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Convergence of the 2D Euler-\(\alpha\) to Euler equations in the Dirichlet case: indifference to boundary layers. (English) Zbl 1364.35277

Summary: 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q31 Euler equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35A35 Theoretical approximation in context of PDEs
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