Uniform estimates of Landau-de Gennes minimizers in the vanishing elasticity limit with line defects

For the Landau-de Gennes functional modeling nematic liquid crystals in dimension three, we prove that, if the energy is bounded by $C(\log \frac{1}{\epsilon }+1)$, then the sequence of minimizers $\{{\mathbf{Q}}_{\epsilon }{\}}_{\epsilon \in (0,1)}$ is relatively compact in $W_{\mathrm{loc}}^{}$ for every $1<p<2$. This extends the classical compactness theorem of Bourgain-Brézis-Mironescu [Publ. Math., IHÉS, 99:1-115, 2004] for complex Ginzburg-Landau minimizers to the $\mathbb{R}{\mathbf{P}}^2$-valued Landau-de Gennes setting. Moreover, We obtain local bounds on the integral of the bulk energy potential that are uniform in $\epsilon$, improving the estimate that follows directly from the assumption.