Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: indifference to boundary layers

Abstract

In this article we consider the Euler-$\alpha$ system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler-$\alpha$ regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler-$\alpha$ system approximate, in a suitable sense, as the regularization parameter $\alpha \to 0$, the initial velocity for the limiting Euler system. For small values of $\alpha$, this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler-$\alpha$ system converge, as $\alpha \to 0$, to the corresponding solution of the Euler equations, in $L^2$ in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the $\alpha \to 0$ limit, which underlies our work.

arXiv comment22pages

References in corpus (12)

• The Camassa-Holm equations as a closure model for turbulent channel flow
• The Navier-Stokes-alpha model of fluid turbulence
• A connection between the Camassa-Holm equations and turbulent flows in channels and pipes
• Acceleration and vortex filaments in turbulence
• On the Clark-alpha model of turbulence: global regularity and long--time dynamics
• Mathematics and Turbulence: where do we stand?
• Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows
• The Three Dimensional Viscous Camassa-Holm Equations, and Their Relation to the Navier-Stokes Equations and Turbulence Theory
• On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Boundary layers and the vanishing viscosity limit for incompressible 2D flow
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

Cited by in corpus (11)

• Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
• Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations
• Weak solutions for the $\alpha$-Euler equations and convergence to Euler
• Geometric Lagrangian averaged Euler-Boussinesq and primitive equations
• Inviscid Limit for Stochastic Second-Grade Fluid Equations
• Global Well-Posedness of 2D Euler-$\alpha$ Equations in Exterior Domain
• Stochastic model error in the LANS-alpha and NS-alpha deconvolution models of turbulence
• Singular limit of 2D second grade fluid past an obstacle
• The limit $\alpha \to 0$ of the $\alpha$-Euler equations in the half plane with no-slip boundary conditions and vortex sheet initial data
• Stability criteria for the 2D $\alpha$-Euler equations
• Uniform time of existence for the alpha Euler equations