Nonstationary flow for the Navier-Stokes equations in a cylindrical pipe

TL;DR

This paper proves the existence of solutions for nonstationary Navier-Stokes flow in a cylindrical pipe with slip boundary conditions, allowing for large inflow and initial velocity but requiring small derivatives of inflow, force, and initial velocity.

Contribution

It establishes existence results for nonstationary Navier-Stokes equations in cylindrical domains without smallness restrictions on inflow or initial velocity, only on derivatives.

Findings

Existence of solutions in $W^{2,1}_2$ space.

No smallness restrictions on inflow or initial velocity.

Smallness condition on derivatives of inflow, force, and initial velocity.

Abstract

In cylindrical domain, we consider the nonstationary flow with prescribed inflow and outflow, modelled with Navier-Stokes equations under the slip boundary conditions. Using smallness of some derivatives of inflow function, external force and initial velocity of the flow, but with no smallness restrictions on the inflow, initial velocity neither force, we prove existence of solutions in $W^{2,1}_2.$