Аннотация:We consider a class of the mean field game (MFG) equations, which have
a structure of the equations of gas dynamics. They can be characterized in terms
of mass, momentum and energy. We study the integral quantities related to the
parabolic and hyperbolic version of the MFG equations and show that the balance
of the energy in the parabolic version is closely connected with the formation of
singularities of solutions to the hyperbolic version.