We study the equations governing the motion of second grade fluids in a bounded domain of R d , d = 2, 3, with Navier-slip boundary conditions with and without viscosity (averaged Euler equations). We show global existence and uniqueness of H 3 solutions in dimension two. In dimension three, we obtain local existence of H 3 solutions for arbitrary initial data and global existence for small initial data and positive viscosity. We close by finding Liapunov stability conditions for stationary solutions for the averaged Euler equations similar to the RayleighArnold stability result for the classical Euler equations.

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International audienceIn 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