Existence and stability of discretely self-similar blowup for a wave maps type equation
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2025
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| author | Glogić, Irfan Hilditch, David Wallauch, David |
| author_facet | Glogić, Irfan Hilditch, David Wallauch, David |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and stability of discretely self-similar blowup for a wave maps type equation Glogić, Irfan Hilditch, David Wallauch, David Analysis of PDEs Mathematical Physics Differential Geometry 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. |
| title | Existence and stability of discretely self-similar blowup for a wave maps type equation |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2512.16623 |