Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations

TL;DR

This paper proves Sobolev stability of steady Prandtl-type shear flows in the Navier-Stokes equations, demonstrating a significant advancement in understanding the inviscid limit in Sobolev spaces for these flows.

Contribution

It establishes the first Sobolev stability result for steady Navier-Stokes shear flows of Prandtl type, contrasting with unsteady cases limited to Gevrey stability.

Findings

Proves $H^1$ stability of shear flows in steady Navier-Stokes.

Shows stability results differ from unsteady flow behavior.

Provides positive resolution to the inviscid limit problem in Sobolev spaces.

Abstract

We show the $H^1$ stability of shear flows of Prandtl type: $U^\nu = (U_s(y/\sqrt{\nu}),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.