1. Introduction

In this paper, we are interested in only homogeneous flow of incompressible isothermal polymer fluids. The Phan-Thien–Tanner (PTT) model originates from the works of Nhan Phan-Thien and Roger I. Tanner in [Reference Phan-Thien44, Reference Phan-Thien and Tanner45]. This model attempts to describe the behaviour of this complex mixture of polymers and fluids, and as such, it presents numerous challenges, simultaneously at the level of their derivation, that of their numerical simulation and mathematical treatment. This flow is governed by the following continuity and momentum balance equations:

(1.1)\begin{equation} \left\{\begin{array}{@{}l} \mathop{\rm div}\nolimits u =0, \\ u_t +u\cdot \nabla u= \mathop{\rm div}\nolimits T-\nabla p, \\ \end{array}\right. \end{equation}

where $u$, $p$ and $T$ represent the velocity field, the isotropic pressure and the stress tensor, respectively. The stress tensor $T$, corresponding to the rates of creation and destruction of junctions which depend on the instantaneous elastic energy of the network, or equivalently, the average extension of the network strand, is often divided into the following two parts:

\[ T=\mu_1\tau +2\mu D(u), \]

where $\tau$ is the polymers contribution to the stress tensor and $D(u):=(1/2) (\nabla u+(\nabla u)^T)$. Two constants $\mu _1>0$ and $\mu >0$ are the elastic coefficient and the solvent viscosity coefficient, respectively.

Throughout this paper, we shall focus on the linear PTT model, which is a well-studied phenomenological constitutive model for polymers. One can deduce that

\[ \lambda_{1} \frac{\mathcal{D} \tau}{\mathcal{D} t}+\frac{\epsilon \lambda_{1}}{\eta}({\rm tr}\, \tau)\tau=2\eta\left(1-\frac{\lambda_{2}}{\lambda_{1}}\right)D(u), \]

and the ‘objective derivative’ ${\mathcal {D} \tau }/{\mathcal {D} t}$ is denoted as follows:

\[ \frac{\mathcal{D} \tau}{\mathcal{D} t}:=\frac{\partial \tau}{\partial t} +u\cdot \nabla \tau+\tau\Omega(u)-\Omega(u)\tau+\lambda (D(u)\tau+\tau D(u)), \]

where $\Omega (u):=(1/2)(\nabla u-(\nabla u)^T)$, $\lambda _1>0$ is the relaxation time of the fluids, $\lambda _2$ $(0 \leq \lambda _2 <\lambda _1)$ is the retardation time of the fluids, $\eta >0$ is the polymer viscosity coefficient, $\varepsilon$ is a parameter controlling the elongational viscosity coefficient and $\lambda$ is a constant that is typically in $[-1,\,1]$. In particular, we call the system a co-rotational case when $\lambda =0$.

For the sake of conciseness, we set $a={1}/{\lambda _{1}}$, $b={\varepsilon }/{\eta }$ and $\mu _2=({2\eta }/{\lambda _{1}}) (1-{\lambda _{2}}/{\lambda _{1}})$. Two constants $a>0$ and $\mu _2>0$ are associated with the Deborah number $De={\mu _2}/{a}$, which indicates the relation between the characteristic flow time and the elastic time (see [Reference Bird, Armstrong and Hassager4]). The constant $b\geq 0$ is related to the rates of creation and destruction for the polymeric network junctions. Using the notations introduced above, we aim to solve the following three-dimensional incompressible PTT system:

(PTT)\begin{equation} \left\{\begin{array}{@{}ll} u_{t}+u\cdot \nabla u-\mu \triangle u+\nabla p=\mu_1\mathop{\rm div}\nolimits \tau, & (t, x) \in \mathbb{R}^+{\times} \mathbb{R}^3, \\ \tau_t+u\cdot \nabla\tau+ (a+b({\rm tr}\,\tau))\tau+Q(\tau,\nabla u)=\mu_2 D(u), & (t, x) \in \mathbb{R}^+{\times} \mathbb{R}^3, \\ \mathop{\rm div}\nolimits u=0, & (t, x) \in \mathbb{R}^+{\times} \mathbb{R}^3, \\ u|_{t=0}=u_{0}(x), \ \tau|_{t=0}=\tau_0(x), & x \in \mathbb{R}^3. \end{array}\right. \end{equation}

