The initial condition  is assumed to be a smooth and divergence-free function (see smooth function) such that, for every multi-index  (see multi-index notation) and any , there exists a constant  such that
:  for all 
The external force  is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well):
:  for all 
For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as .  More precisely, the following assumptions are made:
# There exists a constant  such that  for all 
Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energy of the solution is globally bounded.