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Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space

Joachim Krieger, Jacob Sterbenz

TL;DR

The paper proves global regularity for Yang–Mills equations on high-dimensional Minkowski space under smallness of the critical gauge-covariant Sobolev norm $\dot{H}_A^{(n-4)/2}$ (with even $n\ge 6$). It builds a robust Coulomb gauge framework, uses an explicit non-abelian parametrix with a $G$-valued phase, and reduces Strichartz estimates to a fixed-frequency half-wave parametrix construction. A key a-priori estimate for the curvature, coupled with a detailed bootstrap in angular Besov–type spaces, yields global control of the curvature and connection, ensuring smooth global solutions. The analytic core hinges on intricate Besov and angular decompositions, gauge-covariant energy estimates, and a careful parametrix analysis to transfer local well-posedness into global regularity.

Abstract

We prove that smallness of the critical Sobolev norm implies regularity for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space.

Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space

TL;DR

The paper proves global regularity for Yang–Mills equations on high-dimensional Minkowski space under smallness of the critical gauge-covariant Sobolev norm (with even ). It builds a robust Coulomb gauge framework, uses an explicit non-abelian parametrix with a -valued phase, and reduces Strichartz estimates to a fixed-frequency half-wave parametrix construction. A key a-priori estimate for the curvature, coupled with a detailed bootstrap in angular Besov–type spaces, yields global control of the curvature and connection, ensuring smooth global solutions. The analytic core hinges on intricate Besov and angular decompositions, gauge-covariant energy estimates, and a careful parametrix analysis to transfer local well-posedness into global regularity.

Abstract

We prove that smallness of the critical Sobolev norm implies regularity for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space.

Paper Structure

This paper contains 19 sections, 21 theorems, 553 equations.

Table of Contents

  1. Introduction
  2. Some Basic Notation
  3. Some gauge-theoretic preliminaries
  4. Some analytic preliminaries
  5. Gauge construction for the initial data; Reduction to a second order system and the main a-priori estimate
  6. Construction of the initial frame, and the comparison principle
  7. Local existence in the Coulomb gauge
  8. The second order curvature equation and the main a-priori estimate
  9. Proof of the Main Bootstrapping Estimate
  10. Proof of the Critical Bootstrapping Estimate
  11. Reduction to Approximate Half-Wave Operators
  12. Construction of the half wave operators
  13. Fixed Time $L^2$ Estimates for the Parametrix
  14. Proof of the Accuracy estimate \ref{['half_wave_est3']}
  15. The Dispersive Estimate
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let the number of spatial dimensions be even and such that 6\leqslant n. Then there exists fixed constants 0 < \varepsilon_0 , C such that if (\underline{F}(0),\underline{D}(0) , E(0)) is an admissible data set which satisfies the smallness condition: and there exists constants M_k < \infty, \frac{n-4}{2} < k \in \mathbb{N} such that: then there exists a unique global solution to the field equat

Theorems & Definitions (46)

  • Theorem 1.1: Critical regularity for high dimensional Yang-Mills
  • Remark 1.2
  • Lemma 3.1: Classical Uhlenbeck lemma
  • Lemma 3.2: Uhlenbeck lemma for small $L^n$ perturbations of Coulomb potentials with small $L^\frac{n}{2}$ curvature.
  • Remark 3.3
  • proof : Proof of Lemma \ref{['A_Uhl_lem']}
  • Lemma 4.1
  • Remark 4.2
  • proof : Proof of estimate \ref{['general_besov_embed']}
  • Lemma 5.1: Comparison principle for Sobolev norms on $\mathbb{R}^n$
  • ...and 36 more