Helmholtz-Weyl decomposition on a time dependent domain for time periodic Navier-Stokes flows with large flux

We consider the Helmholtz-Weyl decomposition on a time dependent domain $\Omega(t)$ in $\mathbb{R}^3$ motivated by the time periodic problem of the incompressible Navier-Stokes equations. Especially, we investigate the domain dependence of the harmonic vector fields and vector potentials in the decomposition, i.e., the time continuity and differentiability of them are established. As an application, we construct a time periodic solution with the large boundary data.