Within the classical Eulerian theory of gravitational instability, and in an ) \ 1 universe, we introduce a method for obtaining iterative solutions for the density contrast and peculiar velocity that can be extended to any order of approximation in the quasilinear regime. We compute the solution up to the fourth order (E4) and establish a standard criterion on the validity of Eulerian perturbative studies. The accuracy of E1ÈE4 is analyzed by using a power-law spherical initial proÐle for the

more »

... verage density contrast. Within the standard validity range, predictions of E4 generally have an accuracy of 95% or better. Concordance between perturbative predictions and the exact values depends on the index of the initial proÐle, the nature of the structure (cluster or void), and the magnitude under study (density contrast, d, or relative deviation from the Hubble Ñow, g). From a global point of view, the E2 approximation describes the Ðnal status of clusters quite well (with a relative error of under 10%), while it is poorer in studying the inner parts of voids. So E2 may be considered as a sufficiently good approximation for the study of the evolution of the halo of overdense regions. We also compare the Eulerian approximation with the two main Lagrangian approximations (L1 \ Zeldovich and L2) and show that, in a quasilinear regime and within the standard valid- ity range, E2 is better than L2. There is one exception to this general behavior, for instance, the contrast density for clusters with a steep initial density proÐle or voids with a smooth one. The Zeldovich approximation is particularly inefficient at tracing the evolution of peculiar motions (E1 is clearly better). This L1 approximation leads to an artiÐcial behavior of the inner regions of voids, in disagreement with previous work based on a homogeneous void (top-hat spherical underdensity). Our results warn of the risks of a systematic and indiscriminate use of Lagrangian approximations in the study of the large-scale structures evolution.