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

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 $U_s^{\prime }(0)>0$. 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.

Version 2: typos and bugs fixed. Some linear estimates are modified. Bibliography updated

References in corpus (2)

• Gevrey Stability of Prandtl Expansions for 2D Navier-Stokes
• Formal Derivation and Stability Analysis of Boundary Layer Models in MHD

Cited by in corpus (4)

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• Prandtl-Batchelor flows on a disk
• Stability of the boundary layer expansion for the 3D plane parallel MHD flow
• Global-in-$x$ stability of Prandtl layer expansions for steady magnetohydrodynamics flows over a moving plate