  We prove that the 2d Euler equation is globally well-posed in a space of
vector fields having spatial asymptotic expansion at infinity of any a priori
given order. The asymptotic coefficients of the solutions are holomorphic
functions of $t$, do not involve (spacial) logarithmic terms, and develop even
when the initial data has fast decay at infinity. We discuss the evolution in
time of the asymptotic terms and their approximation properties.
