Weak solutions for theα-Euler equations and convergence to Euler

Abstract

We consider the limit α → 0 for the α-Euler equations in a two-dimensional bounded domain with Dirichlet boundary conditions. Assuming that the vorticity is bounded in L p , we prove the existence of a global solution and we show the convergence towards a solution of the incompressible Euler equation with L p vorticity. The domain can be multiply-connected. We also discuss the case of the second grade fluid when both α and ν go to 0.

The limit $α\to 0$ of the $α$-Euler equations in the half plane with no-slip boundary conditions and vortex sheet initial data

“…In this article we study the limit when α → 0 of solutions to the α-Euler system in the half-plane, with no-slip boundary conditions, to weak solutions of the 2D incompressible Euler equations with non-negative initial vorticity in the space of bounded Radon measures in H −1 . This result extends the analysis done in [4,13]. It requires a substantially distinct approach, analogous to that used for Delort's Theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane.…”

“…This article concerns the limit α → 0 of the α-Euler equations in the half-plane, with no-slip boundary conditions, with initial velocity in L 2 and initial vorticity whose singular part is a nonnegative bounded Radon measure. The present work is a natural continuation of research contained in [4,13], where the respective authors proved convergence, first for initial velocity in H 3 , see [13] and then for initial vorticity in L p , p > 1, see [4], both for flows in bounded, smooth domains. The extension to initial vorticities in the space of Radon measures requires a substantial change in technique.…”

“…In this work we have discussed only the case of flow in the half-plane. It would be interesting to study the α → 0 limit in a smooth, bounded domain, with vortex sheet initial data, thereby complementing and extending the results in [13,4] to less regular initial data.…”

“…The extension to initial vorticities in the space of Radon measures requires a substantial change in technique. The previous results are based on energy estimates and boundary correctors [13] or on the compactness of the velocity sequences obtained from boundedness of the corresponding vorticity in a suitable space [4]. For the present work, a compensated compactness argument is required, involving a subtle cancellation property of the nonlinearity, in the spirit of Delort's celebrated existence result, see [8].…”

Non-conservative solutions of the Euler-$α$ equations

“…Note that by setting α = 0 we formally recover the Euler equations after redefining the pressure. In the case of Euclidean space, the convergence of smooth Euler-α solutions to Euler solutions was demonstrated in [33] (see also [12][13][14]34] for progress in the case of domains with boundary).…”