Here, $Q(\tau,\,\nabla u)$ is a given bilinear form

\[ Q(\tau,\nabla u)=\tau \Omega(u)-\Omega(u)\tau+\lambda(D(u)\tau+\tau D(u)). \]

It can be seen that the PTT system is derived from the coupling of the constitutive equation with the incompressible Navier–Stokes equations. The coupling in Navier–Stokes equations comes from an extra stress tensor contributed from the polymers. Without taking into account the rates of creation and destruction for the polymeric network junctions (i.e. $a>0$ and $b=0$), then the second equation of system (PTT) is replaced by

\[ \tau_t+u\cdot \nabla\tau+ a\tau+Q(\tau,\nabla u)=\mu_2 D(u). \]

One reduces to the well-known Oldroyd-B system, which models the motion of some simple types of steady flow of certain idealized elastico-viscous liquids (see [Reference Oldroyd42]). Note that there is a new nonlinear term $b({\rm tr}\,\tau )\tau$ in the constitutive equation of system (PTT), which is a big difference from the Oldroyd-B system. The limit system as $(a,\,b)\rightarrow (0,\,0)$ formally becomes the following three-dimensional incompressible Oldroyd-B system without damping mechanism:

(OB)\begin{equation} \left\{\begin{array}{@{}ll} \widetilde{u}_{t}+\widetilde{u}\cdot \nabla \widetilde{u}-\mu \triangle \widetilde{u}+\nabla \widetilde{p}=\mu_1\mathop{\rm div}\nolimits \widetilde{\tau}, & (t, x) \in \mathbb{R}^+{\times} \mathbb{R}^3, \\ \widetilde{\tau}_t+\widetilde{u}\cdot \nabla\widetilde{\tau}+Q(\widetilde{\tau},\nabla \widetilde{u})=\mu_2 D(\widetilde{u}), & (t, x) \in \mathbb{R}^+{\times} \mathbb{R}^3, \\ \mathop{\rm div}\nolimits \widetilde{u}=0, & (t, x) \in \mathbb{R}^+{\times} \mathbb{R}^3, \\ \widetilde{u}|_{t=0}=u_{0}(x), \ \widetilde{\tau}|_{t=0}=\tau_0(x), & x \in \mathbb{R}^3. \end{array}\right. \end{equation}

The main objective of this paper is to show the relation of the PTT system to the Oldroyd-B system (with or without damping mechanism). Until now, this question actually remains open. In the sequence, we will give an overview on the study for the PTT and Oldroyd-type models.

To begin, we review some mathematical works dedicated to the PTT model. To the best of our knowledge, it is universally known that the PTT model has been widely studied in numerical analysis (see [Reference Bautista, Sánchez, Arcos and Méndez2, Reference Garduño, Tamaddon-Jahromi, Walters and Webster16, Reference Oliveira and Pinho43]). The first mathematical work devoted to the PTT system is by Masmoudi [Reference Masmoudi41]. There, the global existence of weak solutions was proven. Recently, Chen et al. [Reference Chen, Luo and Yao8] proved the global existence of strong solutions for the periodic PTT system without damping mechanism (i.e. $a=0$ and $b\geq 0$). In a more general context, the global well-posedness of the Cauchy problem for the linear and generalized PTT systems in the whole space was achieved by Chen et al. [Reference Chen, Li, Yao and an Yao6, Reference Chen, Luo and Zhai9] in the critical Besov spaces and [Reference Chen, Li, Yao and Yao7, Reference Chen, Luo and Yao10] in Sobolev spaces. In this paper, we would like to study in a sequel the limit process of the PTT system with $0\leq b\leq Ca$ when coefficients $a$ and $b$ go to $0$ simultaneously (or, only $b$ goes to $0$) and aim to prove in particular that the obtained limit is the solution of the corresponding Oldroyd-B system.

