# A branched transport limit of the Ginzburg-Landau functional

## TL;DR

This paper rigorously derives a simplified branched transportation model from the Ginzburg-Landau functional for type-I superconductors under small magnetic fields, revealing multi-scale flux patterns.

## Contribution

It introduces a rigorous derivation of a branched transport limit from the Ginzburg-Landau model using $ ext{Gamma}$-convergence, clarifying the multi-scale structure of flux patterns.

## Key findings

- Flux patterns are described by a simplified branched transportation functional.
- The derivation is rigorous and based on $ ext{Gamma}$-convergence.
- The analysis reveals the multi-scale nature of the flux patterns.

## Abstract

We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $\Gamma$-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.