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Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions

The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: $α> 0$, corresponding to the elastic response, and $ν> 0$, corresponding to viscosity. Formally setting these parameters to $0$ reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits $α, ν\to 0$ of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-$α$ model ($ν= 0$), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case ($α= 0$), for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided $ν= \mathcal{O}(α^2)$, as $α\to 0$, extending the main result in [19]. Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $ν= \mathcal{O}(α^{6/5})$, $ν/α^2 \to \infty$ as $α\to 0$. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato's classical criterion to the second-grade fluid model, valid if $α= \mathcal{O}(ν^{3/2})$, as $ν\to 0$. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.