On the Euler+Prandtl expansion for the Navier-Stokes equations

TL;DR

This paper proves the validity of the Euler plus Prandtl approximation for Navier-Stokes solutions in the half-plane with boundary conditions, in the vanishing viscosity limit, using an innovative approach that estimates vorticity errors without higher order correctors.

Contribution

It introduces a direct estimation method for the vorticity error in the Euler+Prandtl expansion, requiring only local analyticity near the boundary and Sobolev regularity away from it.

Findings

Validates Euler+Prandtl approximation in the vanishing viscosity limit.

Establishes a new method for error estimation using $L^{1}$-type vorticity norms.

Proves propagation of local analyticity for the Euler equation.

Abstract

We establish the validity of the Euler$+$Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an $L^{1}$-type norm for the vorticity of the error term in the expansion Navier-Stokes$-($Euler$+$Prandtl$)$. An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest.