Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

Version 1 : Received: 30 August 2019 / Approved: 2 September 2019 / Online: 2 September 2019 (04:30:06 CEST)
Version 2 : Received: 12 February 2021 / Approved: 18 February 2021 / Online: 18 February 2021 (10:40:32 CET)

How to cite: Zheng, Y.; Liu, W.; Liu, Y. Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory. Preprints 2019, 2019090008 (doi: 10.20944/preprints201909.0008.v2). Zheng, Y.; Liu, W.; Liu, Y. Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory. Preprints 2019, 2019090008 (doi: 10.20944/preprints201909.0008.v2).

## Abstract

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property.

## Keywords

Navier-Stokes equation; infinite delay; memory kernels

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Comment 1
Commenter: Wenjun Liu
Commenter's Conflict of Interests: Author
Comment: We have changed the sentence at the end of section 2 as "In addition, we define the space $\mathcal{H}^{*}:=BCL_{-\infty}(H)\times L^{2}_{\mu}(\mathbb{R^{+}},V)$ with the following norm
$$\|(\psi,\varphi)\|^{2}_{\mathcal{H}^{*}}=\|\psi\|^{2}_{BCL_{-\infty}(H)}+\|\varphi\|^{2}_{1,\mu}.$$" and other related places.
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