The Optimal Time Decay Rates and Vanishing Limit for Three-Dimensional Incompressible Phan-Thien–Tanner and Oldroyd-B Systems in the Critical Besov Spaces

Abstract

In this article, we study the incompressible Phan-Thien–Tanner (PTT) and Oldroyd-B systems in $R^{3}$ . The interesting features of the Cauchy problem studied in this work are the well-posedness and large-time behavior of global solutions for the PTT and Oldroyd-B systems with or without a damping mechanism, and the relationship between these two systems in the critical Besov spaces. First, we study the well-posedness of global solutions with data having critical regularity in the case of small initial data by proving uniform estimates with respect to the parameters $a$ and $b$ . Second, we prove the optimal time decay rates of global solutions in $L^{q}$ by exploiting harmonic analysis tools, such as non-standard $L^{q}$ product estimates, various Sobolev embeddings, and interpolation inequalities. We only assume that the negative Besov norm at low frequencies of the initial data is bounded and remove the $L^{p}$ ( $1 \leq p < 2$ ) condition, which has been required in previous works on this problem. Finally, we investigate the convergence of global solutions to the PTT system when $b$ tends to $0$ . Moreover, the specific rate of convergence is obtained in some sense.

Keywords

Phan-Thien–Tanner system Oldroyd-B system optimal time decay rates vanishing limit

1. Introduction

The homogeneous flows of incompressible isothermal polymer fluids can be described by the continuity and momentum balance equations. It is universally acknowledged that polymer molecules may be extraordinarily complex objects, but there are some theories and methods to simplify and model them. In particular, the model considered in this article originates from the work in 1977 by N. Phan-Thien and R. I. Tanner in Phan-Thien and Tanner (1977), governed by

\begin{aligned} {\begin{cases} div u = 0, \\ u_{t} + u \cdot \nabla u = div T - \nabla p . \end{cases} \end{aligned}

This model is developed to describe the rheological behavior of viscoelastic fluids. The unknowns

(u, p, T)

are the fluid velocity, the isotropic fluid pressure, and the stress tensor, respectively. Moreover, the symmetric tensor of constraints

T

corresponds to the rates of creation and destruction of junctions, depending on the instantaneous elastic energy of the network or, equivalently, the average extension of the network strand. Then,

T

may be divided into two parts: the elastic part

τ

and the Newtonian part, that is,

\begin{aligned} T = μ_{1} τ + 2 μ D (u), \end{aligned}

where

τ

is the polymer contribution to the stress tensor, the positive real number

μ

is the solvent viscosity coefficient,

μ_{1}

is the elastic coefficient, and

D (u) \overset{def}{=} 1 / 2 (\nabla u + (\nabla u)^{T})

. We denote the objective derivative of the tensor

τ

D τ / D t

, defined as

\begin{aligned} \frac{D τ}{D t} \overset{def}{=} \frac{\partial τ}{\partial t} + u \cdot \nabla τ + τ Ω (u) - Ω (u) τ + λ (D (u) τ + τ D (u)), \end{aligned}

where

λ

is the slip parameter, and

Ω (u) \overset{def}{=} 1 / 2 (\nabla u - (\nabla u)^{T})

. We focus on the linear Phan-Thien–Tanner (PTT) model, which is a well-studied phenomenological constitutive model for polymer fluids. It is given by

\begin{aligned} λ_{1} \frac{D τ}{D t} + (1 + \frac{ϵ λ_{1}}{η} (tr τ)) τ = 2 η (1 - \frac{λ_{2}}{λ_{1}}) D (u), \end{aligned}

where

λ_{1}

is the relaxation time of the fluid,

λ_{2}

(0 \leq λ_{2} \leq λ_{1})

is the retardation time of the fluid,

η

is the polymer viscosity, and

ϵ

is a parameter controlling the elongational viscosity.

We set $a = 1 / λ_{1}$ , $b = ϵ / η$ , and $μ_{2} = 2 η / λ_{1} (1 - (λ_{2} / λ_{1}))$ , to deduce the following linear incompressible PTT system:

\begin{aligned} {\begin{cases} u_{t} + u \cdot \nabla u - μ Δ u + \nabla p = μ_{1} div τ, (t, x) \in R^{+} \times R^{3}, \\ τ_{t} + u \cdot \nabla τ + (a + b (tr τ)) τ + Q (τ, \nabla u) = μ_{2} D (u), (t, x) \in R^{+} \times R^{3}, \\ div u = 0, (t, x) \in R^{+} \times R^{3}, \\ u |_{t = 0} = u_{0} (x), τ |_{t = 0} = τ_{0} (x), x \in R^{3}, \end{cases} \end{aligned}

(1.1)

where the positive real numbers

a

and

μ_{2}

are associated with the Deborah number

D e = μ_{2} / a

, which indicates the relation between the characteristic flow time and the elastic time (see, for instance, Bird et al., 1977), and the non-negative real number

b

is related to the rates of creation and destruction of the polymeric network junctions. Here,

Q (τ, \nabla u)

is a given bilinear form

\begin{aligned} Q (τ, \nabla u) = τ Ω (u) - Ω (u) τ + λ (D (u) τ + τ D (u)), \end{aligned}

where the physical parameter

λ \in [- 1, 1]

. In particular, we call the system a co-rotational case when

λ = 0

. For polymeric fluids, we note that the incompressible PTT system is derived from a constitutive equation, which is coupled to the incompressible Navier–Stokes equations. There is a substantial body of literature concerning interpretations and applications of the PTT system; see, for example, the works (Phan-Thien, 1978; Phan-Thien & Tanner, 1977).

It is interesting to note that system (1.1) reduces to the so-called Oldroyd-B system when $b = 0$ . If there is neither the damping term nor the rates of creation and destruction of the polymeric network junctions, that is, $a = b = 0$ , then system (1.1) reduces to the following Oldroyd-B system without a damping mechanism with the same initial data:

\begin{aligned} {\begin{cases} {\tilde{u}}_{t} + \tilde{u} \cdot \nabla \tilde{u} - μ Δ \tilde{u} + \nabla \tilde{p} = μ_{1} div \tilde{τ}, (t, x) \in R^{+} \times R^{3}, \\ {\tilde{τ}}_{t} + \tilde{u} \cdot \nabla \tilde{τ} + Q (\tilde{τ}, \nabla \tilde{u}) = μ_{2} D (\tilde{u}), (t, x) \in R^{+} \times R^{3}, \\ div \tilde{u} = 0, (t, x) \in R^{+} \times R^{3}, \\ \tilde{u} |_{t = 0} = u_{0} (x), \tilde{τ} |_{t = 0} = τ_{0} (x), x \in R^{3} . \end{cases} \end{aligned}

(1.2)

There is a very rich literature devoted to the so-called Oldroyd-B model, both in mathematics and physics (see Oldroyd, 1958), but we focus only on the relevant results in mathematics. The local existence of strong solutions and the global existence of one-dimensional shear flows were first observed by Guillopé and Saut (1990a, 1990b). In the multidimensional case, the local well-posedness in Sobolev spaces was then solved by Fernández-Cara et al. (1998). J. Chemin and N. Masmoudi essentially proved the local well-posedness in the critical Besov spaces and gave a lower bound for the lifespan in Chemin and Masmoudi (2001). Subsequently, some blow-up criteria were obtained in Feng et al. (2017), Lei et al. (2010). In the co-rotational case, the global result was first obtained by Lions and Masmoudi (2000), where they proved the global existence of weak solutions. The global existence for small smooth perturbations of a stable equilibrium was established by Lin et al. (2005) in the $R^{3}$ framework. Similar results were obtained in several papers using different methods; the reader may refer to Lei et al. (2008), Lei and Zhou (2005), Zhang and Fang (2012), Zi et al. (2014). Y. Zhu considered the Oldroyd-B system without a damping mechanism in Zhu (2018) and proved the global well-posedness result for small data. Later, similar global well-posedness results in the critical Besov spaces were obtained in Chen and Hao (2019), Zhai (2021). More recently, Lin et al. (2022) extended Y. Zhu’s work to the two-dimensional case. However, the question of global existence for general large data in the $R^{n}$ $(n \geq 2)$ framework has remained unsolved. See Fang and Zi (2016), Jiang and Jiang (2021) et al. for partial progress in this direction. Specifically, Fang and Zi (2016) and Jiang and Jiang (2021) removed the smallness condition required in previous papers and showed the global existence of strong solutions for a class of large initial data. Recently, there have been some works dealing with the large-time behavior of global solutions in Hieber et al. (2019), Huang et al. (2022), Wang and Wen (2020). Other problems, such as the initial value problem, were treated in He and Xu (2010), Lin and Zhang (2008), Sun and Zhang (2011). There are also a great deal of works limited to the compressible case; see, for example, the works Hu and Wang (2012), Hu and Wu (2013), Qian and Zhang (2010), Zhou et al. (2018). We mention in passing that results concerning weak and strong solutions for variations of the Navier–Stokes equations coupled with the Fokker-Planck equation can be found in Barrett et al. (2017), Barrett and Süli (2011), Barrett and Süli (2012), Liu et al. (2021), Liu et al. (2023), Lu and Pokorný (2020), Lu and Zhang (2018) and their references.

To the best of our knowledge, systems coupling fluids and polymers are of great interest in many sub-disciplines of applied sciences, such as mathematics, physics, biology, and chemistry, among others. In addition to the Oldroyd-B model, another famous model describing these systems is the PTT model. The PTT model attempts to describe the behavior of the complex mixture of polymers and fluid in more detail, and it poses a wide range of challenges, both in numerical computation and mathematics. It turns out that there are many numerical results for the PTT system (see Bautista et al., 2013; Garduño et al., 2016; Mu et al., 2013; Mu et al., 2012; Oliveira & Pinho, 1999). However, there are few mathematical results for the PTT system. In fact, the additional nonlinear term $(tr τ) τ$ of the PTT system leads to some interesting phenomena that are quite different from those of the Oldroyd-B system. The first significant result was obtained by Masmoudi (2011), where he proved the global existence of weak solutions. Using the characteristic and energy methods, Y. Chen et al. showed in Chen, Luo, and Yao (2019), Chen, Luo, and Zhai (2019), Chen et al. (2023) that global strong solutions exist for the linear PTT system with or without a damping mechanism in both Sobolev spaces and the critical Besov spaces for small initial data. As for the global well-posedness result for the generalized PTT system in both Sobolev spaces and the critical Besov spaces, one may refer to Chen et al. (2021), Chen et al. (2023). We should mention in passing that the sharp time decay rates and stability of large solutions to the two-dimensional and three-dimensional PTT systems were established in Chen et al. (2022), Chen et al. (2023). Until very recently, the vanishing limit for the PTT system in Sobolev spaces was demonstrated in Chen et al. (2024).

In this introductory section, we briefly explain how we consider the problems in the critical Besov spaces. In fact, the systems (1.1) and (1.2) do not have any scaling invariance, but we study the following linear mixed system:

\begin{aligned} {\begin{cases} u_{t} - μ Δ u - μ_{1} Λ (Λ^{- 1} P div τ) = P E, \\ (Λ^{- 1} P div τ)_{t} + a Λ^{- 1} P div τ + μ_{2} Λ u = Λ^{- 1} P div F, \end{cases} \end{aligned}

(1.3)

where

P \overset{def}{=} I - Δ^{- 1} \nabla div

is the Leray projection operator and

Λ^{s} \overset{def}{=} (- Δ)^{s / 2}

. One can easily verify that if

(u, τ, p)

solves system (1.3), then so does

(u_{λ}, τ_{λ}, p_{λ})

, where

\begin{aligned} u_{λ} = λ u (λ^{2} t, λ x), τ_{λ} = τ (λ^{2} t, λ x), p_{λ} = λ^{2} p (λ^{2} t, λ x) . \end{aligned}

Consequently, the linearized system (1.3) can be usefully employed in defining the critical space in this case. The reason for considering the well-posedness in the critical spaces has been fully explained in Chemin and Masmoudi (2001). There are many references concerning the well-posedness in the critical Besov spaces, such as Chen et al. (2010a, 2010b), Danchin (2000), Danchin (2001), Danchin and He (2016) and references therein. The hybrid Besov spaces with different indices of regularity were first established in Chen et al. (2010a).

Before getting into the heart of mathematical results, we have to introduce more notations.