We also review some mathematical works dedicated to the Oldroyd-type models. For all we know, the related Oldroyd-type models have been analysed extensively since 1990. The main breakthrough of the Oldroyd-B system was made by Guillopé and Saut [Reference Guillopé and Saut17, Reference Guillopé and Saut18]. In the above studies, they proved the local existence of strong solutions and the global existence of one-dimensional shear flows. In the multi-dimensional case, the local well-posedness in Sobolev spaces was first proven by Fernández-Cara et al. [Reference Fernández-Cara, Guillén and Ortega15]. Later, Chemin and Masmoudi [Reference Chemin and Masmoudi5] not only proved the local well-posedness in the critical Besov spaces, but gave a low bound for the lifespan as well. Then, Feng et al. and Lei et al. gave some blow up criteria in [Reference Feng, Zhu and Zi14, Reference Lei, Masmoudi and Zhou31]. In [Reference Lions and Masmoudi39], the global existence of weak solutions for general initial data in the co-rotational system was studied by Lions and Masmoudi. In regard to strong solutions, the global existence for small smooth perturbations of a stable equilibrium was first presented in [Reference Lin, Liu and Zhang36]. Surprisingly, there are a number of studies concerning the global existence of strong solutions of the Cauchy problem with small initial data by using different methods, for example, see [Reference Chen and Zhang11, Reference Lei, Liu and Zhou32, Reference Lei and Zhou33, Reference Lin, Liu and Zhang36, Reference Zhang and Fang52, Reference Zhu54] and the references cited therein. In particular, the global existence for the Oldroyd-B system without damping mechanism was discussed in [Reference Zhu54]. We should mention here that He and Xu [Reference He and Xu19], Lin and Zhang [Reference Lin and Zhang37] and Sun and Zhang [Reference Sun and Zhang48] have obtained similar results involving the initial-boundary problem of the incompressible viscoelastic fluid models. For a class of large initial data, Fang and Zi [Reference Fang and Zi13] and Jiang and Jiang [Reference Jiang and Jiang24] made some contributions to the global existence of strong solutions. The optimal time decay estimates have been established recently in [Reference Hieber, Wen and Zi20] for the incompressible Oldroyd-B system and similar results in [Reference Wang and Wen50] for the compressible case. For more results concerning the related compressible viscoelastic fluids models, one may refer to [Reference Barrett, Lu and Süli1, Reference Hu and Wang22, Reference Hu and Wu23, Reference Lu and Zhang40, Reference Qian and Zhang46, Reference Zhou, Zhu and Zi53]. It has been shown that the weak solutions and strong solutions exist globally in time.

In addition, we recall some results related to the limit problems. There has been much progress on the limit problems of various flow models in the last few decades, especially on incompressible limit. The mathematical studies of incompressible limit were stated in [Reference Klainerman and Majda28, Reference Lions and Masmoudi38] for Navier–Stokes equations and in [Reference Hu and Wang21, Reference Jiang, Ju and Li25] for magnetohydrodynamic equations. As for the Oldroyd-B system, incompressible limit problem was investigated in [Reference Lei30, Reference Lei and Zhou33]. Besides, Ju et al. investigated the quasineutral limit for the compressible Navier–Stokes–Poisson equations (for both isentropic and non-isentropic cases) in [Reference Ju, Li and Li26, Reference Ju, Li and Wang27]. As in the previous works, the existence of global classical (or weak) solutions for the incompressible case was shown via incompressible limit. More results on the other limit problems of various flow models may be found in [Reference Bian, Yao and Zhu3, Reference Chen and Zhai12, Reference Lai, Wen and Yao29, Reference Li and Titi34, Reference Li and Zhu35, Reference Wang, Wang and Zhang49, Reference Yao, Zhu and Zi51]. Heuristically, we hope that the process of the vanishing limit for the PTT system selects a physical relevant solution of the corresponding Oldroyd-B system.

