Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X\_{\eta}$ (say) is smooth projective variety and the special fibre $\mathcal X\_0$ (say) is a normal crossing divisor in $\mathcal X$. 


Question: Does there exist a vector bundle $\mathcal E$ over $\mathcal X$ such that $\mathcal E|\_{\_{\mathcal X\_{\eta}}}\cong \Omega^1\_{\_{\mathcal X\_{\eta}}}$ and $\mathcal E|\_{\_{\mathcal X\_0}}\cong \Omega^1\_{\_{\mathcal X\_0}}(\mathrm{Log} D)$, where $D$ is the singular locus of $\mathcal X\_0$?


It will be very helpful if anyone can explain this construction or provide a reference. Thank you.. 



Relative logarithmic cotangent bundle
First of all, it's unclear what you mean by $\Omega^1\_{X\_0}(\log D)$ since $X\_0$ is singular.


Second, if you make up such a definition then most probably such a vector bundle will not exist. Note that the Euler characteristics $\chi(X\_\eta, \Omega^1\_\eta)$ and $\chi(X\_0, \Omega^1\_{X\_0}(\log D)$ have to agree. I suggest checking this for an elliptic curve degenerating to a node. 


Finally, what people usually consider in your situation is the vector bundle $\Omega^1\_{X/S}(\log X\_0)$, so differentials on $X$ with log poles along the components of $X\_0$, divided by the pull-back $t^{-1}dt$ where $t$ is a uniformizer of $S$. This is indeed locally free and restricts to $\Omega^1\_{X\_\eta}$. 


