The Navier-Stokes equations with the Neumann boundary condition in an infinite cylinder

TL;DR

This paper proves the local-in-time existence and uniqueness of smooth solutions to the Navier-Stokes equations with Neumann boundary conditions in an infinite cylindrical domain for initial data in certain Lebesgue spaces.

Contribution

It establishes the first rigorous proof of well-posedness for Navier-Stokes with Neumann boundary conditions in an infinite cylinder.

Findings

Unique local-in-time smooth solutions exist for initial data in L^p, p in [3, ∞).

The solutions are unique and depend smoothly on initial data.

The results extend the understanding of boundary conditions in fluid dynamics.

Abstract

We prove unique existence of local-in-time smooth solutions of the Navier-Stokes equations for initial data in $L^{p}$ and $p \in [3, \infty)$ in an infinite cylinder, subject to the Neumann boundary condition.