$mathcal{L}$-invariants, partially de Rham families and local-global compatibility

Let $F_{wp}$ be a finite extension of $mathbb{Q}_p$. By considering partially de Rham families, we establish a Colmez-Greenberg-Stevens formula (on Fontaine-Mazur $mathcal{L}$-invariants) for (general) $2$-dimensional semi-stable non-crystalline $mathrm{Gal}(overline{mathbb{Q}_p}/F_{wp})$-representations. As an application, we prove local-global compatibility results for completed cohomology of quaternion Shimura curves, and in particular the equality of Fontaine-Mazur $mathcal{L}$-invariants and Breuils $mathcal{L}$-invariants, in critical case.