Global regularity for Oldroyd-B model with only stress tensor dissipation

Abstract

In this paper, we consider the d-dimensional ( $d ⩾ 2$ ) Oldroyd-B model with only dissipation in the equation of stress tensor, and establish a small data global well-posedness result in critical $L^{p}$ framework. Particularly, we give a positive answer to the problem proposed recently by Wu-Zhao (J. Differ. Equ. 316 (2022)) involving the upper bound for the time integral of the lower frequency piece of the stress tensor, and show that it is indeed independent of the time. Moreover, we improve the results in (J. Math. Fluid Mech. 24 (2022)) by relaxing the space dimension $d = 2, 3$ to any $d ⩾ 2$ .

Keywords

Oldroyd-B model tensor dissipation global solutions time independent bound

1. Introduction and main results

In this paper, we study the incompressible Oldroyd-B model of the non-Newtonian fluid in $R^{+} \times R^{d}$ , $\begin{matrix} (1.1) & \{\begin{matrix} \partial_{t} u + (u \cdot \nabla) u - ν Δ u + \nabla P = μ_{1} div τ, \\ \partial_{t} τ + (u \cdot \nabla) τ - η Δ τ + α τ + Q (τ, \nabla u) = μ_{2} D (u), \\ div u = 0, \\ u (0, x) = u_{0} (x), τ (0, x) = τ_{0} (x), \end{matrix} \end{matrix}$ where u denotes the velocity, $τ = τ_{i, j}$ is the non-Newtonian part of the stress tensor (τ is a $d \times d$ symmetric matrix here) and P is a scalar pressure of fluid. $D (u)$ is the symmetric part of the velocity gradient, $\begin{matrix} D (u) = \frac{1}{2} (\nabla u + {(\nabla u)}^{⊤}) . \end{matrix}$ The Q above is a given bilinear form: $\begin{matrix} Q (τ, \nabla u) = τ Ω (u) - Ω (u) τ + b (D (u) τ + τ D (u)), \end{matrix}$ where b is a parameter in $[- 1, 1]$ , $Ω (u)$ is the skew-symmetric part of $\nabla u$ , i.e. $\begin{matrix} Ω (u) = \frac{1}{2} (\nabla u - {(\nabla u)}^{⊤}) . \end{matrix}$ The parameters α, ν, η are non-negative constants and they are specific to the characteristic of the considered material. ν is the well-known viscous coefficient and η is the stress coefficient. Following [14,17], we see that $α = \frac{2 ν}{λ_{1}} (1 - θ)$ , where $θ = \frac{λ_{2}}{λ_{1}}$ , $λ_{1}$ is the retardation time and $λ_{2}$ stands the relaxation time.

The Oldroyd-B model describes the motion of some viscoelastic flows. Formulations about viscoelastic flows of Oldroyd-B type are first established by Oldroyd in [23]. For more detailed physical background and derivations about this model, we refer the readers to [2,9,20,23].

The well-posedness of the system (1.1) has been extensively and continuously studied. Guillopé and Saut [18,19] obtained the local solutions with large initial data and showed that these solutions are global when the coupling and the initial data are small enough. In the co-rotational case, i.e., $b = 0$ , Lions and Masmoudi established the existence of global weak solution in their work [22].

Now let us introduce some global existence results of (1.1) on viscous case and non-viscous case. When $ν \neq 0$ and $η = 0$ , Chemin and Masmoudi [5] first obtained the local solutions and global small solutions in the critical Besov spaces with $α > 0$ . They get the global small solutions when the initial data and coupling parameters are small ( $μ_{1} μ_{2} ⩽ c α ν$ ), which means that coupling effect between the two equations is less important than the viscosity. Inspired by the works [3,7], Zi, Fang and Zhang improved their results in the critical $L^{p}$ framework for the case of non-small coupling parameters in [31].

We point out that the above results of Oldroyd-B model concern the situation that the kinematic dissipation and damping mechanism exist. Some recent results dealt with the case that there is only kinematic dissipation or stress tensor dissipation, i.e., the systems do not have damping term. Zhu [30] got small global smooth solutions of the $3 D$ Oldroyd-B model with $η = 0$ , $α = 0$ by observing the linearization part of the system satisfies the damped wave equation. Inspired by the works of Zhu [30] and Danchin in [11], Chen and Hao [6] extended this small data global solution in Sobolev spaces to the critical Besov spaces. Wu and Zhao [27] and Zhai et al. [29] established the small data global well-posedness in critical Besov spaces for fractional dissipation of velocity, respectively. Moreover, Zhai [28] constructed global solutions for a class of highly oscillating initial velocities by investigating the special structure of the linearization part of system.

