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

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