Notation. For the sake of simplicity, we shall assume throughout this paper that the function space is over $R^{3}$ . The notation $A ≲ B$ stands for $A \leq C B$ for some irrelevant positive constant $C$ , and $A ≳ B$ stands for $A \geq C B$ for some irrelevant positive constant $C$ . The notation $A \sim B$ denotes $A ≲ B ≲ A$ . Denote by $S_{h}^{'} (R^{3})$ the set of tempered distributions. For any $u \in S_{h}^{'} (R^{3})$ , the lower and higher oscillation parts can be expressed as

\begin{aligned} u^{ℓ} \overset{def}{=} \sum_{j \leq N} {\dot{Δ}}_{j} u, u^{h} \overset{def}{=} \sum_{j > N} {\dot{Δ}}_{j} u, \end{aligned}

(1.4)

for some sufficiently large but fixed integer

N

We can now state the main results and ideas of this article.

$∙$ On the one hand, we are concerned with the global well-posedness and optimal time decay rates for the PTT and Oldroyd-B systems with or without a damping mechanism in the critical Besov spaces. Based on the above dissipation structure of $u$ and $Λ^{- 1} P div τ$ in system (1.3), we will show that the local well-posedness for general data and global well-posedness for small data hold for these two systems with or without a damping mechanism in the critical Besov spaces with minimal regularity. However, it does not seem possible to obtain any dissipation of $τ$ when $a = 0$ . The idea is inspired by the method applied in Chen, Luo, and Zhai (2019), Chen et al. (2021), but we then have to prove slightly more accurate estimates of the low- and high-frequency parts; see Appendices B–C for more details.

In previous papers, with the additional $L^{r} (R^{d})$ ( $1 \leq r < 2$ , $d = 2, 3$ ) smallness assumption of the initial data, the following optimal time decay rate was available:

\begin{aligned} ‖ \nabla^{k} (u, τ) ‖_{L^{2}} ≲ (1 + t)^{- \frac{d}{4} (\frac{2}{r} - 1) - \frac{k}{2}} . \end{aligned}

Recently, Chen et al. (2022, 2023) obtained the decay results for the PTT system under the assumptions that the initial data belong to

L^{r} (R^{d}) \cap H^{s} (R^{d})

(

1 \leq r < 2

d = 2, 3

). More precisely, they obtained the

L^{2}

decay rates of the solution and all its spatial derivatives. However, they did not verify whether the solution belongs to

L^{r} (R^{d})

. Motivated by Xin and Xu (2021), we can remove this condition. For the PTT and Oldroyd-B systems with a damping mechanism (i.e.,

a \neq 0

), we can deduce the following energy estimate:

\begin{aligned} \frac{d}{d t} E (t) + ‖ (u^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \leq 0, \end{aligned}

where

E (t) \sim ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{1 / 2}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{(3 / p) - 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{3 / p}}

and

2 \leq p \leq 4

. Under the assumptions that the low frequencies of the initial data belong to

{\dot{B}}_{2, \infty}^{σ}

with

σ \in [3 / 2 - 6 / p, 1 / 2)

, we can prove that

(u^{ℓ}, τ^{ℓ})

belongs to

{\dot{B}}_{2, \infty}^{σ}

. By virtue of some elaborate use of interpolation inequalities, we find that

\begin{aligned} \frac{d}{d t} E (t) + E^{\frac{5 - 2 σ}{1 - 2 σ}} (t) \leq 0. \end{aligned}

It is easy to deduce that

\begin{aligned} E (t) ≲ (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}

Then, with the help of non-standard

L^{q}

product estimates and some elaborate use of Sobolev embedding inequalities, we derive the

L^{q}

decay conclusions for the low- and high-frequency parts of

(u, τ, Λ^{- 1} P div τ)

with

q \geq 2

and

q \geq p

, respectively. However, if we make additional assumptions that the initial data belong to

{\dot{B}}_{2, \infty}^{σ}

with

σ \in [3 / 2 - 6 / p, 1 / 2)

, we also obtain similar results. For the Oldroyd-B system without a damping mechanism (i.e.,

a = b = 0

), we additionally assume that

(u_{0}^{ℓ}, τ_{0}^{ℓ}) \in {\dot{B}}_{2, \infty}^{σ}

with

σ \in [- 3 / 2, 1 / 2)

. We also derive similar

L^{q}

decay conclusions of the low- and high-frequency parts of

(u, Λ^{- 1} P div τ)

. It is worth noting that we only obtain the decay rate of

Λ^{- 1} P div τ

instead of

τ

. As a consequence, we remove the

L^{r}

(

1 \leq r < 2

) condition on the initial data. This is one of the innovations in our article. As a result, we obtain various decay results in many cases.

$∙$ On the other hand, we are concerned with the vanishing limit for the PTT system. Both the Oldroyd-B and PTT models describe fluid-polymer systems, which are of significant interest across many sub-disciplines of applied sciences. While both are widely used, their fundamental difference lies in the nonlinear coupling: the PTT system introduces a quadratic nonlinearity through the term $b (tr τ) τ$ , creating a self-interaction mechanism absent in the Oldroyd-B system. This key distinction motivates our study, which aims to investigate the relationship between the two systems. More precisely, given some initial data, we want to obtain as much information as possible on the convergence of the solution of the PTT system as $b$ tends to $0$ . Heuristically, we want to show that the solution of the PTT system converges to that of the classical Oldroyd-B system as $b$ tends to $0$ . We then focus on the rate of convergence in the critical Besov spaces, whose norm is invariant under the scaling transformation. The main difficulties in the passage to the limit are that new nonlinear terms may arise and $b$ tends to $0$ . To address these challenging terms, we require additional regularity of the high-frequency parts of the initial data and prove uniform estimates with respect to $b$ . Then, we can pass to the limit as $b \to 0$ , and the limit obtained will be the solution to the corresponding Oldroyd-B system. It suffices to obtain the rate of convergence of the PTT system toward the corresponding Oldroyd-B system in the critical Besov spaces. We note that this is the first result studying the relationship between the PTT and Oldroyd-B systems and providing the specific convergence rate in the critical Besov spaces.

2. Main Results

In order to simplify the presentation, we shall assume throughout this article that $μ = μ_{1} = μ_{2} = 1$ . We point out that the PTT system (i.e., $a > 0$ and $b > 0$ ) can be used to describe more physical phenomena than the Oldroyd-B system (i.e., $a > 0$ and $b = 0$ ) we have seen thus far, but it is still not too cumbersome. Whether it is to prove various decay results or the vanishing limit result of the PTT system toward the corresponding Oldroyd-B system, uniform estimates with respect to $a$ and $b$ are required.

2.1. Existence Results

Our main results are based on the following two existence theorems.

Theorem 2.1
Suppose that $a > 0$ and $b \geq 0$ ; the initial data $div u_{0} = 0$ , $(τ_{0})_{i j} = (τ_{0})_{j i}$ , $(u_{0}^{ℓ}, τ_{0}^{ℓ}) \in {\dot{B}}_{2, 1}^{1 / 2}$ , $u_{0}^{h} \in {\dot{B}}_{p, 1}^{(3 / p) - 1}$ , $τ_{0}^{h} \in {\dot{B}}_{p, 1}^{3 / p}$ with $2 \leq p \leq 4$ . There exists a small positive constant $ϵ_{0}$ such that if
$\begin{aligned} E (0) \overset{def}{=} ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \leq ϵ_{0}, \end{aligned}$
then system (1.1) admits a unique global solution $(u, τ)$ satisfying, for any $t \geq 0$ :
$\begin{aligned} \begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} \\ + ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + a ‖ τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} \\ + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \leq \tilde{C} ϵ_{0}, \end{aligned} \end{aligned}$
(2.1)
where $\tilde{C}$ is a positive constant independent of $t$ . If $p = 2$ , then the positive constant $\tilde{C}$ is independent of $t$ and $a$ . If we further assume that $u_{0}^{h} \in {\dot{B}}_{p, 1}^{3 / p}$ and $τ_{0}^{h} \in {\dot{B}}_{p, 1}^{(3 / p) + 1}$ , there exists a positive constant $c_{0}$ (which does not need to be small) such that if
$\begin{aligned} ‖ u_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} \leq c_{0}, \end{aligned}$
then for any $t \geq 0$ :
$\begin{aligned} \begin{aligned} ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \\ + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 2})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \leq \tilde{C} (c_{0} + ϵ_{0}^{2}) . \end{aligned} \end{aligned}$
(2.2)
Remark 2.2
We now verify that the conditions $a > 0$ and $b \geq 0$ are appropriate hypotheses for our problems. In the case where $a = 0$ and $b > 0$ , we shall see that if the initial data satisfies $tr τ_{0} < 0$ , then the solution of the PTT system without a damping mechanism will blow up, as shown in Chen, Luo, and Yao (2019). Therefore, the authors in Chen et al. (2023), Chen, Luo, and Zhai (2019), Chen et al. (2021) consider the global well-posedness for the PTT system without a damping mechanism under the assumptions that the initial data is a small perturbation around some particular solution. This reasoning justifies the assumption that $b ≲ a$ , which is consistent with and naturally enforced by the conditions $a > 0$ and $b \geq 0$ .
Remark 2.3
We point out that for proving the vanishing limit result of the PTT system toward the corresponding Oldroyd-B system, our method requires that we additionally have the estimate (2.2).

For the Oldroyd-B system without a damping mechanism (i.e., $a = b = 0$ ), we only obtain a similar existence result but for the simple case where $p = 2$ . This is because there is no damping term for the stress tensor in system (1.2).
Theorem 2.4
Suppose that $a = b = 0$ . The corresponding system (1.2) admits a unique global solution $(\tilde{u}, \tilde{τ})$ with the same initial data $(u_{0}, τ_{0})$ as in Theorem 2.1 with $p = 2$ . Moreover, the following estimate holds for any $t \geq 0$ :
$\begin{aligned} \begin{aligned} ‖ {\tilde{u}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} \\ + ‖ {\tilde{u}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \\ + ‖ {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \leq \tilde{C} ϵ_{0}, \end{aligned} \end{aligned}$
(2.3)
where $\tilde{C}$ is a positive constant independent of $t$ .
2.2. Decay Results

Next, we devote ourselves to investigating the optimal time decay rates of the global solutions for systems (1.1) and (1.2), respectively.

Theorem 2.5
Suppose that $a > 0$ and $b \geq 0$ . Let $(u, τ)$ be the global solution of system (1.1) given by Theorem 2.1. If, further, $(u_{0}^{ℓ}, τ_{0}^{ℓ}) \in {\dot{B}}_{2, \infty}^{σ}$ with $σ \in [3 / 2 - 6 / p, 1 / 2)$ , then for any $t \geq 0$ :
$\begin{aligned} ‖ (u^{ℓ}, τ^{ℓ}) ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} \leq C ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + C ϵ_{0}, \end{aligned}$
and
$\begin{aligned} ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \leq C (1 + t)^{- \frac{1 - 2 σ}{4}}, \end{aligned}$
where $C$ is a positive constant independent of $t$ . Moreover, for any $q \geq 2$ and $3 / q - 3 / 2 + σ < s \leq 3 / q - 1$ ,
$\begin{aligned} ‖ Λ^{s} (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{L^{q}} \leq C (1 + t)^{- \frac{3}{2} (\frac{1}{2} - \frac{1}{q}) - \frac{s - σ}{2}}, \end{aligned}$
(2.4)
for any $q \geq p$ and $s \leq (3 / q) - 1$ ,
$\begin{aligned} ‖ Λ^{s} (u^{h}, τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{L^{q}} \leq C (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}$
(2.5)
Remark 2.6
To obtain the optimal time decay rate of the global solution in the critical Besov spaces, we require additional boundedness assumptions on the low frequencies of the initial data in ${\dot{B}}_{2, \infty}^{σ}$ , with $σ \in [3 / 2 - 6 / p, 1 / 2)$ . This is in contrast to relying on $L^{p}$ smallness assumptions of the initial data or boundedness assumptions of the solution itself. We prove that the low frequencies of the solution also lie in ${\dot{B}}_{2, \infty}^{σ}$ , provided that the low frequencies of the initial data belong to ${\dot{B}}_{2, \infty}^{σ}$ .

