Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
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2015, Springer Basel. The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: >0, corresponding to the elastic response, and $${ u > 0}$$>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 $${alpha, u o 0}$$,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- model ($${ u = 0}$$=0), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case ( =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 $${ u = mathcal{O}(alpha^2)}$$=O(2), as 0, extending the main result in (Lopes Filho etal., Physica D 292(293):5161, 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 $${ u = mathcal{O}(alpha^{6/5})}$$=O(6/5), $${ u/alpha^{2} o infty}$$/2 as 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 Katos classical criterion to the second-grade fluid model, valid if $${alpha = mathcal{O}( u^{3/2})}$$=O(3/2), as $${ u o 0}$$0. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.