When $ν = 0$ and $η \neq 0$ , Elgindi and Rousset [16] established a global large solution in a certain sense by building a new quantity to avoid singular operators (they still need $α > 0$ ). Later, Liu and Elgindi extend these results to the 3D case in [15]. Constantin, Wu, Zhao and Zhu [10] established the small data global solution in the d-dimensional space for the case of general tensor dissipation and no damping mechanism. Recently, Wu and Zhao [26] investigated the small data global well-posedness in Besov spaces for fractional tensor dissipation. Very recently, when the dimension $d = 2, 3$ , Chen, Liu, Qin and Ye [8] improved their results in the critical $L^{p}$ framework by using the Lagrangian coordinates in the high frequency. About the decay of the solution about the Oldroyd-B model, we suggest the readers to see the works [21,25].

Inspired by above works, in this paper, we study the following d-dimensional ( $d ⩾ 2$ ) Oldroyd-B model with only dissipation in the equation of stress tensor, and try to answer an interesting problem proposed by Wu-Zhao in [27], $\begin{matrix} (1.2) & \{\begin{matrix} \partial_{t} u + (u \cdot \nabla) u + \nabla P = μ_{1} div τ, \\ \partial_{t} τ + (u \cdot \nabla) τ + η Λ^{2 β} τ + Q (τ, \nabla u) = μ_{2} D (u), \\ div u = 0, \\ u (0, x) = u_{0} (x), τ (0, x) = τ_{0} (x), \end{matrix} \end{matrix}$ where $Λ : = {(- Δ)}^{\frac{1}{2}}$ and $β \in (0, 1]$ . In order to state our results later, let us first introduce the Wu-Zhao’s result.

Theorem 1.1 ([27]).

let $d ⩾ 2$ and $μ_{1}, μ_{2}, η > 0$ . Assume $\begin{matrix} either \frac{1}{2} < β ⩽ 1 or β = \frac{1}{2} with η^{2} ⩾ C μ_{1} μ_{2}, \end{matrix}$ where $C > 0$ is a pure constant. Then there exists a small constant ε such that if $τ_{0} \in {\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}$ , $u_{0} \in {\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β} \cap {\dot{B}}_{2, 1}^{\frac{d}{2} + 2 β - 1}$ satisfies $div u_{0} = 0$ and $\begin{matrix} ‖ u_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β} \cap {\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β}} + ‖ τ_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}} ⩽ ε, \end{matrix}$ then ( 1.2 ) has a unique global solution $(u, τ)$ satisfying $\begin{array}{l} u \in C (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β} \cap {\dot{B}}_{2, 1}^{\frac{d}{2} + 2 β - 1}) \cap L^{1} (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}); \\ τ \in C (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β} \cap {\dot{B}}_{2, 1}^{\frac{d}{2}}), τ \in L^{1} (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} + 1} \cap {\dot{B}}_{2, 1}^{\frac{d}{2} + 2 β}) . \end{array}$ In fact, we have $\begin{matrix} \tilde{E} (t) : = {\tilde{E}}^{l} (t) + {\tilde{E}}^{h} (t) ⩽ ε, \end{matrix}$ where $\begin{array}{l} {\tilde{E}}^{l} (t) : = sup_{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β}}^{l} + sup_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β}}^{l} + \int_{0}^{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} + \int_{0}^{t} {‖ Λ^{- 1} P div τ ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}, \\ {\tilde{E}}^{h} (t) : = sup_{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 2 β - 1}}^{h} + sup_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2}}}^{h} + \int_{0}^{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{h} d t^{'} + \int_{0}^{t} {‖ Λ^{- 1} P div τ ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 2 β}}^{h} d t^{'} . \end{array}$ Moreover, one has $\begin{matrix} (1.3) & \int_{0}^{t} {‖ τ (t^{'}) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} ≲ ε + ε t, \end{matrix}$ where $P$ is a standard Leray projection onto divergence free vector fields, i.e. $P = Id - \nabla Δ^{- 1} div$ .

In the paper [27], Wu and Zhao proposed the following interesting problem:

if the upper bound ( 1.3 ) for the time integral of the lower frequency piece of τ can be improved to a time independent bound?

Recently, Chen, Liu, Qin and Ye [8] extended Wu and Zhao’s result to the critical $L^{p}$ framework in $d = 2, 3$ , but the lower frequency piece of τ is still $\int_{0}^{t} ‖ Λ^{- 1} P div τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}$ . Even the difference between the term $Λ^{- 1} P div τ$ and the term τ is a zero degree multiplier, we cannot get the following estimate holds: $\begin{matrix} \int_{0}^{t} {‖ τ (t^{'}) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} ≲ \int_{0}^{t} {‖ Λ^{- 1} P div τ ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} . \end{matrix}$ The reason that we can not get the estimate of the term $\int_{0}^{t} ‖ τ (t^{'}) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}$ can be summarized as follows. When we use the energy estimate to tackle the $E^{l} (t)$ , we cannot properly estimate the term ${sup}_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β}}^{l} + \int_{0}^{t} ‖ τ (t^{'}) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}$ by solely using the second equation, because we just get a trivial estimate $\begin{matrix} sup_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1 - 2 β}}^{l} + \int_{0}^{t} {‖ τ (t^{'}) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} ≲ \tilde{E} (t) . \end{matrix}$ Then we in turn employ the method in [8,27] to estimate the low frequency part of solutions, taking advantage of the following equality, $\begin{matrix} \int_{R^{d}} div τ \cdot u d x + \int_{R^{d}} D (u) : τ = 0, \end{matrix}$ we try to estimate the term $\begin{matrix} sup_{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + sup_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} \end{matrix}$ by adding the variant two equations after some proper $L^{2}$ inner product (cf. [8, (3.16)]). However, the dissipation part $\frac{2^{2 j} ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}}^{2}}{‖ {\dot{Δ}}_{j} u ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}}}$ can not be estimated accurately and thus becomes a useless term, which led to a failure of the expected estimate of $\int_{0}^{t} ‖ τ (t^{'}) ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}$ , it is worth to point out that the essential reason of the invalidity of the method [8,27] is that the terms $div (τ)$ and $D u$ can not be treated as good terms.