In this paper, we aim to establish the vanishing limit of the PTT system for incompressible fluids. Let $0\leq a\leq 1$ and $0\leq b\leq Ca$, under the smallness assumptions on the initial data, our first aim is to prove the global existence of strong solutions for the PTT system with the a priori estimates, uniformly with respect to time $t$ and coefficients $a$ and $b$. Our second purpose is to prove the global existence for the limit system via the vanishing limit. Taking advantage of the uniform energy and decay estimates stated earlier, it is possible to pass to the limit when $a$ and $b$ go to 0 simultaneously (or, only $b$ goes to $0$). This result justifies that the limit is indeed a strong solution of the corresponding Oldroyd-B system and has the required regularity. In addition, we investigate the rate of convergence of the PTT system towards the corresponding Oldroyd-B system for solutions having the above regularity. We shall see that the rate strongly depends on time $t$ and coefficients $a$ and $b$. Moreover, we prove the local a priori estimates in large norm. A similar result holds true for some time interval $[0,\,T]$. As a consequence, this is the first paper devoted to establishing a relation between the PTT system and the Oldroyd-B system with or without damping mechanism.

Notation: We shall introduce some notations. Unless otherwise indicated, we omit $\mathbb {R}^3$ in the rest of the proof. For the sake of simplicity, we restrict our attention here to the case where $\mu =\mu _1=\mu _2=1$. Throughout this paper, $C$ denotes a generic positive constant independent of $t$, $a$ and $b$.

We can now describe our main results. We first state the following theorem, the global well-posedness for system (PTT) with small initial data. In passing, we will establish suitable a priori estimates, uniformly in both time $t$ and coefficients $a$ and $b$.

Theorem 1.1 Let $0\leq a\leq 1$ and $0\leq b\leq Ca$. Assume that the initial data $(u_0,\,\tau _0)$ satisfies $\mathop {\rm div}\nolimits u_0=0$, $(\tau _0)_{ij}=(\tau _0)_{ji}$ and $({|\nabla |^{-1}} u_0,\,{|\nabla |^{-1}}\tau _0)\in H^3(\mathbb {R}^3)$. There exists a sufficiently small positive constant $\delta _0$, such that if

\[ \mathcal{E}(0) :=\|({|\nabla|^{{-}1}} u_0,{|\nabla|^{{-}1}}\tau_0)\|^2_{H^3(\mathbb{R}^3)}\leq \delta_0, \]

then the problem (PTT) admits a unique global classical solution $(u,\, \tau )$, which satisfies

\begin{align*} & \sup_{0\leq s\leq t}\left(\|{|\nabla|^{{-}1}} u(s)\|_{H^3(\mathbb{R}^3)}^2+\|{|\nabla|^{{-}1}}\tau(s)\|_{H^3(\mathbb{R}^3)}^2\right)\\ & \quad+\int_0^t\left(\|u(s)\|_{H^3(\mathbb{R}^3)}^2+a\|{|\nabla|^{{-}1}}\tau(s)\|_{H^3(\mathbb{R}^3)}^2\right)\,{\rm d}s\leq C_1\mathcal{E}(0), \quad \forall\ t\geq0, \end{align*}

where $C_1$ is a positive constant independent of $t$, $a$ and $b$. Moreover, the following time decay estimates hold true$:$

\begin{align*} & \|\nabla u(t)\|_{H^1(\mathbb{R}^3)}^2+\|\mathbb{P}\mathop{\rm div}\nolimits\tau(t)\|_{H^1(\mathbb{R}^3)}^2\leq C_1\mathcal{E}(0)(1+t)^{{-}2},\quad \forall\ t\geq0,\\ & \|{\rm tr}\, \tau(t)\|_{H^2(\mathbb{R}^3)}^2\leq C_1\mathcal{E}(0)\,{\rm e}^{{-}at},\quad\forall\ t\geq0. \end{align*}

