  In this paper we mainly investigate the strong and weak well-posedness of a
class of McKean-Vlasov stochastic (partial) differential equations. The main
existence and uniqueness results state that we only need to impose some local
assumptions on the coefficients, i.e. locally monotone condition both in state
variable and distribution variable, which cause some essential difficulty since
the coefficients of McKean-Vlasov stochastic equations typically are nonlocal.
Furthermore, the large deviation principle is also derived for the
McKean-Vlasov stochastic equations under those weak assumptions. The wide
applications of main results are illustrated by various concrete examples such
as the granular media equations, plasma type models, kinetic equations,
McKean-Vlasov type porous media equations and Navier-Stokes equations. In
particular, we could remove or relax some typical assumptions previously
imposed on those models.
