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 $${\nu > 0}$$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 $${\alpha, \nu \to 0}$$α,ν→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 ($${\nu = 0}$$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 $${\nu = \mathcal{O}(\alpha^2)}$$ν=O(α2), as α → 0, extending the main result in (Lopes Filho et al., Physica D 292(293):51–61, 2015). 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 $${\nu = \mathcal{O}(\alpha^{6/5})}$$ν=O(α6/5), $${\nu/\alpha^{2} \to \infty}$$ν/α2→∞ as α → 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 $${\alpha = \mathcal{O}(\nu^{3/2})}$$α=O(ν3/2), as $${\nu \to 0}$$ν→0. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato