Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

Lin F Liu,Juan J Nieto

doi:10.3390/math8112037

Lin F Liu, Juan J Nieto

Open Access

https://doi.org/10.3390/math8112037

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Abstract

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem.

Highlights

We study the longtime behavior of the following two-dimensional Navier–Stokes equation of fractional order with variable delay on a bounded domain Ω ⊂ R2, Dtαu − ν∆u + (u · ∇)u + ∇p = f (t) + g(t, ut), in (0, T) × Ω, (1)
It is worth pointing out that using a convolution group, Li and Liu [8] introduced a generalized definition of Caputo derivative of order α ∈ (0, 1), and built a convenient framework for studying initial value problems of time fractional differential equations
Motivated by [19], we study the long time behavior of fractional Navier–Stokes equations with variable delay

Summary

Introduction

It is worth pointing out that using a convolution group, Li and Liu [8] introduced a generalized definition of Caputo derivative of order α ∈ (0, 1), and built a convenient framework for studying initial value problems of time fractional differential equations. Estimates of approximate solutions, to apply some compactness criteria—i.e., the Arzelà-Ascoli theorem, etc This method seems not to work for fractional PDEs with variable delay. Because of the appearance of variable delay term, the generalized fractional Gronwall inequality [15] (Theorem 1) is not enough to find some “a priori” estimates of Lyapunov functions. We first prove the existence and uniqueness of weak solutions by Galerkin approximation, and analyze the dissipativity of system (P), namely, we obtain the existence of an absorbing set by fractional Halanay inequalities and generalized comparison principle. Throughout the work, C, c are positive constants, which can be different from line to line, even in the same line

Preliminaries

Existence and Uniqueness of Weak Solutions

Dissipativity

Discussion