Energy dissipation and self-similar solutions for an unforced inviscid   dyadic model
  A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved and decay of energy like $t^{-2}$ is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions. 
D. Barbato, F. Flandoli, F. Morandin