Incompressible Euler as a limit of complex fluid models with Navier boundary conditions

In this article we study the limit α→0 of solutions of the α-Euler equations and the limit α,ν→0 of solutions of the second grade fluid equations in a bounded domain, both in two and in three space dimensions. We prove that solutions of the complex fluid models converge to solutions of the incompressible Euler equations in a bounded domain with Navier boundary conditions, under the hypothesis that there exists a uniform time of existence for the approximations, independent of α and ν. This additional hypothesis is not necessary in 2D, where global existence is known, and for axisymmetric flows without swirl, for which we prove global existence. Our conclusion is strong convergence in L2 to a solution of the incompressible Euler equations, assuming smooth initial data.