Taylor-Green vortex (1937) is a simple perioid initial value problem for the 3D Euler equation:
which leads to a complicated turbulent flow, and has become a standard example in fluid dynamics.
Taylor and Green conjectured that their vortex would develop a finite-time blowup. However, numerical simulations didn’t show a very high increase of maximal vorticity in time. (Only a factor of 4 ?). Nevertheless, the dynamics of the Taylor-Green vortex seems to be extremely complicated.
Some pictures of the Taylor-Green vortex from the internet:
The vortex after some time: