Asymptotic analysis of an $\varepsilon$-Stokes problem connecting Stokes and pressure-Poisson problems

TL;DR

This paper introduces an $ ext{epsilon}$-Stokes problem that bridges the Stokes and pressure-Poisson equations, analyzing how solutions behave as the parameter varies, providing insights into their asymptotic relationships.

Contribution

It formulates an $ ext{epsilon}$-Stokes problem linking Stokes and pressure-Poisson problems and proves solution convergence as the parameter approaches 0 or infinity.

Findings

Solution converges to Stokes problem as $ ext{epsilon} o 0$

Solution converges to pressure-Poisson problem as $ ext{epsilon} o \infty$

Provides a unified framework connecting two fundamental fluid dynamics problems.

Abstract

In this Note, we prepare an $\varepsilon$-Stokes problem connecting the Stokes problem and the corresponding pressure-Poisson equation using one parameter $\varepsilon>0$. We prove that the solution to the $\varepsilon$-Stokes problem, convergences as $\varepsilon$ tends to 0 or $\infty$ to the Stokes and pressure-Poisson problem, respectively.