Spatial asymptotic expansions in the incompressible Euler equation

In this paper we prove that the Euler equation describing the motion of an ideal fluid in ${\mathbb{R}}^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.