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

TL;DR

This paper studies the Helmholtz-Weyl decomposition on moving domains and applies it to construct time-periodic solutions for the Navier-Stokes equations with large fluxes.

Contribution

It analyzes the domain dependence of Helmholtz-Weyl components and constructs time-periodic Navier-Stokes solutions with large boundary fluxes.

Findings

Decomposition components depend continuously on domain movement.

Constructed time-periodic solutions with large boundary fluxes.

Extended Helmholtz-Weyl theory to time-dependent domains.

Abstract

We consider the Helmholtz-Weyl decomposition on a time dependent bounded domain $\Omega(t)$ in $\mathbb{R}^3$. Especially, we investigate the domain dependence of each component in the decomposition, namely, the harmonic vector fields (i.e., $\mathrm{div}$ and $\mathrm{rot}$ free vectors), vector potentials, and scalar potentials equipped with suitable boundary conditions, when $\Omega(t)$ 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.