High order asymptotic inequalities for some dissipative systems

TL;DR

This paper introduces new high order asymptotic inequalities for dissipative systems, revealing long-term behavior of derivatives and improving understanding of fluid dynamics, including Navier-Stokes and micro-rotational fluids.

Contribution

It presents novel inequalities for high order derivatives in dissipative systems, extending existing results and applying to various fluid models including asymmetric incompressible fluids.

Findings

Derived inequalities for high order derivatives in dissipative systems.

Established improved bounds for micro-rotational fluid equations.

Extended the approach to other 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 $L^2$ 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.