Existence and stability of discretely self-similar blowup for a wave maps type equation

TL;DR

This work rigorously identifies discretely self-similar blowup for a wave maps-type equation from Minkowski space to S^1, proving the nonexistence of continuously self-similar blowup and constructing a countable family of DSS profiles with closed-form structure. A central contribution is a comprehensive linear stability analysis carried out in similarity variables, where resolvent constructions via Liouville–Green transforms and Volterra iterations yield sharp semigroup bounds, uniformly in dimension $d\ge2$ and for all $n$. The nonlinear theory then uses a Lyapunov–Perron-type fixed-point argument together with projections away from unstable modes to establish nonlinear stability of all DSS profiles; precise co-dimension of instability matches the number of unstable eigenvalues. In addition, a separate 1D analysis reveals the existence of a stable blowup mechanism in a non-compact target setting, and the results provide a dimension-independent mechanism for DSS blowup in a geometric wave equation, bridging rigorous analysis with phenomena long observed numerically in physics.

Abstract

We study finite-time blowup for a nonlinear wave equation for maps from the Minkowski space $\mathbb{R}^{1+d}$ into the 1-sphere $\mathbb{S}^1$, whose nonlinearity exhibits a null-form structure. We construct, for every dimension $d \geq 1$, a countable family of discretely self-similar blowup solutions, which are even for $d=1$ and radial for $d \geq 2$. The main contribution of the paper is a detailed nonlinear stability analysis of this family of solutions. For $d \geq 2$, we consider radial data, while in $d=1$ we allow for general perturbations. After linearizing around the self-similar profiles in similarity variables, we construct resolvents of the resulting highly non-self-adjoint operators through Liouville-Green transformations and precise Volterra-type asymptotics. The construction itself, which occupies most of the paper, is technically challenging, as it is performed in arbitrary dimensions and for a countable family of operators in each. Combined with a detailed spectral analysis of the linearized operators, this yields sharp semigroup bounds and allows us to establish nonlinear stability of all discretely self-similar profiles in all dimensions, with precise co-dimension determined by the unstable spectrum. To our knowledge, this is the first result on the existence and stability of discretely self-similar blowup for a geometric wave equation.

Theorems & Definitions (117)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10