Navier-Stokes equations on the flat cylinder with vorticity production on the boundary

TL;DR

This paper analyzes the two-dimensional Navier-Stokes equations on a flat cylinder with boundary vorticity production, demonstrating exponential decay of Fourier modes in one direction and power decay in the other.

Contribution

It formulates the problem as an infinite system of ODEs for Fourier components and proves equivalence to the original system, highlighting decay properties of solutions.

Findings

Fourier modes decay exponentially in the periodic direction.

Fourier modes decay polynomially in the non-periodic direction.

Boundary vorticity production is incorporated into the Fourier framework.

Abstract

We study the two-dimensional Navier-Stokes system on a flat cylinder with the usual Dirichlet boundary conditions for the velocity field u. We formulate the problem as an infinite system of ODE's for the natural Fourier components of the vorticity, and the boundary conditions are taken into account by adding a vorticity production at the boundary. We prove equivalence to the original Navier-Stokes system and show that the decay of the Fourier modes is exponential for any positive time in the periodic direction, but it is only power-like in the other direction.