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 is well-posed in a class of functions allowing spatial asymptotic expansions as 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 rapidly decaying initial data develop non-trivial spatial asymptotic expansions of the type considered here.