Motivated by the work of Chen et al. [7] and based on some crucial observations, we succeed in estimating the low frequency part of solutions by the Duhamel’s principle and some detailed estimates. We first present the explicit expression of the Green matrix involving the terms $\overset{ˇ}{A}$ , $\overset{ˇ}{B}$ and $\overset{ˇ}{C}$ (see Lemma 3.1), then, as shown in paper [8], we can obtain the smoothing effect by the key observations of the relationship between these terms and the Green matrix in [7] (cf. Proposition 3.1). Finally, to conquer the difficulties aroused by the estimate of the nonlinear term and the space dimension $d ⩾ 2$ , we apply the Bony decomposition and an important commutator estimate to achieve our aim.

Therefore, we can give an affirmative answer to the important problem proposed by Wu-Zhao [27]. Precisely, we have the following more generalized results involving the small data global existence and regularity of solution for system (1.2).

Theorem 1.2.
Suppose $d ⩾ 2$ and $β = 1$ . Let p satisfy $\begin{matrix} (1.4) & 2 ⩽ p ⩽ min (4, \frac{2 d}{d - 2}) and, additionally, p \neq 4 if d = 2 . \end{matrix}$ Then there exists a positive constant $c_{0}$ , so that for any initial $(u_{0}, τ_{0})$ satisfying ${(u_{0}, τ_{0})}^{l} \in {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}$ , $u_{0}^{h} \in {\dot{B}}_{p, 1}^{\frac{d}{p} + 1}$ , $τ_{0}^{h} \in {\dot{B}}_{p, 1}^{\frac{d}{p}}$ with $div u_{0} = 0$ and $\begin{matrix} {‖ (u_{0}, τ_{0}) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ u_{0} ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h} + ‖ τ_{0} ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h} ⩽ c_{0}, \end{matrix}$ the system ( 1.2 ) admits a unique global solution $(u, τ)$ such that $\begin{array}{l} u^{l} \in C (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}), u^{h} \in C (R^{+}; {\dot{B}}_{p, 1}^{\frac{d}{p} + 1}), u^{l} \in L^{1} (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}), u^{h} \in L^{1} (R^{+}; {\dot{B}}_{p, 1}^{\frac{d}{p} + 1}), \\ τ^{l} \in C (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} - 1}), τ^{h} \in C (R^{+}; {\dot{B}}_{p, 1}^{\frac{d}{p}}), τ^{l} \in L^{1} (R^{+}; {\dot{B}}_{2, 1}^{\frac{d}{2} + 1}), τ^{h} \in L^{1} (R^{+}; {\dot{B}}_{p, 1}^{\frac{d}{p} + 2}) . \end{array}$ Moreover, we obtain $\begin{matrix} sup_{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + sup_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + \int_{0}^{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} + \int_{0}^{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} ⩽ c_{0}, \end{matrix}$ and $\begin{matrix} sup_{t} ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h} + sup_{t} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h} + \int_{0}^{t} ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h} d t^{'} + \int_{0}^{t} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 2}}^{h} d t^{'} ⩽ c_{0} . \end{matrix}$
Remark 1.1.
Compared to the previous result in [8], we get a better estimate of the low frequency part of solution by using the accurate term $\int_{0}^{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}$ instead of the term $\int_{0}^{t} ‖ Λ^{- 1} P div τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}$ . Precisely, we can get $\begin{matrix} (1.5) & \int_{0}^{t} {‖ τ (t^{'}) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} ≲ c_{0} . \end{matrix}$ So we give a positive answer to the problem in [27] when $β = 1$ . We also believe the result is valid for the fractional dissipation of τ via our method. On the other hand, we remove the restriction on the space dimension in [8], i.e., we extend the results in [8] $(d = 2, 3)$ to any dimension $d ⩾ 2$ .

