In this paper, we investigate a Stokes-Magneto system with fractional diffusions. We first deal with the non-resistive case in \({\mathbb {T}}^{d}\) and establish the local and global well-posedness with initial magnetic field \({\varvec{b}}_0\in H^{s}({\mathbb {T}}^d).\) We also show the existence of a unique mild solution of the resistive case with initial data \({\varvec{b}}_0\) in the critical \(L^{p}({\mathbb {R}}^d)\) space. Moreover, we show that \(\Vert {\varvec{b}}(t)\Vert _{L^{p}}\) converges to zero as \(t\rightarrow \infty \) when the initial data is sufficiently small.