For the Oldroyd-B system without a damping mechanism (i.e., $a = b = 0$ ), we only establish a similar result for the simple case $p = 2$ in the following theorem.
Theorem 2.7
Suppose that $a = b = 0$ . Let $(\tilde{u}, \tilde{τ})$ be the global solution of system (1.2) given by Theorem 2.4. If, further, $(u_{0}^{ℓ}, τ_{0}^{ℓ}) \in {\dot{B}}_{2, \infty}^{σ}$ with $σ \in [- 3 / 2, 1 / 2)$ , then for any $t \geq 0$ :
$\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, {\tilde{τ}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 2})} \leq C ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + C ϵ_{0}, \end{aligned}$
and
$\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ {\tilde{u}}^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} \leq C (1 + t)^{- \frac{1 - 2 σ}{4}}, \end{aligned}$
where $C$ is a positive constant independent of $t$ . Moreover, for any $q \geq 2$ and $(3 / q) - (3 / 2) + σ < s \leq (3 / q) - 1$ ,
$\begin{aligned} ‖ Λ^{s} ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{L^{q}} \leq C (1 + t)^{- \frac{3}{2} (\frac{1}{2} - \frac{1}{p}) - \frac{s - σ}{2}}, \end{aligned}$
(2.6)
for any $q \geq 2$ and $s \leq (3 / q) - 1$ ,
$\begin{aligned} ‖ Λ^{s} ({\tilde{u}}^{h}, Λ^{- 1} P div {\tilde{τ}}^{h}) ‖_{L^{q}} \leq C (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}$
(2.7)
Remark 2.8
Here, the time decay rates of the global solutions $(u, τ)$ and $(\tilde{u}, \tilde{τ})$ for systems (1.1) and (1.2), respectively, are called optimal in the sense that they coincide with the time decay rates of the solutions for their linearized systems.
Remark 2.9
The difference between the decay estimates of the global solutions $(u, τ)$ and $(\tilde{u}, \tilde{τ})$ for systems (1.1) and (1.2) lies in whether the optimal time decay rate of the stress tensor itself is obtained. This is due to the missing linear damping term of in system (1.2). Therefore, it follows that we can only establish the optimal time decay rate for $(\tilde{u}, Λ^{- 1} P div \tilde{τ})$ in system (1.2).
Remark 2.10
In contrast to the definitions (1.4) and (A.1), for any $u \in S_{h}^{'} (R^{3})$ , we introduce an alternative decomposition into lower and higher oscillation parts, denoted by $(u^{L}, u^{H})$ , to replace the previous notation $(u^{ℓ}, u^{h})$ . The new decomposition is defined as follows:
$\begin{aligned} u^{L} & \overset{def}{=} \sum_{j \leq N + 4} {\dot{Δ}}_{j} u, u^{H} \overset{def}{=} \sum_{j > N} {\dot{Δ}}_{j} u, \end{aligned}$
(2.8)

$\begin{aligned} ‖ u^{L} ‖_{{\dot{B}}_{p, 1}^{s}} & \overset{def}{=} \sum_{j \leq N + 4} 2^{j s} ‖ {\dot{Δ}}_{j} u ‖_{L^{p}}, ‖ u^{H} ‖_{{\dot{B}}_{p, 1}^{s}} \overset{def}{=} \sum_{j > N} 2^{j s} ‖ {\dot{Δ}}_{j} u ‖_{L^{p}}, \end{aligned}$
(2.9)
where $N$ is a sufficiently large fixed integer. For system (1.2), we can establish similar existence and decay results for the general case $p \in [2, 4]$ . The key to the proof is Corollary B.3, so we omit the details.

Assuming that both the low- and high-frequency parts of the initial data belong to ${\dot{B}}_{2, \infty}^{σ}$ , we obtain the following theorem.
Corollary 2.11
Suppose that $a > 0$ and $b \geq 0$ . Let $(u, τ)$ be the global solution of system (1.1) given by Theorem 2.1. Further assuming that $(u_{0}, τ_{0}) \in {\dot{B}}_{2, \infty}^{σ}$ for some $σ \in [3 / 2 - 6 / p, 1 / 2)$ , we have for all $t \geq 0$ :
$\begin{aligned} ‖ u ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} \leq C ‖ (u_{0}, τ_{0}) ‖_{{\dot{B}}_{2, \infty}^{σ}}, \end{aligned}$
where $C$ is a positive constant independent of $t$ .

For the Oldroyd-B system without a damping mechanism (i.e., $a = b = 0$ ), we establish the following analogous theorem for the case $p = 2$ .
Corollary 2.12
Suppose that $a = b = 0$ . Let $(\tilde{u}, \tilde{τ})$ be the global solution of system (1.2) given by Theorem 2.4. Further assuming that $(u_{0}, τ_{0}) \in {\dot{B}}_{2, \infty}^{σ}$ for some $σ \in [- 3 / 2, 1 / 2)$ , we have for all $t \geq 0$ :
$\begin{aligned} ‖ \tilde{u} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} \leq C ‖ (u_{0}, τ_{0}) ‖_{{\dot{B}}_{2, \infty}^{σ}}, \end{aligned}$
where $C$ is a positive constant independent of $t$ .
Remark 2.13
In Theorems 2.5 and 2.7, we only show that the low frequencies of the solutions to systems (1.1) and (1.2) belong to ${\dot{B}}_{2, \infty}^{σ}$ , provided that the low frequencies of the initial data belong to ${\dot{B}}_{2, \infty}^{σ}$ with $σ \in [3 / 2 - 6 / p, 1 / 2)$ . A natural question is whether this regularity property extends to the solutions themselves. To ensure that the solutions to systems (1.1) and (1.2) belong to ${\dot{B}}_{2, \infty}^{σ}$ , we assume in Corollaries 2.11 and 2.12 that the initial data belong to ${\dot{B}}_{2, \infty}^{σ}$ with $σ \in [3 / 2 - 6 / p, 1 / 2)$ .
2.3. Convergence Result

We note that if $a > 0$ , $b = 0$ , the PTT model reduces to the Oldroyd-B model. Both the PTT and Oldroyd-B models have greater physical significance when a damping mechanism is present. We now state another convergence result as $b \to 0$ . Under the assumptions of Theorem 2.1, we establish the precise rate of convergence for the global solutions of the PTT and Oldroyd-B systems. Specifically, the following assertion holds.

Theorem 2.14
Let $(u, τ)$ and $(\tilde{u}, \tilde{τ})$ be the global solutions, provided by Theorem 2.1, to the PTT (with $a > 0$ , $b > 0$ ) and Oldroyd-B (with $a > 0$ , $b = 0$ ) systems, respectively. Then the following convergence result holds for all $t > 0$ :
$\begin{aligned} ‖ u^{ℓ} - {\tilde{u}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} - {\tilde{τ}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} - {\tilde{u}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{h} - {\tilde{τ}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \leq C b ϵ_{0}^{2} e^{ϵ_{0} + c_{0}}, \end{aligned}$
(2.10)
where $C$ is a positive constant independent of $t$ and $b$ .

The remainder of this article is organized as follows. In Section 3, we prove the global well-posedness of systems (1.1) and (1.2) under the same small perturbation assumption. In Section 4, we prove the decay results stated in Theorems 2.5, 2.7, and Corollaries 2.11, 2.12. Finally, in Section 5, we analyze the rate of convergence of the PTT system to the corresponding Oldroyd-B system. We show that this rate strongly depends on the regularity of the global solutions for both systems. In Appendices A–C, we introduce homogeneous Besov spaces, paradifferential calculus, and tools based on the Littlewood-Paley decomposition.
3. Global Existence

This section focuses on the proof of Theorem 2.1. First, we compare the PTT system with the Oldroyd-B system. We note that the local well-posedness of the PTT system can be proved similarly to the Oldroyd-B system, as the additional term in (1.1) contains no derivatives (see Fernández-Cara et al., 1998, Lin et al., 2005, Zhai, 2021 for details). To prove energy estimates uniformly in time for system (1.1), we define the following basic energies:

\begin{aligned} E_{1} (t) \overset{def}{=} & ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})}, \\ E_{2} (t) \overset{def}{=} & ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} \\ + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + a ‖ τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})}, \\ E_{3} (t) \overset{def}{=} & ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}, \\ E_{4} (t) \overset{def}{=} & ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \\ + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 2})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} . \end{aligned}

We derive the a priori estimates of $E_{1} (t)$ , $E_{2} (t)$ , $E_{3} (t)$ , and $E_{4} (t)$ . We assume that $a > 0$ , $b \geq 0$ , and $b ≲ a$ .

3.1. The Estimate of $E_{1} (t)$

In this subsection, we derive the basic energy estimate for the low frequencies of the solution $(u, τ)$ . Applying ${\dot{Δ}}_{j} P$ to the first equation of (1.1) and ${\dot{Δ}}_{j}$ to the second, and using $div u = 0$ , we obtain

\begin{aligned} {\begin{cases} {\dot{Δ}}_{j} u_{t} + u \cdot \nabla {\dot{Δ}}_{j} u - Δ {\dot{Δ}}_{j} u = {\dot{Δ}}_{j} P div τ + [u \cdot \nabla, {\dot{Δ}}_{j} P] u, \\ {\dot{Δ}}_{j} τ_{t} + u \cdot \nabla {\dot{Δ}}_{j} τ + a {\dot{Δ}}_{j} τ = {\dot{Δ}}_{j} D (u) + [u \cdot \nabla, {\dot{Δ}}_{j}] τ - {\dot{Δ}}_{j} (b (tr τ) τ + Q (τ, \nabla u)) . \end{cases} \end{aligned}

(3.1)

Taking the

L^{2}

inner products of the above equations with

{\dot{Δ}}_{j} u

and

{\dot{Δ}}_{j} τ

, respectively, and integrating by parts, we have

\begin{aligned} \frac{1}{2} \frac{d}{d t} (‖ {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}}^{2}) + ‖ \nabla {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + a ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}}^{2} \\ = \int ({\dot{Δ}}_{j} P div τ \cdot {\dot{Δ}}_{j} u + {\dot{Δ}}_{j} D (u) \cdot {\dot{Δ}}_{j} τ) d x + \int ([u \cdot \nabla, {\dot{Δ}}_{j} P] u \cdot {\dot{Δ}}_{j} u + [u \cdot \nabla, {\dot{Δ}}_{j}] τ \cdot {\dot{Δ}}_{j} τ) d x \\ - \int {\dot{Δ}}_{j} (b (tr τ) τ + Q (τ, \nabla u)) \cdot {\dot{Δ}}_{j} τ d x . \end{aligned}

(3.2)

Thanks to the fact that

τ_{i j} = τ_{j i}

, we conclude that

\begin{aligned} \int ({\dot{Δ}}_{j} P div τ \cdot {\dot{Δ}}_{j} u + {\dot{Δ}}_{j} D (u) \cdot {\dot{Δ}}_{j} τ) d x = 0. \end{aligned}

Integrating in time, multiplying both sides of (3.2) by

2^{(1 / 2) j}

, and summing the resulting inequality for

j \leq N

, we end up with

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} \\ ≲ ‖ u_{0}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ_{0}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j} P] u ‖_{L_{t}^{1} (L^{2})} + \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{2})} \\ + b \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} + \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L_{t}^{1} (L^{2})} . \end{aligned}

(3.3)

Using the commutator estimate (B.2) in Lemma B.1, the product estimate (B.19) in Lemma B.4, and the equality (C.1) in Lemma C.1, it is enough to prove that

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j} P] u ‖_{L_{t}^{1} (L^{2})} & ≲ \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] u ‖_{L_{t}^{1} (L^{2})} + ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div div (u \otimes u)) ‖_{L_{t}^{1} (L^{2})}) \\ ≲ \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] u ‖_{L_{t}^{1} (L^{2})} + ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div (u \cdot \nabla u)) ‖_{L_{t}^{1} (L^{2})}) \\ ≲ \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (1 + c) ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

Using the commutator estimate (B.2) in Lemma B.1 and the product estimate (B.19) in Lemma B.4 again, we have

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{2})} & ≲ \int_{0}^{t} (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s \\ + c \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} d s, \\ \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} & ≲ \int_{0}^{t} (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s, \\ \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L_{t}^{1} (L^{2})} & ≲ \int_{0}^{t} (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

These estimates lead to the following inequality:

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} & ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (1 + c) ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (a + c) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s, \end{aligned}

(3.4)

where we have used the fact that

b ≲ a

. If we take

p = 2

, then the estimate (3.4) also holds for

c = 0

, and its proof is based on the last two estimates in Lemma B.6. Finally, we get

\begin{aligned} E_{1} (t) ≲ E (0) + E_{1}^{2} (t) + E_{2}^{2} (t) + E_{3}^{2} (t) . \end{aligned}

(3.5)

3.2. The Estimate of

E_{2} (t)

In this subsection, we intend to develop a new energy estimate to provide the integrability of the low frequencies of the solution $(u, τ)$ . We apply the operator ${\dot{Δ}}_{j} P$ to the first equation of system (1.1) and ${\dot{Δ}}_{j} Λ^{- 1} P div$ to the second one, to discover that

\begin{aligned} {\begin{cases} {\dot{Δ}}_{j} u_{t} + u \cdot \nabla {\dot{Δ}}_{j} u - Δ {\dot{Δ}}_{j} u - Λ {\dot{Δ}}_{j} ψ = f_{j}, \\ {\dot{Δ}}_{j} ψ_{t} + u \cdot \nabla {\dot{Δ}}_{j} ψ + a {\dot{Δ}}_{j} ψ + \frac{1}{2} Λ {\dot{Δ}}_{j} u = g_{j}, \end{cases} \end{aligned}