Notation. Throughout the paper, we denote the norms of usual Lebesgue space $L^{p} (R^{d})$ by $‖ u ‖_{L^{p}}^{p} = \int_{R^{d}} | u |^{p} d x$ , $for 1 ⩽ p < \infty$ . $C_{i}$ and C denote different positive constants in different places.

The bootstrap argument will be employed in our proof. A rigorous statement of the abstract bootstrap principle can be found in T. Tao’s book (see [24]).

The paper is organized as follows. In Section 2, we give the tools (Littlewood-Paley decomposition and paradifferential calculus) and some useful prior estimates in Besov spaces. In Section 3, we complete the proof of the Theorem 1.2.
2. Preliminaries

In this section, we will recall some properties about the Littlewood-Paley decomposition and Besov spaces. For more details, we refer the readers to the [1].

Proposition 2.1 ([1]).

Let $C$ be the annulus ${ξ \in R^{d} : \frac{3}{4} ⩽ | ξ | ⩽ \frac{8}{3}}$ . There exist radial functions χ and φ, valued in the interval $[0, 1]$ , belonging respectively to $D (B (0, \frac{4}{3}))$ and $D (C)$ , and such that $\begin{array}{l} \forall ξ \in R^{d}, χ (ξ) + \sum_{j ⩾ 0} φ (2^{- j} ξ) = 1, \\ \forall ξ \in R^{d} ∖ {0}, \sum_{j \in Z} φ (2^{- j} ξ) = 1, \\ | j - j^{'} | ⩾ 2 \Rightarrow Supp φ (2^{- j} \cdot) \cap Supp φ (2^{- j^{'}} \cdot) = \emptyset, \\ j ⩾ 1 \Rightarrow Supp χ (\cdot) \cap Supp φ (2^{- j} \cdot) = \emptyset . \end{array}$ The set $\tilde{C} = B (0, \frac{2}{3}) + C$ is an annulus, and we have $\begin{matrix} | j - j^{'} | ⩾ 5 \Rightarrow 2^{j} C \cap 2^{j^{'}} \tilde{C} = \emptyset . \end{matrix}$ Further, one has $\begin{array}{l} \forall ξ \in R^{d}, \frac{1}{2} ⩽ χ^{2} (ξ) + \sum_{j ⩾ 0} φ^{2} (2^{- j} ξ) ⩽ 1, \\ \forall ξ \in R^{d} ∖ {0}, \frac{1}{2} ⩽ \sum_{j \in Z} φ^{2} (2^{- j} ξ) ⩽ 1 . \end{array}$

Definition 2.1 ([1]).

Let u be a tempered distribution in $S^{'} (R^{d})$ and $F$ be the Fourier transform and $F^{- 1}$ be its inverse. For all $j \in Z$ , define $\begin{array}{l} Δ_{j} u = 0 if j ⩽ - 2, Δ_{- 1} u = F^{- 1} (χ F u), \\ Δ_{j} u = F^{- 1} (φ (2^{- j} \cdot) F u) if j ⩾ 0, S_{j} u = \sum_{j^{'} < j} Δ_{j^{'}} u . \end{array}$ Then the nonhomogeneous Littlewood-Paley decomposition is given as follows: $\begin{array}{l} u = \sum_{j \in Z} Δ_{j} u in S^{'} (R^{d}) . \end{array}$ Let $s \in R, 1 ⩽ p, r ⩽ \infty$ . The nonhomogeneous Besov space $B_{p, r}^{s} (R^{d})$ is defined by $\begin{matrix} B_{p, r}^{s} = B_{p, r}^{s} (R^{d}) = {u \in S^{'} (R^{d}) : ‖ u ‖_{B_{p, r}^{s} (R^{d})} = {‖ {(2^{j s} ‖ Δ_{j} u ‖_{L^{p} (S^{d})})}_{j} ‖}_{l^{r} (Z)} < \infty} . \end{matrix}$

Definition 2.2 ([1]).

The homogeneous dyadic blocks ${\dot{Δ}}_{j}$ are defined on the tempered distributions by $\begin{array}{l} {\dot{Δ}}_{j} u = φ (2^{- j} D) u : = F^{- 1} (φ (2^{- j \cdot}) \hat{u}), \\ {\dot{S}}_{j} u = \sum_{j^{'} ⩽ j - 1} {\dot{Δ}}_{j^{'}} u . \end{array}$

Definition 2.3.
We denote by $S_{h}^{'}$ the space of tempered distributions u such that $\begin{matrix} lim_{j \to - \infty} {\dot{S}}_{j} u = 0 in S^{'} . \end{matrix}$

The homogeneous Littlewood-Paley decomposition is defined as $\begin{matrix} u = \sum_{j \in Z} {\dot{Δ}}_{j} u, for u \in S_{h}^{'} . \end{matrix}$
Definition 2.4.
For $s \in R$ , $1 ⩽ p ⩽ \infty$ , the homogeneous Besov space ${\dot{B}}_{p, r}^{s}$ is defined as $\begin{matrix} {\dot{B}}_{p, r}^{s} : = {u \in S_{h}^{'}, ‖ u ‖_{{\dot{B}}_{p, r}^{s}} < \infty}, \end{matrix}$ where the homogeneous Besov norm is given by $\begin{matrix} ‖ u ‖_{{\dot{B}}_{p, r}^{s}} : = {‖ {2^{j s} ‖ {\dot{Δ}}_{j} u ‖_{L^{p}}}_{j} ‖}_{l^{r}} . \end{matrix}$

