High order asymptotic inequalities for some dissipative systems

Abstract

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 algebraic L2 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 improved inequalities were established for the micro-rotational field w. Finally, we show how this technique can be applied to other dissipative systems.