Stability and related zero viscosity limit of steady plane Poiseuille-Couette flows with no-slip boundary condtion

TL;DR

This paper proves the stability of steady plane Poiseuille-Couette flows and establishes the zero viscosity limit of 2D steady Navier-Stokes solutions to Euler equations, including cases with external forces.

Contribution

It demonstrates the existence and stability of smooth solutions near Poiseuille-Couette flow and the zero viscosity limit for these solutions, extending previous results to include external force control.

Findings

Stable smooth solutions near Poiseuille-Couette flow

Zero viscosity limit from Navier-Stokes to Euler equations

Stability under small perturbations and external forces

Abstract

We prove the existence and stability of smooth solutions to the steady Navier-Stokes equations near plane Poiseuille-Couette flow. Consequently, we also provide the zero viscosity limit of the 2D steady Navier-Stokes equations to the steady Euler equations. First, in the absence of any external force, we prove that there exist smooth solutions to the steady Navier-Stokes equations which are stable under infinitesimal perturbations of plane Poiseuille-Couette flow. In particular, if the basic flow is the Couette flow, then we can prove that the flow is stable for any finite perturbation small enough. Moreover, we also show that, if we put a proper external force to control the flow, then we can also obtain a large class of smooth solution of the steady Navier-Stokes equations which is stable for infinitesimal perturbation of the external force. Finally, based on the same linear estimates, we establish the zero viscosity limit of all the solutions obtained above to the solutions of the Euler equations.