In this paper, we use the “time-space” Besov spaces or Chemin–Lerner space first introduced by Chemin and Lerner in [4].
Definition 2.5.
Let $s \in R$ and $0 < T ⩽ + \infty$ . We define $\begin{matrix} ‖ u ‖_{{\tilde{L}}_{T}^{q} (B_{p, 1}^{s})} : = \sum_{j \in Z} 2^{j s} {(\int_{0}^{T} {‖ Δ_{j} u (t) ‖}_{L^{p}}^{q} d t)}^{\frac{1}{q}}, \end{matrix}$ for $p, q \in [1, \infty)$ and with the standard modification for $p, q = \infty$ .

By the Minkowski’s inequality, it is easy to verify that $\begin{matrix} ‖ u ‖_{{\tilde{L}}_{T}^{λ} (B_{p, r}^{s}))} ⩽ ‖ u ‖_{L_{T}^{λ} (B_{p, r}^{s}))} if λ ⩽ r, \end{matrix}$ and $\begin{matrix} ‖ u ‖_{{\tilde{L}}_{T}^{λ} (B_{p, r}^{s}))} ⩾ ‖ u ‖_{L_{T}^{λ} (B_{p, r}^{s}))} if λ ⩾ r . \end{matrix}$

Hybrid Besov spaces. The following hybrid Besov spaces allow different regularity indices for low and high frequencies (see [12]). we will use the notation, let us fixed some integer $j_{0}$ (the value of which will be decided in the proof of Theorem 1.2.) $\begin{matrix} ‖ u ‖_{{\dot{B}}_{p, 1}^{s}}^{l} : = \sum_{j ⩽ j_{0}} 2^{j s} ‖ {\dot{Δ}}_{j} u ‖_{L^{p}} and ‖ u ‖_{{\dot{B}}_{p, 1}^{s}}^{h} : = \sum_{j > j_{0}} 2^{j s} ‖ {\dot{Δ}}_{j} u ‖_{L^{p}} . \end{matrix}$ The following Bernstein’s lemma will be repeatedly used in this paper.
Lemma 2.1.
Let $B$ is a ball and $C$ is a ring of $R^{d}$ . There exists constant C such that for any positive λ, any non-negative integer k, any smooth homogeneous function σ of degree m, any couple $(p, q) \in {[1, \infty]}^{2}$ with $q ⩾ p ⩾ 1$ , and any function $u \in L^{p}$ , there holds $\begin{array}{l} supp \hat{u} \subset λ B \Rightarrow sup_{| a = k |} {‖ \partial^{a} u ‖}_{L^{q}} ⩽ C^{k + 1} λ^{k + d (\frac{1}{p} - \frac{1}{q})} ‖ u ‖_{L^{p}}, \\ supp \hat{u} \subset λ C \Rightarrow C^{- k - 1} λ^{k} ‖ u ‖_{L^{p}} ⩽ sup_{| a = k |} {‖ \partial^{a} u ‖}_{L^{p}} ⩽ C^{k + 1} λ^{k} ‖ u ‖_{L^{p}}, \\ supp \hat{u} \subset λ C \Rightarrow sup_{| a = k |} {‖ σ (D) u ‖}_{L^{p}} ⩽ C_{σ, m} λ^{m + d (\frac{1}{p} - \frac{1}{q})} ‖ u ‖_{L^{p}} . \end{array}$

Next, we will give the paraproducts and product estimates in Besov spaces. Recall the paraproduct decomposition $\begin{matrix} u v = T_{u} v + T_{v} u + R (u, v), \end{matrix}$ where $\begin{matrix} T_{u} v : = \sum_{q} S_{q - 1} u Δ_{v}, R (u, v) : = \sum_{q} Δ_{q} u {\tilde{Δ}}_{q} v, and {\tilde{Δ}}_{q} = Δ_{q - 1} + Δ_{q} + Δ_{q + 1} . \end{matrix}$ The paraproduct T and the remainder R operators satisfy the following continuous properties.
Proposition 2.2.
For all $s \in R$ , $σ > 0$ , and $1 ⩽ p, p_{1}, p_{2}, r, r_{1}, r_{2} ⩽ \infty$ , the paraproduct T is a bilinear, continuous operator from $L^{\infty} \times B_{p, r}^{s}$ to $B_{p, r}^{s}$ and from $B_{p_{1}, r_{1}}^{- σ} \times {\dot{B}}_{p_{2}, r_{2}}^{s}$ to $B_{p, r}^{s - σ}$ with $\frac{1}{r} = min {1, \frac{1}{r_{1}} + \frac{1}{r_{2}}}$ , $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}}$ . The remainder R is bilinear continuous from $B_{p_{1}, r_{1}}^{s_{1}} \times B_{p_{2}, r_{2}}^{s_{2}}$ to $B_{p, r}^{s_{1} + s_{2}}$ with $s_{1} + s_{2} > 0$ , $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} ⩽ 1$ , and $\frac{1}{r} = \frac{1}{r_{1}} + \frac{1}{r_{2}} ⩽ 1$ . In particular, if $r = \infty$ , the continuous property for the remainder R also holds for the case $s_{1} + s_{2} = 0$ , $r = \infty$ , $\frac{1}{r_{1}} + \frac{1}{r_{2}} = 1$ .

