The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~I: Energy estimates and compactness results

We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\Q$-tensor for the liquid crystal component and a magnetisation vector field~$\M$, both of them governed by a Ginzburg-Landau-type energy. The energy, the largest part of which is carried by the $\Q$-component, includes a singular coupling term favouring alignment between~$\Q$ and~$\M$. In this article and in the companion paper~\cite{CDS2}, we analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\eps$ tends to zero. In this paper, we prove that the (rescaled) energy density for the $\Q$-component, concentrates, to leading order, on a finite number of singular points. Moreover, we prove energy estimates and compactness results that will be crucially used in~\cite{CDS2} to determine the structure of the energy concentration set for the $\M$-component as well as the relationship between the two singular sets.