Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

A. Bensouilah, G. K. Duong, T. E. Ghoul

TL;DR

This work constructs non-self-similar (Type II) blow-up solutions for the higher-dimensional Yang-Mills heat flow in dimensions $d\ge 11$ under an $SO(d)$-equivariant symmetry. It develops a time-dependent spectral framework around a moving ground state $Q_b$, introducing blow-up variables and a dynamic operator $\mathscr{L}_b$, and proves both stable and unstable blow-up regimes via a shrinking-set bootstrap, finite-dimensional reduction, and maximum-principle techniques. Central to the approach is the precise diagonalisation of $\mathscr{L}_b$ through matched inner/outer eigenfunctions, yielding blow-up speeds $\lambda_ℓ(t) = C(u_0)(T-t)^{\frac{2ℓ}{\alpha}}$ with stability properties: stable for $\ell=1$ and codimension $\ell-1$ for $\ell\ge 2$. The results also establish the existence and properties of the ground state $Q$ and provide a robust analytical framework that combines spectral analysis with parabolic maximum principles, applicable to other nonlinear parabolic PDEs with Type II blow-up.

Abstract

In this paper, we consider the Yang-Mills heat flow on $\mathbb R^d \times SO(d)$ with $d \ge 11$. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: $$ \partial_t u =\partial_r^2 u +\frac{d+1}{r} \partial_r u -3(d-2) u^2 - (d-2) r^2 u^3, \text{ and } (r,t) \in \mathbb R_+ \times \mathbb R_+. $$ We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for $d \ge 11$ and prove that the asymptotic of the solution is of the form $$ u(r,t) \sim \frac{1}{λ_\ell(t)} \mathcal{Q} \left( \frac{r}{\sqrt{λ_\ell (t)}} \right), \text{ as } t \to T ,$$ where $\mathcal{Q}$ is the ground state with boundary conditions $\mathcal{Q}(0)=-1, \mathcal{Q}'(0)=0$ and the blowup speed $λ_\ell$ verifies $$λ_\ell (t) = \left( C(u_0) +o_{t\to T}(1) \right) (T-t)^{\frac{2\ell }α} \text{ as } t \to T,~~ \ell \in \mathbb{N}^*_+, ~~α>1.$$ In particular, when $\ell = 1$, this asymptotic is stable whereas for $ \ell \ge 2$ it becomes stable on a space of codimension $\ell-1$. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates.

Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

TL;DR

This work constructs non-self-similar (Type II) blow-up solutions for the higher-dimensional Yang-Mills heat flow in dimensions under an -equivariant symmetry. It develops a time-dependent spectral framework around a moving ground state , introducing blow-up variables and a dynamic operator , and proves both stable and unstable blow-up regimes via a shrinking-set bootstrap, finite-dimensional reduction, and maximum-principle techniques. Central to the approach is the precise diagonalisation of through matched inner/outer eigenfunctions, yielding blow-up speeds with stability properties: stable for and codimension for . The results also establish the existence and properties of the ground state and provide a robust analytical framework that combines spectral analysis with parabolic maximum principles, applicable to other nonlinear parabolic PDEs with Type II blow-up.

Abstract

In this paper, we consider the Yang-Mills heat flow on with . Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for and prove that the asymptotic of the solution is of the form where is the ground state with boundary conditions and the blowup speed verifies In particular, when , this asymptotic is stable whereas for it becomes stable on a space of codimension . Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates.
Paper Structure (28 sections, 26 theorems, 686 equations)

This paper contains 28 sections, 26 theorems, 686 equations.

Table of Contents

  1. Mathematical setting
  2. Strategy of the proof
  3. Spectral analysis
  4. Proof in the stable case without technical details
  5. Shrinking set
  6. Preparing initial data
  7. The proof of Theorem \ref{['theorem-existence-Type-II-blowup']}
  8. Finite dimensional system
  9. Control of the infinite dimensional part
  10. Energy estimate
  11. $L^\infty$ bounds
  12. The existence of unstable blowup solutions
  13. Shrinking set for $\ell \ge 2$
  14. Preparing initial data
  15. Finite dimensional reduction
  16. ...and 13 more sections

Key Result

Theorem 1

Let $d \ge 11$ be an integer. Then, there exist initial data $u_0 \in C^\infty_0 (\mathbb{R}_+, \mathbb{R})$ such that the corresponding solution to equa-Yang-Mills- blows up in finite time $T(u_0)$. Moreover, the following decomposition holds true where $Q$ is the ground state of equa-Yang-Mills- satisfying $Q(0)= -1$ and $Q'(0) =0$; and the error $\tilde{u} (r,t)$ satisfies and the blowup spee

Theorems & Definitions (56)

  • Theorem 1: Existence of stable blowup solution
  • Theorem 2: Existence of unstable blowup solutions
  • Remark 3: Related blowup results for PDE's problem
  • Remark 4: Novelty of the paper
  • Remark 5: Structure of the paper
  • Proposition 3.1: Diagonalisation of $\mathscr{L}_\infty^\beta$, HVunpublished-92, BSIMRN19, CMRJAMS20
  • Proposition 3.2: Diagonalisation of $\mathscr{L}_b$
  • proof
  • Definition 4.1: Shrinking set
  • Lemma 4.2: Pointwise estimates
  • ...and 46 more