Remark 1.5 Actually, besides the assumptions of theorem 1.1, we assume that $(u_0,\,\tau _0)\in L^1(\mathbb {R}^3)$. For any fixed constant coefficients $a\geq 0$ and $0\leq b\leq Ca$, we deduce not only that there exists a unique global classical solution $(u,\,\tau )$, but that the following time decay estimates hold true:

\[ \|\nabla u(t)\|_{H^1(\mathbb{R}^3)}+\|\mathbb{P}\mathop{\rm div}\nolimits\tau(t)\|_{H^1(\mathbb{R}^3)}\leq C_1(1+t)^{-(5/4) },\quad\forall\ t\geq0, \]

where $C_1$ is a positive constant which does not depend on $t$, $a$ and $b$. This conclusion may be proved by using standard energy method, the proof of which is similar to that of this paper. We omit the details, since this is not what we focus on in this paper.

Remark 1.6 Furthermore, if we also assume that $(u_0,\,\tau _0)\in L^1(\mathbb {R}^3)$. For any fixed constant coefficients $a>0$ and $0\leq b\leq Ca$, we can infer that the following optimal time decay estimates, for $k=0,\,1,\,2$:

\[ \|\nabla^ku(t)\|_{L^2(\mathbb{R}^3)}+\|\nabla^k\tau(t)\|_{L^2(\mathbb{R}^3)}\leq C_2(1+t)^{-(3/4)- (k/2) },\quad\forall\ t\geq0, \]

where $C_2$ is a positive constant independent of $t$, but may depend on $a$ and $b$. The proof is similar to that of the work in [Reference Hieber, Wen and Zi20].

We are concerned with the vanishing limit for system (PTT) in the sequel. According to theorem 1.1, we shall prove the convergence of the solution $(u,\,\tau )$ to system (PTT) with $0\leq b\leq Ca$ when $(a,\,b)$ tends to $(0,\,0)$. The aim is to show that when $(a,\,b)$ tends to $(0,\,0)$, the global solution $(u,\,\tau )$ of system (PTT) with $0\leq b\leq Ca$ converges to some limit solution $(\widetilde {u},\,\widetilde {\tau })$, which is a global solution of system (OB).

Theorem 1.7 For any $T>0$, let $(u,\,\tau )$ be a strong solution of system (PTT) in the time interval $[0,\,T]$ given by theorem 1.1. Then there exists $(\widetilde {u},\,\widetilde {\tau })$ such that for $0\leq b\leq Ca$, if $(a,\,b)\rightarrow (0,\,0)$, it holds true for any $s \in (0,\,1/2)$,

\begin{align*} & {|\nabla|^{{-}1}} u\rightarrow {|\nabla|^{{-}1}}\widetilde{u}\quad {\rm{strongly}}\; {\rm{in}}\;\mathcal{C}([0,T];H_{loc}^{3-s}(\mathbb{R}^3)),\\ & {|\nabla|^{{-}1}}\tau\rightarrow {|\nabla|^{{-}1}}\widetilde{\tau} \quad {\rm{strongly}}\; {\rm{in}}\; \mathcal{C}([0,T];H_{loc}^{3-s}(\mathbb{R}^3)). \end{align*}

