The main purpose of this paper is to study the stability and
error estimates of the local discontinuous Galerkin (LDG) methods
coupled with multi-step implicit-explicit (IMEX) time
discretization schemes,
for solving time-dependent incompressible fluid flows.
We will give theoretical analysis for the Oseen equation,
and assess the performance of the schemes for
incompressible Navier-Stokes equations numerically. For the
Oseen equation, using first order IMEX time discretization
as an example,
we show that the IMEX-LDG scheme is unconditionally stable
for $\Qcal_k$ elements on cartesian meshes, in the sense that
the time-step $\dt$ is only
required to be {bounded from above}
by a positive constant independent
of the spatial mesh size $h$.
Furthermore, by the aid of the \emph{Stokes
projection} and an elaborate energy analysis, we
obtain the $L^{\infty}(L^2)$ optimal error estimates for both
the velocity and
the stress (gradient of velocity), in both space and time.
By the \emph{inf-sup} argument, we also obtain
the $L^{\infty}(L^2)$ optimal error estimates
for the pressure. Numerical experiments are given to
validate our main results.