It is well known that the 2d incompressible Navier-Stokes equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions. 


I tried searching for a proof that the 2d incompressible Euler equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions, but couldn't find one. Is this still an open problem? 



2d incompressible Euler equations under periodic boundary conditions
A famous result of Beale, Kato and Majda shows that the maximum norm of the vorticity has to become infinite if smooth solutions do not exist globally. This is true even in three dimensions. In two dimensions, the $L^\infty$ norm of the vorticity is conserved, so global existence follows immediately.


