  We consider a rather general class of evolutionary PDEs involving dissipation
(of possibly fractional order), which competes with quadratic nonlinearities on
the regularity of the overall equation. This includes as prototype models,
Burgers' equation, the Navier-Stokes equations, the surface quasi-geostrophic
equations and the Keller-Segel model for chemotaxis. Here we establish a
Petrowsky type parabolic estimate of such equations which entail a precise time
decay of higher-order Sobolev norms for this class of equations. To this end,
we introduce as a main new tool, an "infinite order energy functional", E(t): =
\Sigma_n \alpha_n t^n |(-\Delta)^{n\theta/2} u(*,t)|_{H^{\beta_c}} for
appropriate critical regularity index \beta_c. It captures the regularizing
effect of all higher order derivatives of u(*,t), by proving --- for a
carefully, problem-dependent choice of weights {\alpha_n}, that E(t) is
non-increasing in time.
