Axisymmetric solutions to coupled Navier-Stokes/Allen-Cahn equations?

We investigate a family of axisymmetric solutions to a coupling of Navier-Stokes and Allen-Cahn equations in R³. First, a one-dimensional system of equations is derived from the method of separation of variables, which approximates the three-dimensional system along its symmetry axis. Then based on them, by adding perturbation terms, we construct finite energy solutions to the three-dimensional system. We prove the global regularity of the constructed solutions in both large viscosity and small initial data cases. These solutions can be considered as perturbations near infinite-energy solutions.