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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2603.24900 |
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| _version_ | 1866915896351522816 |
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| author | Zhao, Jason |
| author_facet | Zhao, Jason |
| contents | Given an initial data configuration $(A^{\mathrm{in}}, ϕ^{\mathrm{in}})$ on $\mathbb R^2$ such that the self-dual abelian Higgs energy is near the minimum energy within its topological class, we prove that its evolution under the self-dual abelian Higgs gradient flow in temporal gauge converges exponentially as $t \to \infty$ with respect to the $(H^1 \times L^2)$-metric to a minimiser of the energy. Furthermore, we show that the convergence of the scalar field $ϕ$ may be upgraded to the $H^1$-metric provided the additional assumption on the potential that $A^{\mathrm{in}} \in L^p (\mathbb R^2)$ for $2 < p < \infty$. As a corollary, we obtain a quantitative stability for the self-dual abelian Higgs energy which improves upon the previous result of Halavati (arXiv:2310.04866) and partially resolves the open problem posed in his article. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of the self-dual abelian Higgs gradient flow Zhao, Jason Analysis of PDEs Given an initial data configuration $(A^{\mathrm{in}}, ϕ^{\mathrm{in}})$ on $\mathbb R^2$ such that the self-dual abelian Higgs energy is near the minimum energy within its topological class, we prove that its evolution under the self-dual abelian Higgs gradient flow in temporal gauge converges exponentially as $t \to \infty$ with respect to the $(H^1 \times L^2)$-metric to a minimiser of the energy. Furthermore, we show that the convergence of the scalar field $ϕ$ may be upgraded to the $H^1$-metric provided the additional assumption on the potential that $A^{\mathrm{in}} \in L^p (\mathbb R^2)$ for $2 < p < \infty$. As a corollary, we obtain a quantitative stability for the self-dual abelian Higgs energy which improves upon the previous result of Halavati (arXiv:2310.04866) and partially resolves the open problem posed in his article. |
| title | Convergence of the self-dual abelian Higgs gradient flow |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.24900 |