Convergence of the self-dual abelian Higgs gradient flow

Given an initial data configuration $(A^{\mathrm{in}}, \phi^{\mathrm{in}})$ 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 $\phi$ 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.