Combining the above proposition with Lemma 2.1 yields the following product estimates:
Corollary 2.1.
Let a and b be in $L^{\infty} \cap B_{p, r}^{s}$ for some $s > 0$ and $(p, r) \in {[1, \infty]}^{2}$ . Then there exists a constant C depending only on d, p and such that $\begin{matrix} ‖ a b ‖_{B_{p, r}^{s}} ⩽ C (‖ a ‖_{L^{\infty}} ‖ b ‖_{B_{p, r}^{s}} + ‖ b ‖_{L^{\infty}} ‖ a ‖_{B_{p, r}^{s}}) . \end{matrix}$

We note that the above Proposition and Corollary are valid in the homogeneous framework (i.e., use ${\dot{Δ}}_{j}$ instead of $Δ_{j}$ and use homogeneous Besov norms instead of nonhomogeneous ones), the readers can refer the [1]. Finally, we introduce a useful commutator estimate. Lemma 2.2 ([13,28]).

Let $A (D)$ be a zero-order Fourier multiplier. Set $j_{0} \in Z$ , $s < 1$ , $σ \in R$ , $1 ⩽ p_{1}$ , $p_{2} ⩽ \infty$ , and $\frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}}$ . Then, there exists a positive constant C depending only on $j_{0}$ and the regularity parameters such that $\begin{matrix} {‖ [{\dot{S}}_{j_{0}} A (D), T_{u}] v ‖}_{{\dot{B}}_{p, 1}^{σ + s}} ⩽ C ‖ \nabla u ‖_{{\dot{B}}_{p_{1}, 1}^{s - 1}} ‖ v ‖_{{\dot{B}}_{p_{2}, 1}^{σ}} . \end{matrix}$ For the $s = 1$ , one has $\begin{matrix} {‖ [{\dot{S}}_{j_{0}} A (D), T_{u}] v ‖}_{{\dot{B}}_{p, 1}^{σ + 1}} ⩽ C ‖ \nabla u ‖_{L^{p_{1}}} ‖ v ‖_{{\dot{B}}_{p_{2}, 1}^{σ}} . \end{matrix}$

3. A priori estimate

In this section, we devote to prove the main global energy inequality, namely, $\begin{array}{l} (3.1) & \begin{matrix} E (t) ⩽ & C E (0) + C E^{2} (t) + C 2^{3 N_{0}} e^{C V (t)} E^{2} (t) \\ + C 2^{3 N_{0}} e^{C V (t)} E^{h} (0) + C e^{C V (t)} (2^{- N_{0}} + 2^{3 N_{0}} (e^{C V (t)} - 1)) E^{h} (t) \\ + C 2^{3 N_{0}} e^{C V (t)} (e^{C V (t)} - 1) E^{l} (t) + C 2^{3 N_{0}} e^{C V (t)} (e^{C V (t)} - 1) E^{h} (t), \end{matrix} \\ \begin{matrix} E^{l} (t) ≜ & sup_{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + sup_{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + \int_{0}^{t} ‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'} + \int_{0}^{t} ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} d t^{'}, \end{matrix} \\ E^{h} (t) ≜ sup_{t} ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h} + sup_{t} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h} + \int_{0}^{t} ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h} d t^{'} + \int_{0}^{t} ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 2}}^{h} d t^{'}, \end{array}$ where $E (t) : = E^{l} (t) + E^{h} (t)$ , $V (t) : = \int_{0}^{t} ‖ \nabla u (t^{'}) ‖_{L^{\infty}} d t^{'}$ , and $N_{0}$ is a positive constant.

The proof will be divided into two parts, the estimates to low frequency part and high frequency part of the solutions.

In order to estimate the low frequency part of the solutions, we do not adopt the energy estimate, we shall use the method similar to the skill that tackle the high frequency part in [8].

