  We obtain some important fundamental inequalities concerning the long time
behavior of high order derivatives for solutions of some dissipative systems in
terms of their $L^2$ algebraic decay. Some of these inequalities have not been
observed in the literature even for the fundamental Navier-Stokes equations.To
illustrate this new approach, we derive bounds for the asymmetric
incompressible fluids equations, where an improved inequalities established for
the micro-rotational field. Finally, we show how this technique can be applied
for other dissipative systems.
