  We show that a weak version of the canonical bundle formula holds for
fibrations of relative dimension one. We provide various applications thereof,
for instance, using the recent result of Xu and Zhang, we prove the log
non-vanishing conjecture for three-dimensional klt pairs over any algebraically
closed field $k$ of characteristic $p>5$. We also show the log abundance
conjecture for threefolds over $k$ when the nef dimension is not maximal, and
the base point free theorem for threefolds over the algebraic closure of any
finite field of characteristic $p>2$.