Step 1 (The low frequency estimates of the solutions). Without loss of generality, we let $η = 1$ , $μ_{1} = 1$ , $μ_{2} = 2$ in (1.2), $\begin{matrix} \{\begin{matrix} \partial_{t} u + u \cdot \nabla u + \nabla P = div τ, \\ \partial_{t} τ + u \cdot \nabla τ + Λ^{2} τ + Q (τ, \nabla u) = 2 D (u) . \end{matrix} \end{matrix}$ We study the following linearization part of equation (1.2): $\begin{matrix} (3.2) & \{\begin{matrix} \partial_{t} u - Λ (Λ^{- 1} P div τ) = 0, \\ \partial_{t} Λ^{- 1} P div τ + Λ u + Λ^{2} (Λ^{- 1} P div τ) = 0 . \end{matrix} \end{matrix}$ By using of the Fourier transformation to the equation (3.2), we have $\begin{matrix} \frac{d}{d t} [\begin{matrix} \hat{u} \\ \hat{Λ^{- 1} P div τ} \end{matrix}] = [\begin{matrix} 0 & | ξ | \\ - | ξ | & - | ξ |^{2} \end{matrix}] [\begin{matrix} \hat{u} \\ \hat{Λ^{- 1} P div τ} \end{matrix}] ≜ X (ξ) [\begin{matrix} \hat{u} \\ \hat{Λ^{- 1} P div τ} \end{matrix}] \end{matrix}$

Lemma 3.1 ([8]).

Let G be the Green matrix of system ( 3.2 ). And denote $\hat{G} (t, ξ) = e^{X (ξ) t}$ as following $\begin{matrix} \hat{G} (t, ξ) = [\begin{matrix} A & B \\ - B & - C \end{matrix}], where \{\begin{matrix} A (t, ξ) = \frac{λ_{+} e^{λ_{-} t} - λ_{-} e^{λ_{+} t}}{λ_{+} - λ_{-}}, \\ B (t, ξ) = \frac{e^{λ_{+} t} - e^{λ_{-} t}}{λ_{+} - λ_{-}} | ξ |, \\ C (t, ξ) = \frac{λ_{-} e^{λ_{-} t} - λ_{+} e^{λ_{+} t}}{λ_{+} - λ_{-}}, \end{matrix} \end{matrix}$ with the $λ_{\pm}$ are two roots of the corresponding characteristic equation $\begin{matrix} (3.3) & λ^{2} + | ξ |^{2} λ + | ξ |^{2} = 0 . \end{matrix}$ Then the explicit expression of $\hat{G}$ can be given by the following formulae: $\begin{matrix} \hat{u} (t, ξ) = A (t, ξ) {\hat{u}}_{0} + B (t, ξ) {\hat{Λ^{- 1} P div τ}}_{0}, \end{matrix}$ and $\begin{matrix} {\hat{τ}}^{j, k} (t, ξ) = & e^{- | ξ |^{2} t} {\hat{τ}}_{0}^{j, k} + i B (t, ξ) (ξ_{j} {\hat{Λ^{- 1} u_{0}}}^{k} + ξ_{k} {\hat{Λ^{- 1} u_{0}}}^{j}) \\ + i (e^{- | ξ |^{2} t} + C (t, ξ)) \frac{(ξ_{j} {\hat{P div τ}}_{0}^{k} + ξ_{k} {\hat{P div τ_{0}}}^{j})}{| ξ |^{2}} . \end{matrix}$

The proof of the Lemma 3.1 can be found in [8], so we omit it here. Inspired by the Proposition 4.4 in [7], next, we give an key proposition to show the smoothing effects of $\overset{ˇ}{A}$ , $\overset{ˇ}{B}$ and $\overset{ˇ}{C}$ . Our purpose is to establish the following lemma.

Lemma 3.2.

For some fixed positive constant R, there exist two positive constants c and C only depending on R, such that for any $| ξ | ⩽ R$ , we have $\begin{array}{l} | A (t, ξ) | ⩽ C e^{- c | ξ |^{2} t}, \\ | B (t, ξ) | ⩽ C e^{- c | ξ |^{2} t}, \\ | C (t, ξ) | ⩽ C e^{- c | ξ |^{2} t} . \end{array}$

Proof.