Moreover, $(\widetilde {u},\,\widetilde {\tau })$ is a strong solution of system (OB) in the time interval $[0,\,T]$, and satisfies

\begin{align*} & {|\nabla|^{{-}1}}\widetilde{u}\in \mathcal{C} ([0,T];H^{3}(\mathbb{R}^3))\cap \mathcal{C}^1 ([0,T];H^{1}(\mathbb{R}^3)),\quad \widetilde{u} \in L^{2}((0,T);H^3(\mathbb{R}^3)),\\ & {|\nabla|^{{-}1}}\widetilde{\tau}\in \mathcal{C} ([0,T];H^{3}(\mathbb{R}^3))\cap \mathcal{C}^1 ([0,T];H^{2}(\mathbb{R}^3)). \end{align*}

In addition, there exists a positive constant $C_1$, independent of $t$, $a$ and $b$, such that

\[ \|{|\nabla|^{{-}1}} \widetilde{u}(t)\|_{H^3(\mathbb{R}^3)}^2+\|{|\nabla|^{{-}1}}\widetilde{\tau}(t)\|_{H^3(\mathbb{R}^3)}^2+\int_0^t\|\widetilde{u}(s)\|_{H^3(\mathbb{R}^3)}^2\,{\rm d}s \leq C_1\mathcal{E}(0), \quad \forall\ t\geq0. \]

Finally, we investigate the rate of convergence of system (PTT) towards system (OB) for solutions having the above regularity. We shall see that the rate strongly depends on $t$, $a$ and $b$.

Theorem 1.8 Let $(u,\,\tau )$ be a strong solution of system (PTT) given by theorem 1.1 and $(\widetilde {u},\,\widetilde {\tau })$ be a strong solution of system (OB) given by theorem 1.7. Then there exists a positive constant $C_1$, independent of $t$, $a$ and $b$, such that

\[ \|(u-\widetilde{u},\tau-\widetilde{\tau})(t)\|_{H^1(\mathbb{R}^3)}^2 \leq C_1 \left(a^2 +b^2\right)t\,{\rm e}^{C_1(t+1)}, \quad \forall\ t\geq0. \]

Remark 1.9 On the one hand, there is no derivative in the additional term $b({\rm tr}\,\tau )\tau$ of system (PTT), in contrast to the Oldroyd-B system. Then the proof of local well-posedness for system (PTT) is similar to that of the Oldroyd-B system. For any fixed constant coefficients $0\leq a\leq 1$ and $0\leq b\leq Ca$, let $(u,\,\tau )$ be a local solution of system (PTT) in the time interval $[0,\,T]$ with the general initial data, there exists $(\widetilde {u},\,\widetilde {\tau })$ be a local solution of system (OB) in the time interval $[0,\,T]$ if $(a,\,b)\rightarrow (0,\,0)$. Further, then there exist some $T>0$ and a positive constant $\widetilde C_1$, independent of $a$ and $b$, but may depend on $T$, such that for any $0\leq t\leq T$, we have

\[ \|(u-\widetilde{u},\tau-\widetilde{\tau})(t)\|_{H^1(\mathbb{R}^3)}^2 \leq \widetilde C_1 \left(a^2 +b^2\right). \]

The proof of the above estimate may be obtained by arguing as in theorem 1.8, which will not be explained in detail in this paper. In fact, it turns out to be a simpler problem.

In the last part of this section, we present the main ideas of the proofs of theorems 1.1, 1.7 and 1.8. From the point of view of the limit system (OB), it will be difficult to establish the a priori estimates, uniformly in both time $t$ and coefficients $a$ and $b$, because system (OB) (i.e. $a=0$ and $b=0$) loses damping influence about $\tau$. In order to overcome this difficulty, motivated by the result presented in [Reference Zhu54], we study the following linearized system:

\[ \left\{\begin{array}{@{}l} u_{t}-\triangle u=\mathbb{P} \mathop{\rm div}\nolimits \tau, \\ \tau_t=D(u), \end{array} \right. \]

where $\mathbb {P}:=\mathbb {I}-\triangle ^{-1}\nabla \mathop {\rm div}\nolimits$ is the Leray projection operator. We decouple the linearized system and find that both $u$ and $\mathbb {P} \mathop {\rm div}\nolimits \tau$ satisfy the following damped wave equation:

\[ W_{tt}-\triangle W_t-\frac{1}{2}\triangle W=0. \]

Therefore, we could obtain the time decay estimates of $u$ and $\mathbb {P} \mathop {\rm div}\nolimits \tau$, which are useful to deal with the nonlinear terms.

