2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165668We 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.25 pagesAnalysis of PDEs35Q30, 35Q35, 76Dxx, 76Bxxtext