On the stability of shear flows of Prandtl type for the steady Navier-Stokes equations

TL;DR

This paper proves the stability of Prandtl-type shear flows in steady Navier-Stokes equations using a spectral assumption and a novel combination of energy and compact methods to analyze the Orr-Sommerfeld equation.

Contribution

It introduces a new stability proof for shear flows of Prandtl type under spectral assumptions, combining energy and compact methods for the Orr-Sommerfeld equation.

Findings

Stability of shear flows of Prandtl type established.

Development of a combined energy and compact method.

Application of spectral assumptions to Navier-Stokes stability.

Abstract

In this paper, we prove the stability of shear flows of Prandtl type as $ \big(U(y/\sqrt{\nu}),0\big)$ for the steady Navier-Stokes equations under a natural spectral assumption on the linearized NS operator. We develop a direct energy method combined with compact method to solve the Orr-Sommerfeld equation.