In this talk I will present a recent result in collaboration with Thomas Alazard (CNRS-École Polytechnique). While it is well known that the Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data, we show that for a dense $G_\delta$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.