(3.6)

with

\begin{aligned} ψ & \overset{def}{=} Λ^{- 1} P div τ, f_{j} \overset{def}{=} [u \cdot \nabla, {\dot{Δ}}_{j} P] u, \\ g_{j} & \overset{def}{=} [u \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] τ - b {\dot{Δ}}_{j} Λ^{- 1} P div ((tr τ) τ) - {\dot{Δ}}_{j} Λ^{- 1} P div Q (τ, \nabla u) . \end{aligned}

Taking the $L^{2}$ inner products of the above equations with $(1 - η) {\dot{Δ}}_{j} u$ and $2 {\dot{Δ}}_{j} ψ$ , respectively, then we have

\begin{aligned} \frac{1}{2} \frac{d}{d t} ((1 - η) ‖ {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + 2 ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2}) + (1 - η) ‖ \nabla {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + 2 a ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2} + η \int_{R^{3}} Λ {\dot{Δ}}_{j} u \cdot {\dot{Δ}}_{j} ψ d x \\ ≲ ‖ f_{j} ‖_{L^{2}} ‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + ‖ g_{j} ‖_{L^{2}} ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}} . \end{aligned}

Note that $φ \overset{def}{=} 2 Λ ψ - u$ satisfies the following equation:

\begin{aligned} {\dot{Δ}}_{j} φ_{t} + u \cdot \nabla {\dot{Δ}}_{j} φ + (1 + 2 a) Λ {\dot{Δ}}_{j} ψ = 2 Λ g_{j} - f_{j} + 2 [u \cdot \nabla, Λ] {\dot{Δ}}_{j} ψ . \end{aligned}

Take the

L^{2}

inner product of the above equation with

{\dot{Δ}}_{j} φ

, we get that

\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ {\dot{Δ}}_{j} φ ‖_{L^{2}}^{2} + 2 (1 + 2 a) ‖ Λ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2} - (1 + 2 a) \int_{R^{3}} Λ {\dot{Δ}}_{j} ψ \cdot {\dot{Δ}}_{j} u d x \\ ≲ (‖ f_{j} ‖_{L^{2}} + 2^{j} ‖ g_{j} ‖_{L^{2}} + ‖ [u \cdot \nabla, Λ] {\dot{Δ}}_{j} ψ ‖_{L^{2}}) (‖ Λ {\dot{Δ}}_{j} ψ ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} u ‖_{L^{2}}) . \end{aligned}

Together with the above two estimates, we gather that

\begin{aligned} \frac{1}{2} \frac{d}{d t} A_{j}^{2} + (1 - η) ‖ \nabla {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + 2 a ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2} + 2 η ‖ Λ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2} \\ ≲ (‖ f_{j} ‖_{L^{2}} + (1 + 2^{j}) ‖ g_{j} ‖_{L^{2}} + ‖ [u \cdot \nabla, Λ] {\dot{Δ}}_{j} ψ ‖_{L^{2}}) (‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + (1 + 2^{j}) ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}), \end{aligned}

where

\begin{aligned} A_{j} \overset{def}{=} \sqrt{(1 - η) ‖ {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + 2 ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2} + \frac{η}{1 + 2 a} ‖ {\dot{Δ}}_{j} φ ‖_{L^{2}}^{2}} . \end{aligned}

For any

j \leq N

, we can choose

η > 0

small enough (independent of

N

) such that

\begin{aligned} \frac{1}{2} (‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}) & \leq A_{j} \leq \sqrt{(1 - η) ‖ {\dot{Δ}}_{j} u ‖_{L^{2}}^{2} + 2 ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}^{2} + η ‖ {\dot{Δ}}_{j} φ ‖_{L^{2}}^{2}} \\ \leq C_{N} (‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}}) . \end{aligned}

Bernstein’s inequality asserts that

\begin{aligned} \frac{d}{d t} A_{j} + 2^{2 j} ‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + 2^{2 j} ‖ {\dot{Δ}}_{j} ψ ‖_{L^{2}} ≲ (1 + 2^{j}) (‖ f_{j} ‖_{L^{2}} + (1 + 2^{j}) ‖ g_{j} ‖_{L^{2}} + ‖ [u \cdot \nabla, Λ] {\dot{Δ}}_{j} ψ ‖_{L^{2}}) . \end{aligned}

Integrating in time, multiplying both sides of the above inequality by $2^{(1 / 2) j}$ , and summing the resulting inequality for $j \leq N$ , we obtain that

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ ψ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ ψ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} \\ ≲ ‖ u_{0}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ ψ_{0}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ f_{j} ‖_{L_{t}^{1} (L^{2})} + ‖ g_{j} ‖_{L_{t}^{1} (L^{2})} + ‖ [u \cdot \nabla, Λ] {\dot{Δ}}_{j} ψ ‖_{L_{t}^{1} (L^{2})}) . \end{aligned}

(3.7)

We readily have

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ f_{j} ‖_{L^{2}} ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (1 + c) ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Thanks to the estimate (B.15) in Corollary B.2 and the estimate (B.19) in Lemma B.4, we verify that

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ g_{j} ‖_{L^{2}} & ≲ \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] τ ‖_{L^{2}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L^{2}}) \\ ≲ (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + b ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + b ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + c (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (a + c) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}), \end{aligned}

where we have used the fact that

b ≲ a

. The application of the estimates (B.2) and (B.4) in Lemma B.1, the estimate (B.19) in Lemma B.4, and the equality (C.4) in Lemma C.1, gives that

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ [u \cdot \nabla, Λ] {\dot{Δ}}_{j} ψ ‖_{L^{2}} & ≲ \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] Λ ψ ‖_{L^{2}} + ‖ Λ ([u \cdot \nabla, {\dot{Δ}}_{j}] ψ) ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} (Λ u \cdot \nabla ψ) ‖_{L^{2}}) \\ ≲ (‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + c (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + c ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Plugging the above estimates into (3.7) yields that

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (1 + c) ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (a + c) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

(3.8)

Recalling the equation (3.1)

_{2}

, we find that

\begin{aligned} \frac{d}{d t} ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}} + a ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}} ≲ 2^{j} ‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L^{2}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} (Q (τ, \nabla u)) ‖_{L^{2}} . \end{aligned}

Integrating in time, multiplying both sides of the above inequality by

2^{(3 / 2) j}

, and summing the resulting inequality for

j \leq N

, we obtain that

\begin{aligned} ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + a ‖ τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} & ≲ E (0) + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + \sum_{j \leq N} 2^{\frac{3}{2} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{2})} \\ + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} + ‖ {\dot{Δ}}_{j} (Q (τ, \nabla u)) ‖_{L_{t}^{1} (L^{2})}) . \end{aligned}

Hence, we have

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} \\ + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + a ‖ τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (1 + c) ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + (a + c) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

(3.9)

If we take

p = 2

, then the estimate (3.9) also holds for

c = 0

. This means that

\begin{aligned} E_{2} (t) ≲ E (0) + E_{1}^{2} (t) + E_{2}^{2} (t) + E_{3}^{2} (t) . \end{aligned}

(3.10)

3.3. The Estimates of

E_{3} (t)

and

E_{4} (t)

In this subsection, we are committed to deriving the energy estimate of the high frequencies of the solution $(u, τ)$ . We observe that the equation for $u$ contains the linear term $Δ (u - Λ^{- 1} ψ)$ , where $Λ^{- 1} ψ$ is a low-order derivative of $ψ$ , while the equation for $Λ^{- 1} ψ$ depends linearly only on $u$ and itself. This structure implies that the evolution of $u - Λ^{- 1} ψ$ is governed by $u$ and $Λ^{- 1} ψ$ . In the high-frequency regime, $u$ is controlled by its dissipation and $Λ^{- 1} ψ$ via the $ψ$ equation. Firstly, considering a new unknown quantity $Γ \overset{def}{=} u - Λ^{- 1} ψ$ , it is obvious that

\begin{aligned} {\dot{Δ}}_{j} Γ_{t} + u \cdot \nabla {\dot{Δ}}_{j} Γ - Δ {\dot{Δ}}_{j} Γ = a {\dot{Δ}}_{j} Λ^{- 1} ψ + \frac{1}{2} {\dot{Δ}}_{j} u - [u \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} ψ + f_{j} - Λ^{- 1} g_{j} . \end{aligned}

(3.11)

By the standard

L^{p}

estimate, we get that

\begin{aligned} \frac{d}{d t} ‖ {\dot{Δ}}_{j} Γ ‖_{L^{p}} + 2^{2 j} ‖ {\dot{Δ}}_{j} Γ ‖_{L^{p}} ≲ a ‖ {\dot{Δ}}_{j} Λ^{- 1} ψ ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} u ‖_{L^{p}} + ‖ [u \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} ψ ‖_{L^{p}} + ‖ f_{j} ‖_{L^{p}} + ‖ Λ^{- 1} g_{j} ‖_{L^{p}} . \end{aligned}

The combination of the estimates (B.3) and (B.5) in Lemma B.1, the other one (B.23) in Lemma B.4, and the equality (C.5) in Lemma C.1, ensure that

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ [u \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} ψ ‖_{L^{p}} \\ ≲ \sum_{j > N} 2^{(\frac{3}{p} - 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] Λ^{- 1} ψ ‖_{L^{p}} + ‖ Λ^{- 1} ([u \cdot \nabla, {\dot{Δ}}_{j}] ψ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Λ^{- 1} u \cdot \nabla ψ) ‖_{L^{p}}) \\ ≲ \sum_{j > N} 2^{\frac{3}{p} j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] Λ^{- 1} ψ ‖_{L^{p}} + \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] ψ ‖_{L^{p}} + \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ {\dot{Δ}}_{j} (Λ^{- 1} u \cdot \nabla ψ) ‖_{L^{p}} \\ ≲ ‖ ψ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} (‖ \nabla ψ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla ψ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) \\ ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

With the help of the estimate (B.3) in Lemma B.1, the other one (B.20) in Lemma B.4, and the equality (C.1) in Lemma C.1, one can obtain that

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ f_{j} ‖_{L^{p}} & ≲ \sum_{j > N} 2^{(\frac{3}{p} - 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] u ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div div (u \otimes u)) ‖_{L^{p}}) \\ ≲ \sum_{j > N} 2^{(\frac{3}{p} - 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] u ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div (u \cdot \nabla u)) ‖_{L^{p}}) \\ ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Since $Λ^{- 1} P div$ is a zero-order Fourier multiplier, in view of the estimate (B.16) in Corollary B.2 and the other one (B.20) in Lemma B.4, we arrive at

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ Λ^{- 1} g_{j} ‖_{L^{p}} ≲ & \sum_{j > N} 2^{(\frac{3}{p} - 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] τ ‖_{L^{p}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L^{p}}) \\ ≲ & (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}), \end{aligned}

where we have used the fact that

b ≲ a

. Together with the above estimates, we gather that

\begin{aligned} ‖ Γ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ Γ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \\ ≲ ‖ Γ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + 2^{- 2 N} (\frac{1}{2} ‖ Γ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + (\frac{1}{2} + a) ‖ ψ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}) \\ + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

(3.12)

We rewrite the second equation of (3.6) as

\begin{aligned} {\dot{Δ}}_{j} ψ_{t} + u \cdot \nabla {\dot{Δ}}_{j} ψ + (\frac{1}{2} + a) {\dot{Δ}}_{j} ψ = g_{j} - \frac{1}{2} Λ {\dot{Δ}}_{j} Γ . \end{aligned}

(3.13)

By virtue of the standard

L^{p}

estimate, we get that

\begin{aligned} \frac{d}{d t} ‖ {\dot{Δ}}_{j} ψ ‖_{L^{p}} + (\frac{1}{2} + a) ‖ {\dot{Δ}}_{j} ψ ‖_{L^{p}} ≲ ‖ g_{j} ‖_{L^{p}} + ‖ Λ {\dot{Δ}}_{j} Γ ‖_{L^{p}}, \end{aligned}

which leads to

\begin{aligned} ‖ ψ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + (\frac{1}{2} + a) ‖ ψ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} & ≲ ‖ ψ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + \sum_{j > N} 2^{\frac{3}{p} j} (‖ g_{j} ‖_{L_{t}^{1} (L^{p})} + ‖ Λ {\dot{Δ}}_{j} Γ ‖_{L_{t}^{1} (L^{p})}) \\ ≲ ‖ ψ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Γ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + \sum_{j > N} 2^{\frac{3}{p} j} ‖ g_{j} ‖_{L_{t}^{1} (L^{p})} . \end{aligned}

