  We study a class of McKean-Vlasov type stochastic differential equations
(SDEs) which arise from the random vortex dynamics and other physics models. By
introducing a new approach we resolve the existence and uniqueness of both the
weak and strong solutions for the McKean-Vlasov stochastic differential
equations whose coefficients are defined in terms of singular integral kernels
such as the Biot-Savart kernel. These SDEs which involve the distributions of
solutions are in general not Lipschitz continuous with respect to the usual
distances on the space of distributions such as the Wasserstein distance.
Therefore there is an obstacle in adapting the ordinary SDE method for the
study of this class of SDEs, and the conventional methods seem not appropriate
for dealing with such distributional SDEs which appear in applications such as
fluid mechanics.
