On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet

TL;DR

This paper analyzes a linearized Stokes problem related to free liquid jet flow, demonstrating that the associated operator generates an analytic semigroup, which is crucial for proving local existence and uniqueness of solutions to the nonlinear Navier-Stokes problem.

Contribution

The paper introduces a sectorial operator framework for the linearized problem, enabling the use of Fourier methods to establish key estimates for the nonlinear analysis.

Findings

The Stokes operator is sectorial and generates an analytic semigroup.

Fourier methods provide estimates on solutions.

The results are essential for proving local existence and uniqueness.

Abstract

In this work, we analyze a Stokes problem arising in the study of the Navier-Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.