  We study the curves of coherence for the Bell-diagonal states including
$l_{1}$-norm of coherence and relative entropy of coherence under the Markovian
channels in the first subsystem once. For a special Bell-diagonal state under
bit-phase flip channel, we find frozen coherence under $l_{1}$ norm occurs, but
relative entropy of coherence decrease. It illustrates that the occurrence of
frozen coherence depends on the type of the measure of coherence. We study the
coherence evolution of Bell-diagonal states under Markovian channels in the
first subsystem $n$ times and find coherence under depolarizing channel
decreases initially then increases for small $n$ and tend to zero for large
$n$. We discuss the dynamics of coherence of the Bell-diagonal state under two
independent same type local Markovian channels. We depict the dynamic behaviors
of relative entropy of coherence for Bell-diagonal state under the bi-side
different Markovian channel. We depict the dynamic behaviors of relative
entropy of coherence for Bell-diagonal state under the bi-side different
Markovian channel.
