Hydrodynamical models are descriptions of fluids based on transport equations for macroscopic quantities such as flow field, temperature, density, etc. Two distinct terms are found in these equations: advection terms, which describe transport by the fluid flow itself and are reversible and independent on the nature of the fluid, and diffusion terms, which describe the irreversible transport due to disordered molecular motion.
In many interesting cases, after non-dimensionalizing the equations, one finds that diffusion coefficients are extremely small. For example, the momentum diffusion coefficient which applies when pouring a glass of water is about 10^{-5}! When dealing with such cases, although it is very tempting to omit diffusion terms, we know that the water quickly comes to rest, its momentum being irreversibly dissipated. How could this process be described without dissipative terms in the equations??
We will show that asking this simple question leads to the mathematical concept of a dissipative weak solution, and to some of the most difficult open problems in mathematical fluid dynamics.