Some regularity results for a double time-delayed 2D-Navier-Stokes model

Abstract

In this paper we analyze some regularity properties of a double time-delayed 2D-Navier-Stokes model, that includes not only a delay force but also a delay in the convective term. The interesting feature of the model-from the mathematical point of view-is that being in dimension two, it behaves similarly as a 3D-model without delay, and extra conditions in order to have uniqueness were required for well-posedness. This model was previously studied in several papers, being the existence of attractor in the L 2-framework obtained by the authors [Discrete Contin. Dyn. Syst. 34 (2014), 4085-4105]. Here regularization properties of the solutions and existence of (regular) attractors for several associated dynamical systems are established. Moreover, relationships among these objects are also provided.

Dynamics of a globally modified Navier–Stokes model with double delay

“…Moreover, such delay can produce a regularizing effect in the 3D Navier-Stokes equations and allows to prove the uniqueness of weak solutions in [2], where they also showed those regularized solutions converge to a weak solution of the 3D Navier-Stokes equations when the delay tends to zero. In addition, Navier-Stokes equations with double delay have been investigated, one can refer to [14,15,30] for 2D Navier-Stokes equations with double delay and to [16,32] for 3D Navier-Stokes equations with double delay. Motivated by the above literature, we will investigate the long term dynamics of the globally modified Navier-Stokes equations with double delay (1.1).…”