Due to $Λ^{- 1} P div$ is a zero-order Fourier multiplier, we get, by the estimate (B.18) in Corollary B.2 and the estimate (B.22) in Lemma B.4, that

\begin{aligned} \sum_{j > N} 2^{\frac{3}{p} j} ‖ g_{j} ‖_{L^{p}} & ≲ \sum_{j > N} 2^{\frac{3}{p} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] τ ‖_{L^{p}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L^{p}}) \\ ≲ (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Hence, we have

\begin{aligned} ‖ ψ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + (\frac{1}{2} + a) ‖ ψ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} & ≲ ‖ ψ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Γ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + \int_{0}^{t} (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

(3.14)

Additionally, combining (3.12) with (3.14), if we take

N \in N^{+}

to be sufficiently large, then

\begin{aligned} ‖ Γ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ ψ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Γ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ ψ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

We deduce from $u = Γ + Λ^{- 1} ψ$ that

\begin{aligned} ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ ψ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ ψ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ ‖ Γ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ ψ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Γ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ ψ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}, \end{aligned}

whence we get that

\begin{aligned} ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s . \end{aligned}

(3.15)

Recalling the equation (3.1)

_{2}

, we have

\begin{aligned} \frac{d}{d t} ‖ {\dot{Δ}}_{j} τ ‖_{L^{p}} + a ‖ {\dot{Δ}}_{j} τ ‖_{L^{p}} ≲ 2^{j} ‖ {\dot{Δ}}_{j} u ‖_{L^{p}} + ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L^{p}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Q (τ, \nabla u)) ‖_{L^{p}}, \end{aligned}

which gives rise to

\begin{aligned} ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ E (0) + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + \sum_{j > N} 2^{\frac{3}{p} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{p})} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{p})} + ‖ {\dot{Δ}}_{j} (Q (τ, \nabla u)) ‖_{L_{t}^{1} (L^{p})}) . \end{aligned}

This, together with the estimate (B.5) in Lemma B.1 and the estimate (B.22) in Lemma B.4, and (3.15), yields immediately that

\begin{aligned} ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) d s, \end{aligned}

(3.16)

which implies that

\begin{aligned} E_{3} (t) ≲ E (0) + E_{1}^{2} (t) + E_{2}^{2} (t) + E_{3}^{2} (t) . \end{aligned}

(3.17)

If we assume, in addition, that $u_{0}^{h} \in {\dot{B}}_{p, 1}^{3 / p}$ and $τ_{0}^{h} \in {\dot{B}}_{p, 1}^{(3 / p) + 1}$ , then combining the above proof, we easily get that

\begin{aligned} ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \\ + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 2})} + a ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \\ ≲ ‖ u_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + \sum_{j > N} 2^{\frac{3}{p} j} (‖ [u \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} ψ ‖_{L_{t}^{1} (L^{p})} + ‖ f_{j} ‖_{L_{t}^{1} (L^{p})} \\ + ‖ Λ^{- 1} g_{j} ‖_{L_{t}^{1} (L^{p})}) + \sum_{j > N} 2^{(\frac{3}{p} + 1) j} ‖ g_{j} ‖_{L_{t}^{1} (L^{p})} + \sum_{j > N} 2^{(\frac{3}{p} + 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{p})} \\ + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{p})} + ‖ {\dot{Δ}}_{j} (Q (τ, \nabla u)) ‖_{L_{t}^{1} (L^{p})}) . \end{aligned}

(3.18)

Similarly, we apply the estimate (B.3) in Lemma B.1, the other estimate (B.21) in Lemma B.4 and the equality (C.5) in Lemma C.1 to discover that

\begin{aligned} \sum_{j > N} 2^{\frac{3}{p} j} ‖ [u \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} ψ ‖_{L^{p}} & ≲ \sum_{j > N} 2^{\frac{3}{p} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] Λ^{- 1} ψ ‖_{L^{p}} + ‖ Λ^{- 1} ([u \cdot \nabla, {\dot{Δ}}_{j}] ψ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Λ^{- 1} u \cdot \nabla ψ) ‖_{L^{p}}) \\ ≲ ‖ ψ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla ψ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) . \end{aligned}

And by virtue of the estimate (B.5) in Lemma B.1 and the other one (B.21) in Lemma B.4, we find that

\begin{aligned} \sum_{j > N} 2^{\frac{3}{p} j} ‖ f_{j} ‖_{L^{p}} & ≲ \sum_{j > N} 2^{\frac{3}{p} j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] u ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div div (u \otimes u)) ‖_{L^{p}}) \\ ≲ ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ≲ (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}), \end{aligned}

and

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} + 1) j} ‖ g_{j} ‖_{L^{p}} & ≲ \sum_{j > N} 2^{(\frac{3}{p} + 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] τ ‖_{L^{p}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L^{p}}) \\ ≲ ‖ \nabla τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla^{2} u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + a ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) \\ + (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 2}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) . \end{aligned}

Using the estimate (B.5) in Lemma B.1 and the other estimate (B.21) in Lemma B.4 again, we have

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} + 1) j} (‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L^{p}} + b ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} + ‖ {\dot{Δ}}_{j} (Q (τ, \nabla u)) ‖_{L^{p}}) \\ ≲ (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) \\ + (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 2}} + a ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + a ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) . \end{aligned}

Finally, together with these estimates implies that

\begin{aligned} E_{4} (t) ≲ ‖ u_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + E_{1}^{2} (t) + E_{2}^{2} (t) + E_{3}^{2} (t) + (E_{1} (t) + E_{2} (t) + E_{3} (t)) E_{4} (t) . \end{aligned}

(3.19)

3.4. The Proof of Theorem 2.1

As a final step, we need to combine the estimates for $E_{1} (t)$ , $E_{2} (t)$ and $E_{3} (t)$ to prove Theorem 2.1. Therefore, we deduce that for any $0 \leq t \leq T$ ,

\begin{aligned} E (t) \overset{def}{=} E_{1} (t) + E_{2} (t) + E_{3} (t) \leq C^{*} E (0) + C^{*} (E_{1}^{2} (t) + E_{2}^{2} (t) + E_{3}^{2} (t)) . \end{aligned}

(3.20)

Due to the local existence theory, there exists a positive time

T

such that

\begin{aligned} E (t) \leq 2 C^{*} E (0), \forall t \in [0, T], \end{aligned}

(3.21)

where

C^{*}

is a positive constant independent of

t

. If we take

p = 2

, from the estimates (3.4), (3.9) and (3.16) with

c = 0

, then we can assume that

C^{*}

is a positive constant independent of

t

and

a

. Let

T^{*}

be the largest possible time of existence of the solution with (3.21) holds. By virtue of total energy (3.20) and the smallness assumption on the initial data, we get that

\begin{aligned} E (t) \leq C^{*} E (0) + C E^{2} (0) < 2 C^{*} E (0), \forall t \in [0, T^{*}] . \end{aligned}

Using the standard continuity argument, we can show that

T^{*} = + \infty

, provided that

ϵ_{0}

is small enough. Moreover, if we assume that

u_{0}^{h} \in {\dot{B}}_{p, 1}^{3 / p}

and

τ_{0}^{h} \in {\dot{B}}_{p, 1}^{(3 / p) + 1}

, from the estimate for

E_{4} (t)

, we find that

\begin{aligned} E_{4} (t) \leq C (‖ u_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) + C E^{2} (0) + C E (0) E_{4} (t) . \end{aligned}

Then, we have

\begin{aligned} E_{4} (t) \leq C (‖ u_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ_{0}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ϵ_{0}^{2}), \end{aligned}

provided that

ϵ_{0}

is small enough. The proof of Theorem 2.1 is therefore complete.

3.5. The Proof of Theorem 2.4

If $a = b = 0$ , then the proof of Theorem 2.4 is very similar to that of the previous one. Let $p = 2$ , the idea is to take advantage of the last two estimates (B.27) and (B.28) in Lemma B.6 instead of the estimates (B.2) and (B.4) in Lemma B.1 (i.e., $c = 0$ ). Obviously, it suffices to show that

\begin{aligned} {\tilde{E}}_{1} (t) & \overset{def}{=} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) d s, \\ {\tilde{E}}_{2} (t) & \overset{def}{=} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ u^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) d s, \\ {\tilde{E}}_{3} (t) & \overset{def}{=} ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ Λ^{- 1} P div τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \\ ≲ E (0) + \int_{0}^{t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) \\ \times (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) d s . \end{aligned}

We then infer that

\begin{aligned} \tilde{E} (t) \overset{def}{=} {\tilde{E}}_{1} (t) + {\tilde{E}}_{2} (t) + {\tilde{E}}_{3} (t) \leq {\tilde{C}}^{*} E (0) + {\tilde{C}}^{*} ({\tilde{E}}_{1}^{2} (t) + {\tilde{E}}_{2}^{2} (t) + {\tilde{E}}_{3}^{2} (t)) . \end{aligned}

Under the smallness assumption on the initial data, we can conclude that the solution exists on

[0, + \infty)

, whence we complete the proof of Theorem 2.4 immediately.

4. Large-Time Behavior

In this section, we study the precise large-time behavior of the solutions to systems (1.1) and (1.2), as given by Theorems 2.1 and 2.4, respectively. Specifically, we prove Theorems 2.5, 2.7, and Corollaries 2.11, 2.12. To this end, we employ harmonic analysis tools, including non-standard $L^{p}$ product estimates, Sobolev embeddings, and interpolation inequalities. We derive the optimal time decay rates for the solutions $(u, τ)$ and $(\tilde{u}, \tilde{τ})$ to systems (1.1) and (1.2), respectively.

4.1. The Proof of Theorem 2.5

Let’s start by proving Theorem 2.5. For simplicity, we shall prove this theorem only in the case where $a = b = 1$ . From the above section, we deduce from the estimate (3.4) that

\begin{aligned} \frac{d}{d t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}}) ≲ ϵ_{0} (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

While in view of the estimate (3.9), we write

\begin{aligned} \frac{d}{d t} (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ (2 P div τ - u)^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}}) + ‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} \\ ≲ ϵ_{0} (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

According to the estimate (3.16), we have

\begin{aligned} \frac{d}{d t} (‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ ϵ_{0} (‖ \nabla u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Hence, there exists a positive constant $c_{1}$ such that

\begin{aligned} \frac{d}{d t} (‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} \\ + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) + c_{1} (‖ (u^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} \\ + ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \leq 0, \end{aligned}

(4.1)

provided that

ϵ_{0}

is small enough, and we have used the fact that

\begin{aligned} ‖ (2 P div τ - u)^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} ≲ ‖ Λ^{- 1} P div τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} . \end{aligned}

Next, we claim that

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ≲ ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + ϵ_{0} . \end{aligned}

(4.2)

The proof of the claim (4.2) will be given in the end of this subsection. Yet by virtue of Lemma A.3 and the estimate (4.2), for any

σ < 1 / 2

, we have

\begin{aligned} ‖ (u^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} \\ ≲ ‖ (u^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}}^{\frac{4}{5 - 2 σ}} ‖ (u^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}}^{\frac{1 - 2 σ}{5 - 2 σ}} \\ ≲ ‖ (u^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}}^{\frac{1 - 2 σ}{5 - 2 σ}}, \end{aligned}

(4.3)

and

\begin{aligned} ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} ≲ ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, \infty}^{σ}}^{\frac{2}{3 - 2 σ}} ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}^{\frac{1 - 2 σ}{3 - 2 σ}} ≲ ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}^{\frac{1 - 2 σ}{3 - 2 σ}} ≲ ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}^{\frac{1 - 2 σ}{5 - 2 σ}}, ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} ≲ ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}^{\frac{1 - 2 σ}{5 - 2 σ}}, \end{aligned}

(4.4)

where we have used the fact that

‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} ≲ ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} ≲ ϵ_{0}

, which have been proven in Theorem 2.1. We also point out, for any

β > 0

, that

\begin{aligned} \begin{aligned} ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} & ≳ ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} ≳ ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}^{1 + β}, \\ ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} & ≳ ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}^{1 + β} . \end{aligned} \end{aligned}

(4.5)

We substitute the above estimates (4.3)–(4.5) into (4.1), and choose

β = 4 / (1 - 2 σ) > 0

, there then exists a positive constant

c_{2}

such that

\begin{aligned} \frac{d}{d t} E (t) + c_{2} E {(t)}^{\frac{5 - 2 σ}{1 - 2 σ}} \leq 0, \end{aligned}

where

\begin{aligned} E (t) \overset{def}{=} ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}, (2 P div τ - u)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ (u - Λ^{- 2} P div τ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} . \end{aligned}

