  A finite-state Markov chain is introduced in the noise terms of the
three-dimensional stochastic Navier-Stokes equations in order to allow for
transitions between two types of multiplicative noises. We call such systems as
stochastic Navier-Stokes equations with Markov switching. To solve such a
system, a family of regularized stochastic systems is introduced. For each such
regularized system, the existence of a unique strong solution (in the sense of
stochastic analysis) is established by the method of martingale problems and
pathwise uniqueness. The regularization is removed in the limit by obtaining a
weakly convergent sequence from the family of regularized solutions, and
identifying the limit as a solution of the three-dimensional stochastic
Navier-Stokes equation with Markov switching.
