On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions

TL;DR

This paper proves existence and uniqueness of weak solutions for a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions, supporting vorticity-based numerical methods.

Contribution

It establishes well-posedness results for a vorticity-velocity formulation with specific boundary conditions, advancing mathematical understanding for vorticity-based simulations.

Findings

Existence of weak solutions under small data conditions

Uniqueness of solutions in the stationary case

Supports development of vorticity-based numerical methods

Abstract

We study well-posedness of a velocity-vorticity formulation of the Navier--Stokes equations, supplemented with no-slip velocity boundary conditions, a no-penetration vorticity boundary condition, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier--Stokes equations.