It is obvious that

\begin{aligned} ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ≲ E (t) ≲ (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}

(4.6)

For any

q \geq 2

and

(3 / q) - (3 / 2) + σ < s \leq (3 / q) - 1

, we deduce that

\begin{aligned} ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{q, 1}^{s}} & ≲ ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{s + \frac{3}{2} - \frac{3}{q}}} \\ ≲ ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}}^{θ} ‖ (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}}^{1 - θ} \\ ≲ (1 + t)^{- \frac{3}{2} (\frac{1}{2} - \frac{1}{q}) - \frac{s - σ}{2}}, \end{aligned}

where

θ = ((3 / q) - 1 - s) / ((1 / 2) - σ)

. Since

{\dot{B}}_{q, 1}^{0} ↪ L^{q}

, we have

\begin{aligned} ‖ Λ^{s} (u^{ℓ}, τ^{ℓ}, Λ^{- 1} P div τ^{ℓ}) ‖_{L^{q}} ≲ (1 + t)^{- \frac{3}{2} (\frac{1}{2} - \frac{1}{q}) - \frac{s - σ}{2}} . \end{aligned}

For any $q \geq p$ and $s \leq 3 / q - 1$ , it is easy to check that

\begin{aligned} ‖ (u^{h}, τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{q, 1}^{s}} & ≲ ‖ (u^{h}, τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{s + \frac{3}{p} - \frac{3}{q}}} \\ ≲ ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ (τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ (1 + t)^{- \frac{1 - 2 σ}{4}}, \end{aligned}

which implies that

\begin{aligned} ‖ Λ^{s} (u^{h}, τ^{h}, Λ^{- 1} P div τ^{h}) ‖_{L^{q}} ≲ (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}

Finally, there is the claim (4.2) left to prove. In view of the estimate (3.3), we also get that

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} & ≲ ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + sup_{j \leq N} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j} P] u ‖_{L_{t}^{1} (L^{2})} + sup_{j \leq N} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{2})} \\ + sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} + sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L_{t}^{1} (L^{2})} . \end{aligned}

(4.7)

The application of the estimate (B.1) in Lemma B.1, the estimate (B.25) in Lemma B.5, and the equality (C.1) in Lemma C.1 gives that

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j} P] u ‖_{L_{t}^{1} (L^{2})} & ≲ sup_{j \leq N} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] u ‖_{L_{t}^{1} (L^{2})} + sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div div (u \otimes u)) ‖_{L_{t}^{1} (L^{2})} \\ ≲ (‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})}) ‖ \nabla u ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + c ‖ u ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} ‖ u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ + (‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})}) ‖ u ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p} + 1})} \\ ≲ (‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})}) ‖ \nabla u ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} . \end{aligned}

With the help of the estimate (B.1) in Lemma B.1, we find that

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{2})} \\ ≲ (‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})}) ‖ \nabla u ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + c ‖ u ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ (‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ u^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}) (‖ \nabla u ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}) . \end{aligned}

We apply the estimate (B.24) in Lemma B.5, to discover that

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} & ≲ ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ‖ τ ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ τ ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} ‖ τ^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ (‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}) ‖ τ ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}, \end{aligned}

and

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (τ, \nabla u) ‖_{L_{t}^{1} (L^{2})} & ≲ ‖ \nabla u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ‖ τ ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ τ ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p} - 1})} ‖ \nabla u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} \\ ≲ (‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ τ^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}) (‖ τ ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + ‖ \nabla u^{h} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}) . \end{aligned}

Substituting the above estimates into (4.7), one can obtain that

\begin{aligned} ‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ≲ ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + ϵ_{0} (‖ u^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})}) + ϵ_{0}^{2}, \end{aligned}

where we have used the estimate (2.1). By choosing

ϵ_{0}

small enough, we can now conclude that the claim (4.2) holds. The proof is therefore complete.

4.2. The Proof of Theorem 2.7.

In this subsection, we will give the proof of Theorem 2.7. Since there is no damping term about $\tilde{τ}$ in system (1.2), the proof of Theorem 2.7 is different from that of Theorem 2.5. For system (1.2), we first claim that

\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, {\tilde{τ}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\tilde{L}}_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 2})} ≲ ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + ϵ_{0} . \end{aligned}

(4.8)

The proof of the claim (4.8) will be given in the end of this subsection. Let

p = 2

, the idea is to take advantage of the last two estimates (B.27) and (B.28) in Lemma B.6 instead of the estimates (B.2) and (B.4) in Lemma B.1 (i.e.,

c = 0

). Using the estimate (2.3), we deduce from the estimates (3.8) and (3.15) with

a = c = 0

that

\begin{aligned} \frac{d}{d t} (‖ {\tilde{u}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ (2 P div \tilde{τ} - \tilde{u})^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}}) + ‖ {\tilde{u}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} \\ ≲ ϵ_{0} (‖ \nabla {\tilde{u}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla {\tilde{u}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}), \end{aligned}

and

\begin{aligned} \frac{d}{d t} (‖ (\tilde{u} - Λ^{- 2} P div \tilde{τ})^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) + ‖ (\tilde{u} - Λ^{- 2} P div \tilde{τ})^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} \\ ≲ ϵ_{0} (‖ \nabla {\tilde{u}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla {\tilde{u}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Along the similar line as the estimate (4.6), we can get that

\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ {\tilde{u}}^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} ≲ \tilde{E} (t) ≲ (1 + t)^{- \frac{1 - 2 σ}{4}}, \end{aligned}

where

\tilde{E} (t) \overset{def}{=} ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}, (2 P div \tilde{τ} - \tilde{u})^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{1 / 2}} + ‖ (\tilde{u} - Λ^{- 2} P div \tilde{τ})^{h} ‖_{{\dot{B}}_{2, 1}^{1 / 2}} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\dot{B}}_{2, 1}^{3 / 2}}

. Similarly, for any

q \geq 2

and

(3 / q) - (3 / 2) + σ < s \leq (3 / q) - 1

, we also have

\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\dot{B}}_{q, 1}^{s}} ≲ (1 + t)^{- \frac{3}{2} (\frac{1}{2} - \frac{1}{q}) - \frac{s - σ}{2}}, \end{aligned}

and for any

q \geq 2

and

s \leq (3 / q) - 1

\begin{aligned} ‖ ({\tilde{u}}^{h}, Λ^{- 1} P div {\tilde{τ}}^{h}) ‖_{{\dot{B}}_{q, 1}^{s}} ≲ ‖ ({\tilde{u}}^{h}, Λ^{- 1} P div {\tilde{τ}}^{h}) ‖_{{\dot{B}}_{2, 1}^{s + \frac{3}{2} - \frac{3}{q}}} ≲ ‖ ({\tilde{u}}^{h}, Λ^{- 1} P div {\tilde{τ}}^{h}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} ≲ (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}

Due to ${\dot{B}}_{q, 1}^{0} ↪ L^{q}$ , it is not hard to check that

\begin{aligned} ‖ Λ^{s} ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{L^{q}} & ≲ ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\dot{B}}_{q, 1}^{s}} ≲ (1 + t)^{- \frac{3}{2} (\frac{1}{2} - \frac{1}{q}) - \frac{s - σ}{2}}, \\ ‖ Λ^{s} ({\tilde{u}}^{h}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{L^{q}} & ≲ ‖ ({\tilde{u}}^{h}, Λ^{- 1} P div {\tilde{τ}}^{h}) ‖_{{\dot{B}}_{q, 1}^{s}} ≲ (1 + t)^{- \frac{1 - 2 σ}{4}} . \end{aligned}

In order to complete the proof of Theorem 2.7, we need to prove the claim (4.8). Along the similar line as the estimates (3.3) and (3.7), we find that

\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, {\tilde{τ}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 2})} \\ ≲ ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} P] \tilde{u} ‖_{L_{t}^{1} (L^{2})} + sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] \tilde{τ} ‖_{L_{t}^{1} (L^{2})} \\ + sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (\tilde{τ}, \nabla \tilde{u}) ‖_{L_{t}^{1} (L^{2})} + sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] \tilde{τ} ‖_{L_{t}^{1} (L^{2})} \\ + sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} Λ^{- 1} P div Q (\tilde{τ}, \nabla \tilde{u}) ‖_{L_{t}^{1} (L^{2})} + sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, Λ] {\dot{Δ}}_{j} Λ^{- 1} P div \tilde{τ} ‖_{L_{t}^{1} (L^{2})} . \end{aligned}

(4.9)

We apply the estimate (B.25) in Lemma B.5, the estimate (B.26) in Lemma B.6 and the equality (C.1) in Lemma C.1, to find that

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} P] \tilde{u} ‖_{L_{t}^{1} (L^{2})} & ≲ sup_{j \leq N} 2^{σ j} (‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] \tilde{u} ‖_{L_{t}^{1} (L^{2})} + ‖ {\dot{Δ}}_{j} (Δ^{- 1} \nabla div div (\tilde{u} \otimes \tilde{u})) ‖_{L_{t}^{1} (L^{2})}) \\ ≲ (‖ {\tilde{u}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ {\tilde{u}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})}) ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} . \end{aligned}

The application of the estimate (B.26) in Lemma B.6 yields that

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] \tilde{τ} ‖_{L_{t}^{1} (L^{2})} & ≲ (‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})}) ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \\ ≲ (‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} . \end{aligned}

By virtue of the estimate (B.29) in Lemma B.7, we have

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} Λ^{- 1} P div Q (\tilde{τ}, \nabla \tilde{u}) ‖_{L_{t}^{1} (L^{2})} & ≲ sup_{j \leq N} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (\tilde{τ}, \nabla \tilde{u}) ‖_{L_{t}^{1} (L^{2})} \\ ≲ ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} (‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})} + ‖ \nabla {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) \\ ≲ (‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) (‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})} + ‖ \nabla {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) . \end{aligned}

We deduce from the estimate (B.26) in Lemma B.6, the estimates (B.29) and (B.30) in Lemma B.7, and the equality (C.3) in Lemma C.1 that

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] \tilde{τ} ‖_{L_{t}^{1} (L^{2})} \\ ≲ sup_{j \leq N} 2^{σ j} (‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] Λ^{- 1} P div \tilde{τ} ‖_{L_{t}^{1} (L^{2})} + ‖ {\dot{Δ}}_{j} (\tilde{u} \cdot \nabla Λ^{- 1} P div \tilde{τ}) ‖_{L_{t}^{1} (L^{2})} \\ + ‖ {\dot{Δ}}_{j} Λ^{- 1} P div (\nabla \tilde{u} \cdot \tilde{τ} + \tilde{u} \cdot P div \tilde{τ} - \nabla \tilde{u} \cdot Δ^{- 1} div div \tilde{τ}) ‖_{L_{t}^{1} (L^{2})}) \\ ≲ (‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})}) ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \\ + ‖ {\tilde{u}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} (‖ \nabla Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ \nabla Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{1}{2}})}) \\ + ‖ \nabla Λ^{- 1} P div \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} ‖ {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} (‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})} + ‖ \nabla {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) \\ ≲ (‖ {\tilde{u}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \\ + ‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) (‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{5}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) \\ + (‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) ‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})} . \end{aligned}

In view of the estimate (B.26) in Lemma B.6, the estimate (B.29) in Lemma B.7, and the equality (C.4) in Lemma C.1, we arrive at

\begin{aligned} sup_{j \leq N} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, Λ] {\dot{Δ}}_{j} Λ^{- 1} P div \tilde{τ} ‖_{L_{t}^{1} (L^{2})} \\ ≲ sup_{j \leq N} 2^{σ j} (‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] P div \tilde{τ} ‖_{L_{t}^{1} (L^{2})} + ‖ Λ ([\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] Λ^{- 1} P div \tilde{τ}) ‖_{L_{t}^{1} (L^{2})} \\ + ‖ {\dot{Δ}}_{j} (Λ \tilde{u} \cdot \nabla Λ^{- 1} P div \tilde{τ}) ‖_{L_{t}^{1} (L^{2})}) \\ ≲ (‖ P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ P div {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})}) \\ \times ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} + ‖ \nabla \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} (‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})} + ‖ \nabla {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) \\ ≲ (‖ Λ^{- 1} P div {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ Λ^{- 1} P div {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})} \\ + (‖ {\tilde{τ}}^{ℓ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{1}{2}})} + ‖ {\tilde{τ}}^{h} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) (‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})} + ‖ \nabla {\tilde{u}}^{h} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}) . \end{aligned}

Substituting these estimates into (4.9), we find that

\begin{aligned} ‖ ({\tilde{u}}^{ℓ}, {\tilde{τ}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ ({\tilde{u}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 2})} \\ ≲ ‖ (u_{0}^{ℓ}, τ_{0}^{ℓ}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + ϵ_{0} (‖ ({\tilde{u}}^{ℓ}, {\tilde{τ}}^{ℓ}, Λ^{- 1} P div {\tilde{τ}}^{ℓ}) ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ \nabla {\tilde{u}}^{ℓ} ‖_{L_{t}^{1} ({\dot{B}}_{2, \infty}^{σ + 1})}) + ϵ_{0}^{2}, \end{aligned}

where we have used the estimate (2.3). By choosing

ϵ_{0}

small enough, we get the claim (4.8) immediately. As a consequence, the proof of Theorem 2.7 has been completed.

