  In this paper we propose a quaternion formulation for the orientation
variable in the three dimensional Kobayashi--Warren model for the dynamics of
polycrystals. We obtain existence of solutions to the $L^2$-gradient descent
flow of the constrained energy functional via several approximating problems.
In particular, we use a Ginzburg-Landau type approach and some extra
regularizations. Existence of solutions to the approximating problems is shown
by the use of nonlinear semigroups. Coupled with good a-priori estimates, this
leads to successive passages to the limit up to finally showing existence of
solutions to the proposed model. Moreover, we also obtain a maximum principle
for the orientation variable.
