Globally stable blowup profile for supercritical wave maps in all dimensions

TL;DR

This paper proves the global nonlinear stability of a known self-similar blowup solution for supercritical wave maps from Minkowski space to spheres in all dimensions d ≥ 3, confirming a conjecture about generic blowup behavior.

Contribution

It introduces a new stability analysis method using similarity variables on the entire space, applicable to all dimensions d ≥ 3, for wave map blowup profiles.

Findings

Confirmed global stability of the blowup profile for all d ≥ 3

Developed a novel similarity variable stability analysis approach

Provided a general framework for stability analysis of self-similar blowup in nonlinear wave equations

Abstract

We consider wave maps from the $(1+d)$-dimensional Minkowski space into the $d$-sphere. It is known from the work of Bizo\'n and Biernat \cite{BizBie15} that in the energy-supercritical case, i.e., for $d \geq 3$, this model admits a closed-form corotational self-similar blowup solution. We show that this blowup profile is globally nonlinearly stable for all $d \geq 3$, thereby verifying a perturbative version of the conjecture posed in \cite{BizBie15} about the generic large data blowup behavior for this model. To accomplish this, we develop a novel stability analysis approach based on similarity variables posed on the whole space $\mathbb{R}^d$. As a result, we draw a general road map for studying spatially global stability of self-similar blowup profiles for nonlinear wave equations in the radial case for arbitrary dimension $d \geq 3$.