More specifically, the standard energy method be performed in system (PTT) then gives a bound on

\[ \|{|\nabla|^{{-}1}} u\|_{L^\infty_t(H^3)}+\|{|\nabla|^{{-}1}}\tau\|_{L^\infty_t(H^3)}+\|u\|_{L^2_t(H^3)}+a\|{|\nabla|^{{-}1}} \tau\|_{L^2_t(H^3)}. \]

Due to the presence of coefficient $a$ in the dissipative term for $\tau$, the standard energy estimates above alone is not enough to prove the global-in-time estimates, uniformly in coefficients $a$ and $b$. Indeed, the nonlinear terms $u\cdot \nabla u$, $u\cdot \nabla \tau$, $Q(\tau,\,\nabla u)$ and $b({\rm tr}\,\tau )\tau$ cannot be dealt with here. For this, we use the equation for $u$ in system (PTT) to get the dissipation estimate for $\mathbb {P}\mathop {\rm div}\nolimits \tau$. In fact, due to the loss of the dissipation estimate for $\tau$, we will establish the time decay estimates of $u$ and $\mathbb {P} \mathop {\rm div}\nolimits \tau$ to deal with the nonlinear terms $u\cdot \nabla u$, $u\cdot \nabla \tau$ and $Q(\tau,\,\nabla u)$. It suffices to prove a bound on

\begin{align*} & \sup_{0\leq s \leq t}(1+s)^{2}\left(\|\nabla u(s)\|_{H^1}^2+\|\mathbb{P} \mathop{\rm div}\nolimits\tau(s)\|_{H^1}^2\right)\\ & \quad+\int_0^t (1+s)^{2}\left(\|\nabla^2u(s)\|_{H^1}^2+\|\nabla\mathbb{P} \mathop{\rm div}\nolimits\tau(s)\|_{L^2}^2+a\|\mathbb{P} \mathop{\rm div}\nolimits\tau(s)\|_{H^1}^2\right)\,{\rm d}s. \end{align*}

Note that, however, it does not seem possible to handle the nonlinear term $b({\rm tr}\,\tau )\tau$. This is another difficult task. To achieve this, we may use a method as in [Reference Chen, Li, Yao and Yao7, Reference Chen, Luo and Yao8, Reference Chen, Luo and Yao10]. We shall make full use of the structure of the constitutive equation of system (PTT) to obtain the time decay estimate of the unknown good quantity ${\rm tr}\,\tau$. As a consequence, we prove the global-in-time estimates for system (PTT) with small initial data, uniformly in coefficients $a$ and $b$ (see theorem 1.1). Moreover, we prove the convergence of the global solution $(u,\,\tau )$ to system (PTT) with $0\leq b\leq Ca$ when $(a,\,b)\rightarrow (0,\,0)$, and the global solution $(\widetilde {u},\,\widetilde {\tau })$ is obtained by passing to the limit. Hence, $(\widetilde {u},\,\widetilde {\tau })$ be a global solution of system (OB) (see theorem 1.7). In addition, a rate of convergence involving explicit norms will be obtained by using energy estimate (see theorem 1.8). On the one hand, according to the local well-posedness for the PTT system with general initial data, we also obtain the similar results (see remark 1.9). We mention in passing that in the case $a>0$ and only set $b\rightarrow 0$, similar results are also true (see remark 1.10).

The rest of this paper is arranged as follows. In §2, we recall a few basic lemmas. Section 3 is devoted to proving the suitable a priori estimates for the PTT system, which will be useful for proving our main results. The proofs of theorems 1.1, 1.7 and 1.8 are presented in § 4.