Pullback dynamics of a 3D modified Navier-Stokes equations with double delays

Abstract

<p style='text-indent:20px;'>This paper is concerned with the tempered pullback dynamics for a 3D modified Navier-Stokes equations with double time-delays, which includes delays on external force and convective terms respectively. Based on the property of monotone operator and some suitable hypotheses on the external forces, the existence and uniqueness of weak solutions can be shown in an appropriate functional Banach space. By using the energy equation technique and weak convergence method to achieve asymptotic compactness for the process, the existence of minimal family of pullback attractors has also been derived.</p>

A discrete retarded Gronwall–Bellman type inequality and its applications to difference equations

“…By developing some new techniques the authors obtained several uniform decay estimates for the functions 𝑦 satisfying the inequality, which turns out to be very helpful in studying the dynamical behavior of delay differential equations (see, for example, previous studies 19,20 ). In this work, we give a discrete counterpart of (1.1), namely, we extend the main results in Li et al 19 to the following discrete inequality:…”