  Note that some classic fluid dynamical systems such as the Navier-Stokes
equations, Magnetohydrodynamics (MHD), Boussinesq equations, and etc are
observably different from each other but obey some energy inequalities of the
similar type. In this paper, we attempt to axiomatize the extending mechanism
of solutions to these systems, merely starting from several basic axiomatized
conditions such as the local existence, joint property of solutions and some
energy inequalities. The results established have nothing to do with the
concrete forms of the systems and, thus, give the extending mechanisms in a
unified way to all systems obeying the axiomatized conditions. The key tools
are several new multiplicative interpolation inequalities of Besov type, which
have their own interests.
