  We study behaviors of scalar quantities near the possible blow-up time, which
is made of smooth solutions of the Euler equations, Navier-Stokes equations and
the surface quasi-geostrophic equations. Integrating the dynamical equations of
the scaling invariant norms, we derive the possible blow-up behaviors of the
above quantities, from which we obtain new type of blow-up criteria and some
necessary conditions for the blow-up.
