  We study some partially de Rham representations of $\mathrm{Gal}(\bar{L}/L)$
for a finite unramified extension $L$ of $\mathbb{Q}_p$. We study some related
subspaces of Galois cohomology and of cohomology of $B$-pairs. As an
application, we associate a locally $\mathbb{Q}_p$-analytic representation of
$\mathrm{GL}_2(L)$ to a $2$-dimensional trianguline representation of
$\mathrm{Gal}(\bar{L}/L)$, by generalizing Breuil's theory in the crystalline
case.
