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 bounded domain $\mathrm{\Omega }(t)$ in ${\mathbb{R}}^3$. Especially, we investigate the domain dependence of each component in the decomposition, namely, the harmonic vector fields (i.e., $\mathrm{d}\mathrm{i}\mathrm{v}$ and $\mathrm{r}\mathrm{o}\mathrm{t}$ free vectors), vector potentials, and scalar potentials equipped with suitable boundary conditions, when $\mathrm{\Omega }(t)$ moves along to $t\in \mathbb{R}$. As an application, we construct a time periodic solution of the incompressible Navier-Stokes equations for some large boundary data with non-zero fluxes.