# Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- (g1)
- For any $\xi \in {C}_{H}$, the mapping $[0,T]\ni t\mapsto g(t,\xi )\in {({L}^{2}(\Omega ))}^{2}$ is measurable.
- (g2)
- $g(\xb7,0)=0$.
- (g3)
- There exists ${L}_{g}>0$ such that, for any $t\in [0,T]$ and all $\xi ,\eta \in {C}_{H}$,$$\begin{array}{c}\hfill |g(t,\xi )-g(t,\eta )|\le {L}_{g}{\parallel \xi -\eta \parallel}_{{C}_{H}}.\end{array}$$

**Remark**

**1.**

**Example**

**1.**

**A forcing term with bounded variable delay**. Let $G:[0,T]\times {\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ be a measurable function satisfying $G(t,0)=0$ for all $t\in [0,T]$, and assume that there exists ${L}_{G}>0$ such that

**Example**

**2.**

**A forcing term with finite distributed delay**. Let $G:[0,T]\times [-h,0]\times {\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ be a measurable function satisfying $G(t,s,0)=0$ for all $(t,s)\in [0,T]\times [-h,0]$, and there exists a function $\beta (s)\in {L}^{1}(-h,0)$ such that

**Definition**

**6.**

**Lemma**

**1.**

**Lemma**

**2.**

- (i)
- There exists ${r}_{1}\in [1,\infty )$ and $C>0$ such that $\forall u\in W$,$$\begin{array}{c}\hfill \underset{t\in (0,T)}{sup}{I}^{\alpha}{(\parallel u\parallel}_{M}^{{r}_{1}})=\underset{t\in (0,T)}{sup}\frac{1}{\Gamma (\alpha )}{\int}_{0}^{t}{(t-s)}^{\alpha -1}{\parallel u\parallel}_{M}^{{r}_{1}}(s)ds\le C.\end{array}$$
- (ii)
- There exists ${p}_{1}\in (p,\infty ]$, such that, W is bounded in ${L}^{{p}_{1}}((0,T);X)$.
- (iii)
- There exists ${r}_{2}\in [1,\infty )$, $C>0$ such that for any $u\in W$ with right limit ${u}_{0}$ at $t=0$, it holds that$$\begin{array}{c}\hfill \parallel {D}_{t}^{\alpha}{u\parallel}_{{L}^{{r}_{2}}((0,T);Y)}\le C.\end{array}$$Then, W is relatively compact in ${L}^{p}((0,T);X).$

**Proposition**

**1.**

- (i)
- There is a subsequence such that ${u}_{0,n}$ converges weakly to some value ${u}_{0}\in Y.$
- (ii)
- If $r>1,$ there exists a subsequence such that ${D}_{t}^{\alpha}{u}^{nk}$ converges weakly to v and ${u}_{0,{n}_{k}}$ converges weakly to ${u}_{0}$. Moreover, v is the Caputo derivative of u with initial value ${u}_{0}$ so that$$\begin{array}{c}\hfill u(t)={u}_{0}+\frac{1}{\Gamma (\alpha )}{\int}_{0}^{t}{(t-s)}^{\alpha -1}v(s)ds.\end{array}$$Further, if $r\ge 1$, then, $u({0}_{+})={u}_{0}$ in Y is the sense of Definition 2.

**Proof.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

**Proposition**

**3.**

**Proof.**

**Remark**

**4.**

**Lemma**

**3.**

**Remark**

**5.**

## 3. Existence and Uniqueness of Weak Solutions

**Theorem**

**1.**

**Proof.**

**Step 1.**The Galerkin approximation. By the definition of $A=-\Delta $ and the classical spectral theory of elliptic operators, it follows that A possesses a sequence of eigenvalues ${\left\{{\lambda}_{j}\right\}}_{j\ge 1}$ and a corresponding family of eigenfunctions ${\left\{{w}_{j}\right\}}_{j\ge 1}\subset V$, which form a Hilbert basis of H, dense on V. We consider the subspace ${V}_{m}=\mathrm{span}\{{w}_{1},{w}_{2},\cdots ,{w}_{m}\}$, and the projector ${P}_{m}:H\to {V}_{m}$ given by ${P}_{m}u={\sum}_{j=1}^{m}(u,{w}_{j}){w}_{j}$, and define ${u}^{(m)}(t)={\sum}_{j=1}^{m}{\gamma}_{m,j}(t){w}_{j}$, where the superscript m will be used instead of $(m)$, for short, since no confusion is possible with powers of u, and where the coefficients ${\gamma}_{m,j}(t)$ are required to satisfy the Cauchy problem

**Step 2.**A priori estimates. Multiplying (18) by ${\gamma}_{m,j}(t)$, $j=1,\dots ,m,$ summing up, and using Lemma 1, Cauchy–Schwartz and Young’s inequalities, we obtain

**Step 3.**Approximation of initial datum in ${C}_{H}$. Let us check

**Step 4.**Compactness results. By (21) and (22), the compact imbedding $V\hookrightarrow H$, and the generalized Aubin–Lions Lemma 2 as well as Proposition 1, for any $\alpha \in (0,1)$, we obtain there exist a subsequence still relabeled as $\left\{{u}^{m}\right\}$ and a function $u\in C([-h,T);H)\cap {L}^{2}((0,T);V)$ for all $T>0$, with $u(t)=\varphi (t)$ in $[-h,0]$, $u({0}_{+})={u}_{0}$, and ${D}_{t}^{\alpha}u\in {L}^{2}((0,T);{V}^{\prime})$ for all $T>0$, and an element $\chi \in {L}^{\infty}((0,T);H)$ such that

**Step 5.**Uniqueness of solution. Let $u(t;\varphi ),v(t;\varphi )$ be two solutions of $(P)$ with the same initial values—i.e., $u(t)=v(t)=\varphi (t)$, $t\in [-h,0].$ Set $w(t)=u(t)-v(t)$, $t\ge 0$, then $w(t)=0$, for all $t\in [-h,0].$ For $w(t)$, we have

**Remark**

**6.**

**Remark**

**7.**

**Theorem**

**2.**

**Proof.**

## 4. Dissipativity

**Definition**

**7.**

**Theorem**

**3.**

**Proof.**

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Liu, L.F.; Nieto, J.J.
Dissipativity of Fractional Navier–Stokes Equations with Variable Delay. *Mathematics* **2020**, *8*, 2037.
https://doi.org/10.3390/math8112037

**AMA Style**

Liu LF, Nieto JJ.
Dissipativity of Fractional Navier–Stokes Equations with Variable Delay. *Mathematics*. 2020; 8(11):2037.
https://doi.org/10.3390/math8112037

**Chicago/Turabian Style**

Liu, Lin F., and Juan J. Nieto.
2020. "Dissipativity of Fractional Navier–Stokes Equations with Variable Delay" *Mathematics* 8, no. 11: 2037.
https://doi.org/10.3390/math8112037