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Global oscillatory solutions for the Yang-Mills heat flow

Yannick Sire, Juncheng Wei, Youquan Zheng, Yifu Zhou

Abstract

We investigate the long-time dynamics for the global solution of the $SO(4)$-equivariant Yang-Mills heat flow (YMHF) with structure group $SU(2)$ in space dimension $4$. For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as $t\to \infty$.

Global oscillatory solutions for the Yang-Mills heat flow

Abstract

We investigate the long-time dynamics for the global solution of the -equivariant Yang-Mills heat flow (YMHF) with structure group in space dimension . For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as .
Paper Structure (8 sections, 8 theorems, 171 equations)

This paper contains 8 sections, 8 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The inner outer gluing scheme and estimates for the linear problems
  3. Global Scaling Law
  4. The outer problem
  5. Solving the scaling parameter
  6. Solving the inner problem: Proof of Theorem \ref{['thm']}
  7. Convolution estimates in the outer problem
  8. Radial heat kernel

Key Result

Theorem 1.1

Let $\Theta(r,t)$ solve Then for any smooth initial data $\Theta_0(r)$ satisfying problem (YMSO4-6Dheat) has a solution with the following form where $U(\rho) = \frac{2}{\rho^2+1}$, and $\|\Psi_*(\cdot,t)\|_{L^\infty}=O(t^{-1}(\log t)^{-a-\epsilon})$ for some $0<\epsilon<1$. The global scaling law of $\lambda$ is described precisely by Here $C(t_0, t)$ is a smooth bounded function of $t\in (t_

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 7 more