4.3. The Proof of Corollaries 2.11–2.12

For the system (1.2), it is easy to check that

\begin{aligned} ‖ \tilde{u} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} & ≲ ‖ (u_{0}, τ_{0}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + sup_{j} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} P] \tilde{u} ‖_{L_{t}^{1} (L^{2})} \\ + sup_{j} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] \tilde{τ} ‖_{L_{t}^{1} (L^{2})} + sup_{j} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (\tilde{τ}, \nabla \tilde{u}) ‖_{L_{t}^{1} (L^{2})} . \end{aligned}

(4.10)

We apply the estimate (B.31) in Lemma B.8, the estimate (B.33) in Lemma B.9 and the equality (C.1) in Lemma C.1, to obtain that

\begin{aligned} sup_{j} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} P] \tilde{u} ‖_{L_{t}^{1} (L^{2})} & ≲ sup_{j} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] \tilde{u} ‖_{L_{t}^{1} (L^{2})} + sup_{j} 2^{(σ + 1) j} ‖ {\dot{Δ}}_{j} (\tilde{u} \otimes \tilde{u}) ‖_{L_{t}^{1} (L^{2})} \\ ≲ ‖ \tilde{u} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} . \end{aligned}

The application of the estimate (B.31) in Lemma B.8 gives that

\begin{aligned} sup_{j} 2^{σ j} ‖ [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] \tilde{τ} ‖_{L_{t}^{1} (L^{2})} ≲ ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} . \end{aligned}

By virtue of the estimate (B.32) in Lemma B.9, we have

\begin{aligned} sup_{j} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (\tilde{τ}, \nabla \tilde{u}) ‖_{L_{t}^{1} (L^{2})} ≲ ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} . \end{aligned}

Substituting these estimates into (4.10), we find that

\begin{aligned} ‖ \tilde{u} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ≲ ‖ (u_{0}, τ_{0}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + (‖ \tilde{u} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ \tilde{τ} ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})}) ‖ \nabla \tilde{u} ‖_{L_{t}^{1} ({\dot{B}}_{2, 1}^{\frac{3}{2}})}, \end{aligned}

which together with the energy estimate (2.3), and by using the fact that

ϵ_{0}

is small enough, we finish the proof of Corollary 2.12.

Along the similar lines, for the system (1.1), we need to treat the extra nonlinear term $(tr τ) τ$ , that is,

\begin{aligned} sup_{j} 2^{σ j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} ≲ ‖ τ ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} ‖ τ ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} . \end{aligned}

Similarly, we have

\begin{aligned} ‖ u ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} & ≲ ‖ (u_{0}, τ_{0}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + sup_{j} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j} P] u ‖_{L_{t}^{1} (L^{2})} + sup_{j} 2^{σ j} ‖ [u \cdot \nabla, {\dot{Δ}}_{j}] τ ‖_{L_{t}^{1} (L^{2})} \\ + sup_{j} 2^{σ j} ‖ {\dot{Δ}}_{j} Q (u, \nabla τ) ‖_{L_{t}^{1} (L^{2})} + b sup_{j} 2^{σ j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L_{t}^{1} (L^{2})} \\ ≲ ‖ (u_{0}, τ_{0}) ‖_{{\dot{B}}_{2, \infty}^{σ}} + (‖ u ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})} + ‖ τ ‖_{{\tilde{L}}_{t}^{\infty} ({\dot{B}}_{2, \infty}^{σ})}) (‖ \nabla u ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})} + a ‖ τ ‖_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{3}{p}})}), \end{aligned}

which together with the energy estimate (2.1), by using the fact that

ϵ_{0}

is small enough, we complete the proof of Corollary 2.11.

5. Convergence Rate of the PTT System to the Oldroyd-B System

In Section 3, we established the existence of global solutions $(u, τ)$ and $(\tilde{u}, \tilde{τ})$ for the PTT system (with $a > 0$ , $b > 0$ ) and the Oldroyd-B system (with $a > 0$ , $b = 0$ ), respectively. A natural question is whether $(u, τ)$ converges to $(\tilde{u}, \tilde{τ})$ as $b \to 0$ , and if so, what is the convergence rate. This section addresses this question, and our goal is to prove Theorem 2.14. More precisely, we study the rate of convergence of the global solution $(u, τ)$ for the PTT system to the global solution $(\tilde{u}, \tilde{τ})$ for the Oldroyd-B system as $b \to 0$ .

Let $(u, τ)$ and $(\tilde{u}, \tilde{τ})$ be two global solutions of the PTT system (with $a > 0$ , $b > 0$ ) and the corresponding Oldroyd-B system (with $a > 0$ , $b = 0$ ) given by Theorem 2.1, respectively. In order to show that these two global solutions coincide, we shall give the uniform estimate for $(v, σ) \overset{def}{=} (u - \tilde{u}, τ - \tilde{τ})$ . For notational simplicity, we set $a = 1$ . We shall consider the following error system satisfied by $(v, σ)$ :

\begin{aligned} \begin{aligned} {\begin{cases} v_{t} + \tilde{u} \cdot \nabla v + v \cdot \nabla u - Δ v + \nabla (p - \tilde{p}) = div σ, \\ σ_{t} + \tilde{u} \cdot \nabla σ + v \cdot \nabla τ + σ + b ((tr τ)) τ + Q (τ, \nabla v) + Q (σ, \nabla \tilde{u}) = D (v), \\ div v = div u = div \tilde{u} = 0, \\ v |_{t = 0} = 0, σ |_{t = 0} = 0. \end{cases} \end{aligned} \end{aligned}

(5.1)

The proof of Theorem 2.14 will be based on Theorem 2.1, and the key of the proof is to deal with the nonlinear term

v \cdot \nabla τ

. We broaden our analysis in Section 3 to cover the new system (5.1). Applying the operator

{\dot{Δ}}_{j} P

to the first equation of system (5.1) and

{\dot{Δ}}_{j}

to the second one, yields that

\begin{aligned} {\begin{cases} {\dot{Δ}}_{j} v_{t} + \tilde{u} \cdot \nabla {\dot{Δ}}_{j} v - Δ {\dot{Δ}}_{j} v - {\dot{Δ}}_{j} P div σ = {\tilde{f}}_{j}, \\ {\dot{Δ}}_{j} σ_{t} + \tilde{u} \cdot \nabla {\dot{Δ}}_{j} σ + {\dot{Δ}}_{j} σ - {\dot{Δ}}_{j} D (v) = - b {\dot{Δ}}_{j} ((tr τ) τ) + {\tilde{g}}_{j}, \end{cases} \end{aligned}

where

\begin{aligned} {\tilde{f}}_{j} & = [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} P] v - {\dot{Δ}}_{j} P (v \cdot \nabla u) = [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] v + {\dot{Δ}}_{j} (Δ^{- 1} \nabla div div (\tilde{u} \otimes v)) - {\dot{Δ}}_{j} P (v \cdot \nabla u), \\ {\tilde{g}}_{j} & = [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j}] σ + {\dot{Δ}}_{j} (v \cdot \nabla τ) - {\dot{Δ}}_{j} (Q (τ, \nabla v) + Q (σ, \nabla \tilde{u})) . \end{aligned}

It is easy to check that

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\tilde{f}}_{j} ‖_{L^{2}} ≲ (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) + ‖ \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}, \end{aligned}

and

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\tilde{g}}_{j} ‖_{L^{2}} & ≲ (‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) ‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ + (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) ‖ \nabla v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} (‖ \nabla τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ \nabla τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) \\ ≲ (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + (‖ \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Now following the similar argument used in Subsection 3.1, we conclude that

\begin{aligned} \frac{d}{d t} (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}}) & ≲ b \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{2}} + \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ {\tilde{f}}_{j} ‖_{L^{2}} + ‖ {\tilde{g}}_{j} ‖_{L^{2}}) \\ ≲ b (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ {\tilde{f}}_{j} ‖_{L^{2}} + ‖ {\tilde{g}}_{j} ‖_{L^{2}}) \\ ≲ b ϵ_{0} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ϵ_{0} (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) + (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} \\ + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

(5.2)

Applying the operator

{\dot{Δ}}_{j} P

to the first equation of system (5.1) and

{\dot{Δ}}_{j} Λ^{- 1} P div

to the second one, gives that

\begin{aligned} {\begin{cases} {\dot{Δ}}_{j} v_{t} + \tilde{u} \cdot \nabla {\dot{Δ}}_{j} v - Δ {\dot{Δ}}_{j} v - Λ {\dot{Δ}}_{j} Λ^{- 1} P div σ = {\tilde{f}}_{j}, \\ {\dot{Δ}}_{j} Λ^{- 1} P div σ_{t} + \tilde{u} \cdot \nabla {\dot{Δ}}_{j} Λ^{- 1} P div σ + {\dot{Δ}}_{j} Λ^{- 1} P div σ + \frac{1}{2} Λ {\dot{Δ}}_{j} v = - b {\dot{Δ}}_{j} Λ^{- 1} P div ((tr τ) τ) + {\tilde{h}}_{j}, \end{cases} \end{aligned}

where

\begin{aligned} {\tilde{h}}_{j} = [\tilde{u} \cdot \nabla, {\dot{Δ}}_{j} Λ^{- 1} P div] σ + {\dot{Δ}}_{j} Λ^{- 1} P div (v \cdot \nabla τ) - {\dot{Δ}}_{j} Λ^{- 1} P div (Q (τ, \nabla v) + Q (σ, \nabla \tilde{u})) . \end{aligned}

After a few calculations, we get that

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ [v \cdot \nabla, Λ] {\dot{Δ}}_{j} Λ^{- 1} P div σ ‖_{L^{2}} \\ ≲ \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ [v \cdot \nabla, {\dot{Δ}}_{j}] P div σ ‖_{L^{2}} + ‖ Λ ([v \cdot \nabla, {\dot{Δ}}_{j}] Λ^{- 1} P div σ) ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} (Λ v \cdot \nabla Λ^{- 1} P div σ) ‖_{L^{2}}) \\ ≲ (‖ Λ^{- 1} P div σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) + (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ {\tilde{τ}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ {\tilde{τ}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + (‖ u^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ u^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ {\tilde{u}}^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ {\tilde{u}}^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}, \end{aligned}

and

\begin{aligned} \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\tilde{h}}_{j} ‖_{L^{2}} & ≲ (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + (‖ \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

Following the similar argument used in Subsection 3.2, we have

\begin{aligned} \frac{d}{d t} (‖ (v^{ℓ}, Λ^{- 1} P div σ^{ℓ}, (2 P div σ - v)^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}}) + ‖ (v^{ℓ}, Λ^{- 1} P div σ^{ℓ}) ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} \\ ≲ b \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} Λ^{- 1} P div ((tr τ) τ) ‖_{L^{2}} + \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ {\tilde{f}}_{j} ‖_{L^{2}} + ‖ {\tilde{h}}_{j} ‖_{L^{2}} \\ + ‖ [v \cdot \nabla, Λ] {\dot{Δ}}_{j} Λ^{- 1} P div σ ‖_{L^{2}}) + b \sum_{j \leq N} 2^{\frac{3}{2} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{2}} + \sum_{j \leq N} 2^{\frac{3}{2} j} ‖ {\tilde{g}}_{j} ‖_{L^{2}} \\ ≲ b \sum_{j \leq N} 2^{\frac{1}{2} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{2}} + \sum_{j \leq N} 2^{\frac{1}{2} j} (‖ {\tilde{f}}_{j} ‖_{L^{2}} + ‖ {\tilde{g}}_{j} ‖_{L^{2}} + ‖ {\tilde{h}}_{j} ‖_{L^{2}} + ‖ [v \cdot \nabla, Λ] {\dot{Δ}}_{j} Λ^{- 1} P div σ ‖_{L^{2}}) \\ ≲ b ϵ_{0} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ϵ_{0} (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) + (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} \\ + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

(5.3)

