, 307 equations)

This paper contains 17 sections, 20 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. On the linearized operator $\mathscr{L}$
  3. Decomposition, kernel and computation of the inverse of $\mathscr{L}$
  4. Coercivity of $\mathscr{L}$
  5. Leibniz rule for the iteration of $\mathscr{L}$
  6. Construction of an approximate solution
  7. Definition and properties of degree and homogeneous admissable functions
  8. Estimate of the approximate profile and error terms
  9. Localized approximation
  10. Linearization of $\{b_k\}_{k=1}^L$
  11. Decomposition of the solution and coercivity-determined estimates on the remainder term
  12. Decomposition of the solution
  13. Coercivity-determined estimates on $q$
  14. Description of initial data and bootstrap assumption
  15. Modulation estimates and improved bounds
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d>10,$$\gamma$ as in (def of gamma), let $l$ be any positive integer, denoting given $L\gg l$ a large integer and defining $\Bbbk:=L+\hbar+1.$ Then there exists a smooth radial initial data $u_0$ such that the corresponding solution to (eq:Y-M heat) has the decomposition where and Moreover, the blow-up solution is $(l-1)$-codimension stable.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Proof
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 31 more