On stable self-similar blowup for corotational wave maps and equivariant Yang-Mills connections

TL;DR

This work proves nonlinear asymptotic stability of explicit self-similar blowup solutions for corotational wave maps and equivariant Yang-Mills connections in all energy-supercritical dimensions, extending stability beyond backward light cones into regions $\\Omega^{1,n}_T(\\kappa)$ with $-1\\le\\kappa<1$. Central to the approach is a general functional-analytic framework in adapted similarity coordinates, which yields a first-order autonomous system with a linear generator $\\mathbf{L}$ that splits into a one-dimensional unstable part (arising from time-translation symmetry) and a stable subspace. Through spectral analysis, Lyapunov-Perron theory, and sharp nonlinear estimates in radial Sobolev spaces, the authors construct stabilised evolutions that converge to the self-similar blowup profile with quantitative decay rates, while finite speed of propagation allows extension to the full Cauchy problem. The results provide a unified, cross-dimensional stability mechanism for large data near explicit self-similar blowups and establish a robust framework applicable to both wave maps and Yang-Mills in arbitrary energy-supercritical settings. This advances the understanding of universal blowup dynamics and offers precise, region-wide stability guarantees with potential implications for related geometric wave equations.

Abstract

We consider corotational wave maps from Minkowski spacetime into the sphere and the equivariant Yang-Mills equation for all energy-supercritical dimensions. Both models have explicit self-similar finite time blowup solutions, which continue to exist even past the singularity. We prove the nonlinear asymptotic stability of these solutions in spacetime regions that approach the future light cone of the singularity. For this, we develop a general functional analytic framework in adapted similarity coordinates that allows to evolve the stable wave flow near a self-similar blowup solution in such spacetime regions.

Figures (2)

• Figure 1: Sketch of similarity coordinates centered around $(T,0) \in \mathbb{R}^{1,n}$ for $n=1$. The gray-shaded part depicts $\Omega^{1,n}_{T}(\kappa)$, which exhausts the exterior of the future light cone of $(T,0) \in \mathbb{R}^{1,n}$ as $\kappa \to 1$. The region covered by the coordinate grid is the image region $\mathrm{X}^{1,n}_{T}(\kappa)$. Here, the spacelike curves correspond to the hypersurfaces $\Sigma^{1,n}_{T}(t)$ whose flat part is determined by the parameter $r>0$ in condition \ref{['h1']}. The dotted region shows $\Lambda^{1,n}_{T}(\kappa)$.
• Figure 2: Spacetime regions involved in the construction of the solution. The gray region illustrates the image region $\mathrm{X}^{1,d}_{T}(\kappa)$ of similarity coordinates from \ref{['SimilarityCoordinatesFlat']}. This is where the stable solution $u$ has been constructed from initial data prescribed along the initial hypersurface $\Sigma^{1,d}_{T}(0)$. The zigzag line marks the support of the perturbations $f,g$ in the initial data. In the dotted region $\mathrm{X}^{1,d}_{T} \cap \Lambda^{1,d}_{T}(r)$, \ref{['FiniteSpeedOfPropagation']} yields that $\psi^{\ast}_{1}$ is the unique solution.

Theorems & Definitions (114)

• Definition 1.1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.1: Stable blowup for corotational wave maps
• Theorem 1.2: Stable blowup for equivariant Yang-Mills connections
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7