On energy conservation in fluid dynamics

Conserved quantities, like the energy, are at the heart  
of the study of many evolutionary PDEs. We will discuss the problem of  
energy conservation for some equations of fluid mechanics - starting  
with the celebrated Onsager's conjecture in the realm of ideal  
incompressible fluids, and moving on to more recent results. In  
particular we consider the compressible Euler equations with vacuum.  
Time permitting, we will discuss some results concerning a general  
class of conservation laws.