Global Existence for the Seiberg-Witten Flow

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in $C^{\mathrm{\infty }}$ up to gauge to a critical point of the Seiberg-Witten functional.