In this talk I will show 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 $|x|\to\infty$ 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 rapidly decaying initial data develop non-trivial spatial asymptotic expansions of the type considered here. This is a joint work with R.McOwen.