Motivated by the auxiliary variable

Γ \overset{def}{=} u - Λ^{- 1} ψ

, we estimate the combination

v - Λ^{- 2} P div σ

in an analogous way. We mimic as much as possible the analysis from Subsection 3.3, then show that

\begin{aligned} \frac{d}{d t} (C_{1} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + C_{3} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + C_{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + C_{1} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + C_{3} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + C_{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ b C_{1} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ {\dot{Δ}}_{j} Λ^{- 2} P div ((tr τ) τ) ‖_{L^{p}} + C_{1} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} (‖ {\tilde{f}}_{j} ‖_{L^{p}} + ‖ Λ^{- 1} {\tilde{h}}_{j} ‖_{L^{p}} + ‖ [v \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} Λ^{- 1} P div σ ‖_{L^{p}}) \\ + b C_{2} \sum_{j > N} 2^{\frac{3}{p} j} ‖ {\dot{Δ}}_{j} Λ^{- 1} P div ((tr τ) τ) ‖_{L^{p}} + C_{2} \sum_{j > N} 2^{\frac{3}{p} j} ‖ {\tilde{h}}_{j} ‖_{L^{p}} + b C_{3} \sum_{j > N} 2^{\frac{3}{p} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} + C_{3} \sum_{j > N} 2^{\frac{3}{p} j} ‖ {\tilde{g}}_{j} ‖_{L^{p}}, \end{aligned}

by choosing some positive constants

C_{1} ≫ C_{2} ≫ C_{3}

. The estimate (B.5) in Lemma B.1 leads us to get that

\begin{aligned} \sum_{j > N} 2^{\frac{3}{p} j} ‖ {\dot{Δ}}_{j} ((tr τ) τ) ‖_{L^{p}} ≲ ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}^{2} . \end{aligned}

In view of the estimate (B.3) in Lemma B.1, the other estimate (B.23) in Lemma B.4 and the equality (C.5) in Lemma C.1, we find that

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ [v \cdot \nabla, Λ^{- 1}] {\dot{Δ}}_{j} Λ^{- 1} P div σ ‖_{L^{p}} \\ ≲ ‖ Λ^{- 1} P div σ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} ‖ \nabla v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} (‖ \nabla Λ^{- 1} P div σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) \\ ≲ (‖ σ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

With the help of the estimate (B.3) in Lemma B.1, the estimate (B.20) in Lemma B.4, and the equality (C.1) in Lemma C.1, one can obtain that

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ {\tilde{f}}_{j} ‖_{L^{p}} ≲ (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) . \end{aligned}

It then follows from the estimate (B.5) in Lemma B.1, the estimates (B.20) and (B.22) in Lemma B.4, and the equality (C.3) in Lemma C.1 that

\begin{aligned} \sum_{j > N} 2^{(\frac{3}{p} - 1) j} ‖ Λ^{- 1} {\tilde{h}}_{j} ‖_{L^{p}} & ≲ ‖ σ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}), \\ \sum_{j > N} 2^{\frac{3}{p} j} (‖ {\tilde{g}}_{j} ‖_{L^{p}} + ‖ {\tilde{h}}_{j} ‖_{L^{p}}) & ≲ ‖ σ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} (‖ \nabla τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ ≲ ‖ σ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ‖ \nabla v ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} (‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} \\ \times ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + 2^{- N} (‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} . \end{aligned}

This implies that

\begin{aligned} \frac{d}{d t} (C_{1} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + C_{3} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + C_{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + C_{1} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + C_{3} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + C_{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ b ϵ_{0} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + (C_{2} + C_{3}) c_{0} 2^{- N} (‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + ϵ_{0} (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) . \end{aligned}

This leads us to get that,

\begin{aligned} \frac{d}{d t} (C_{1} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + C_{3} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + C_{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + \frac{C_{1}}{2} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + C_{3} ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + \frac{C_{2}}{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ b ϵ_{0} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ϵ_{0} (‖ \nabla v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ \nabla v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ + (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}), \end{aligned}

(5.4)

where we have used the fact that

\begin{aligned} (C_{2} + C_{3}) c_{0} 2^{- N} (‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ < \frac{C_{1}}{2} ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + \frac{C_{2}}{2} ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}, \end{aligned}

provided that

C_{1} ≫ C_{2} ≫ C_{3}

and

N

large enough. We point out that here, the smallness assumption on

c_{0}

is not needed. Thus, from (5.2)–(5.4), we deduce that

\begin{aligned} \frac{d}{d t} (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ (2 P div σ - v)^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} \\ + ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) + ‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} \\ + ‖ Λ^{- 1} P div σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{5}{2}}} + ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ b ϵ_{0} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + (‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}}) \\ \times (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) . \end{aligned}

To prove the above inequality it suffices to use the fact that

ϵ_{0}

is sufficiently small. The Gronwall lemma ensures that, for all

t \geq 0

\begin{aligned} ‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ Λ^{- 1} P div σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ (2 P div σ - v)^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} \\ + ‖ (v - Λ^{- 2} P div σ)^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} \\ ≲ b ϵ_{0} (\int_{0}^{t} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} d s) \exp \int_{0}^{t} (‖ \nabla \tilde{u} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ τ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{3}{2}}} + ‖ τ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} + 1}}) d s ≲ b ϵ_{0}^{2} e^{ϵ_{0} + c_{0}}, \end{aligned}

which leads to

\begin{aligned} ‖ v^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ Λ^{- 1} P div σ^{ℓ} ‖_{{\dot{B}}_{2, 1}^{\frac{1}{2}}} + ‖ v^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p} - 1}} + ‖ σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} + ‖ Λ^{- 1} P div σ^{h} ‖_{{\dot{B}}_{p, 1}^{\frac{3}{p}}} ≲ b ϵ_{0}^{2} e^{ϵ_{0} + c_{0}} . \end{aligned}

The proof of Theorem 2.14 is therefore complete.

Footnotes

ORCID iDs

Yuhui Chen

Minling Li

Zheng-an Yao

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Yuhui Chen is supported by National Key R&D Program of China (Grant No. 2023YFA1010300) and Shenzhen Science and Technology Program (Grant No. JCYJ20240813151513017). Minling Li is supported by Postdoctoral Fellowship Program of CPSF (Grant No. GZB20240024) and China Postdoctoral Science Foundation (Grant No. 2024M760057 and 2025T180840).

Competing Interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Appendix A. Homogeneous Besov Spaces

In this section, we introduce the definition of homogeneous Besov spaces (see Bahouri et al., 2011 for more details). The homogeneous Littlewood-Paley decomposition relies upon a dyadic partition of unity. Let $C$ be an annulus ${ξ \in R^{3} | 3 / 4 \leq | ξ | \leq 8 / 3} .$ There exists a radial function $φ$ , belonging to $D (C)$ , valued in the interval $[0, 1]$ , and such that

\begin{aligned} \sum_{j \in Z} φ (2^{- j} ξ) = 1, \forall ξ \in R^{3} ∖ {0}, \end{aligned}

\begin{aligned} | j - j^{'} | \geq 2 \Rightarrow Supp φ (2^{- j} \cdot) \cap Supp φ (2^{- j^{'}} \cdot) = \emptyset . \end{aligned}

Defining

\begin{aligned} χ (ξ) \overset{def}{=} 1 - \sum_{j \geq 0} φ (2^{- j} ξ), h (x) \overset{def}{=} F^{- 1} (φ) (x), \tilde{h} (x) \overset{def}{=} F^{- 1} (χ) (x) . \end{aligned}

The homogeneous dyadic blocks

{\dot{Δ}}_{j}

are defined by

\begin{aligned} {\dot{Δ}}_{j} u \overset{def}{=} φ (2^{- j} D) u = 2^{3 j} \int_{R^{3}} h (2^{j} y) u (x - y) d y, \forall j \in Z, \end{aligned}

the homogeneous low-frequency cut-off operators

{\dot{S}}_{j}

are defined by

\begin{aligned} {\dot{S}}_{j} u \overset{def}{=} χ (2^{- j} D) u = 2^{3 j} \int_{R^{3}} \tilde{h} (2^{j} y) u (x - y) d y, \forall j \in Z . \end{aligned}

We denote by

S_{h}^{'} (R^{3})

the space of tempered distributions

u

such that

\begin{aligned} lim_{λ \to \infty} ‖ θ (λ D) u ‖_{L^{\infty}} = 0, \forall θ \in D (R^{3}) . \end{aligned}

Let

u

be in

S_{h}^{'} (R^{3})

, we have the following formal homogeneous Littlewood-Paley decomposition:

\begin{aligned} u = \sum_{j \in Z} {\dot{Δ}}_{j} u . \end{aligned}

Furthermore, the above dyadic decomposition has nice properties of quasi-orthogonality. With our choice of

φ

, we have

\begin{aligned} | i - j | \geq 2 \Rightarrow {\dot{Δ}}_{i} {\dot{Δ}}_{j} u \equiv 0 and | i - j | \geq 5 \Rightarrow {\dot{Δ}}_{i} (S_{j - 1} u {\dot{Δ}}_{j} u) \equiv 0 . \end{aligned}

Next, we introduce the Bernstein’s inequality and an interpolation inequality, which will be frequently used in this article.

In order to study the way the product acts on homogeneous Besov spaces, we shall use the following homogeneous Bony’s decomposition.

Obviously, we do not obtain an estimate in a space of type $L^{ρ} ([0, T]; {\dot{B}}_{p, r}^{s})$ since the time integration has been performed before the summation. This naturally leads to the following definition. Definition A.5

Let $s \in R$ , $(ρ, p, r) \in [1, + \infty]^{3}$ , and $t \in (0, + \infty]$ , we set

\begin{aligned} ‖ u ‖_{{\tilde{L}}_{t}^{ρ} ({\dot{B}}_{p, r}^{s})} \overset{def}{=} ‖ 2^{j s} ‖ {\dot{Δ}}_{j} u ‖_{L_{t}^{ρ} (L^{p})} ‖_{ℓ^{r}} . \end{aligned}

Noticing the Minkowski inequality yields

‖ u ‖_{L_{t}^{ρ} ({\dot{B}}_{p, r}^{s})} \leq ‖ u ‖_{{\tilde{L}}_{t}^{ρ} ({\dot{B}}_{p, r}^{s})}

ρ \geq r

‖ u ‖_{{\tilde{L}}_{t}^{ρ} ({\dot{B}}_{p, r}^{s})} \leq ‖ u ‖_{L_{t}^{ρ} ({\dot{B}}_{p, r}^{s})}

ρ \leq r

Appendix B. Some Useful Lemmas

We prove several key lemmas in this section, which will be used in the proofs of our main theorems. The first lemma concerns commutator estimates.

Following the same approach as in the proof of Lemma B.1, we derive the following two corollaries.

The following lemma concerns product estimates in Besov spaces, which are crucial for analyzing the bilinear terms in systems (1.1) and (1.2).

In Theorem 2.5, we prove that the low frequencies of the solutions lie in ${\dot{B}}_{2, \infty}^{σ}$ , provided that the low frequencies of the initial data belong to ${\dot{B}}_{2, \infty}^{σ}$ . To this end, we use the following lemma to handle the nonlinear terms.

When proving Theorems 2.4 and 2.7, handling the nonlinear terms is challenging due to the absence of the damping term for $\tilde{τ}$ . To address this issue, we focus on the simpler case $p = 2$ . The proof relies on the following two key lemmas. The first lemma focuses on commutator estimates.

The following lemma concerns product estimates.

For the general case $p \in [2, 4]$ , we assume in Theorems 2.11 and 2.12 that both the low- and high-frequency parts of the initial data belong to ${\dot{B}}_{2, \infty}^{σ}$ . To address this, we prove the following two key lemmas for handling the commutator and nonlinear product terms. This allows us to prove that the solutions lie in ${\dot{B}}_{2, \infty}^{σ}$ .

Appendix C. Some Useful Calculations

In this section, we perform some elementary calculations, which will be useful for handling commutators in the proofs of our main theorems.

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The Optimal Time Decay Rates and Vanishing Limit for Three-Dimensional Incompressible Phan-Thien–Tanner and Oldroyd-B Systems in the Critical Besov Spaces

Abstract

Keywords

1. Introduction

2.1. Existence Results

3.1. The Estimate of E 1 ( t )

4. Large-Time Behavior

4.1. The Proof of Theorem 2.5

Footnotes

ORCID iDs

Funding

Competing Interest

Data Availability

Appendix A. Homogeneous Besov Spaces

Appendix B. Some Useful Lemmas

Appendix C. Some Useful Calculations

References

3.1. The Estimate of $E_{1} (t)$