According to the characteristic equation (3.3), one has $\begin{matrix} λ_{\pm} = \frac{- | ξ |^{2} \pm \sqrt{| ξ |^{4} - 4 | ξ |^{2}}}{2} . \end{matrix}$ If $| ξ |^{4} - 4 | ξ |^{2} ⩾ 0$ , i.e., $| ξ | ⩾ 2$ , we obtain $\begin{matrix} (3.4) & λ_{-} ⩽ λ_{+} = \frac{- | ξ |^{2} + \sqrt{| ξ |^{4} - 4 | ξ |^{2}}}{2} = \frac{- 2 | ξ |^{2}}{| ξ |^{2} + \sqrt{| ξ |^{4} - 4 | ξ |^{2}}} ⩽ - c_{1} | ξ |^{2}, for | ξ | ⩽ R, \end{matrix}$ where $\begin{matrix} c_{1} = \frac{2}{R^{2} + max {0, \sqrt{R^{4} - 4 R^{2}}}} . \end{matrix}$ If $| ξ |^{4} - 4 | ξ |^{2} < 0$ , i.e., $0 ⩽ | ξ | < 2$ , one has $\begin{matrix} (3.5) & Re (λ_{\pm}) = - \frac{| ξ |^{2}}{2} . \end{matrix}$ Therefore, combining with (3.4) and (3.5), we can conclude that for $c_{2} = min {c_{1}, \frac{1}{2}}$ , $\begin{matrix} (3.6) & Re (λ_{\pm}) ⩽ - c_{2} | ξ |^{2}, and | e^{λ_{\pm} t} | ⩽ e^{- c_{2} | ξ |^{2} t} . \end{matrix}$ Then for any $| ξ | ⩽ R$ , one can deduce that $E (t, ξ) ≜ \frac{e^{λ_{+} t} - e^{λ_{-} t}}{λ_{+} - λ_{-}}$ satisfies $\begin{matrix} | E (t, ξ) | & = | \frac{e^{λ_{+} t} - e^{λ_{-} t}}{λ_{+} - λ_{-}} | = t | \frac{e^{λ_{+} t} - e^{λ_{-} t}}{λ_{+} t - λ_{-} t} | \\ ⩽ t sup_{s \in [0, 1]} (exp (Re (s λ_{+} t - (1 - s) λ_{-} t))) \\ ⩽ t e^{- c_{2} | ξ |^{2} t} ⩽ C e^{- \frac{c_{2}}{2} | ξ |^{2} t} . \end{matrix}$ Finally, we can finish the proof by the following equalities and choosing a suitable constant c, $\begin{matrix} A (t, ξ) = - λ_{+} E (t, ξ) + e^{λ_{+} t}, B (t, ξ) = | ξ | E (t, ξ), C (t, ξ) = - λ_{+} E (t, ξ) - e^{λ_{-} t} . \end{matrix}$ □

Proposition 3.1.

Let $C$ be a ring centered at 0 in $R^{d}$ . Then there exist positive constant $R_{0}$ , C, and c such that, if $supp \hat{u} \subset λ C$ and $λ ⩽ R_{0}$ , one has $\begin{array}{l} ‖ \overset{ˇ}{A} * u ‖_{L^{2}} ⩽ C e^{- c λ^{2} t} ‖ u ‖_{L^{2}}, \\ ‖ \overset{ˇ}{B} * u ‖_{L^{2}} ⩽ C e^{- c λ^{2} t} ‖ u ‖_{L^{2}}, \\ ‖ \overset{ˇ}{C} * u ‖_{L^{2}} ⩽ C e^{- c λ^{2} t} ‖ u ‖_{L^{2}} . \end{array}$

Proof.

By using the Lemma 3.2 and a routine calculation, when $ξ \in supp \hat{u} \subset λ C$ and $λ ⩽ R_{0}$ , we obtain $\begin{matrix} | \hat{G} (t, ξ) | ⩽ C e^{- c | ξ |^{2} t} ⩽ C e^{- c λ^{2} t} . \end{matrix}$ Thanks to the Plancherel theorem, one has $\begin{matrix} ‖ G * u ‖_{L^{2}} = {‖ \hat{G} (ξ) \hat{u} (ξ) ‖}_{L^{2}} ⩽ C {‖ e^{- c | ξ |^{2} t} \hat{u} (ξ) ‖}_{L^{2}} ⩽ C e^{- c λ^{2} t} ‖ u ‖_{L^{2}}, \end{matrix}$ So we complete the proof. □

The estimate of the low frequency part of solutions are based on the following frequency localized system: $\begin{matrix} \{\begin{matrix} \partial_{t} {\dot{Δ}}_{j} u - P div {\dot{Δ}}_{j} τ = {\dot{Δ}}_{j} F, \\ \partial_{t} {\dot{Δ}}_{j} τ - 2 D ({\dot{Δ}}_{j} u) - Δ {\dot{Δ}}_{j} τ = {\dot{Δ}}_{j} H, \end{matrix} \end{matrix}$ where $\begin{array}{l} F : = - P (u \cdot \nabla u), \\ H : = - u \cdot \nabla τ - Q (τ, \nabla u) . \end{array}$ By Duhamel’s principle and Lemma 3.1, we can express the $({\dot{Δ}}_{j} u, {\dot{Δ}}_{j} τ)$ as $\begin{matrix} {\dot{Δ}}_{j} u = & \overset{˘}{A} * {\dot{Δ}}_{j} u_{0} + \int_{0}^{t} \overset{ˇ}{A} (t - t^{'}, \cdot) * {\dot{Δ}}_{j} F d t^{'} \\ + \overset{˘}{B} * P Λ^{- 1} div {\dot{Δ}}_{j} τ_{0} + \int_{0}^{t} \overset{ˇ}{B} (t - t^{'}, \cdot) * (P Λ^{- 1} div {\dot{Δ}}_{j} H) d t^{'}, \end{matrix}$ and $\begin{matrix} {\dot{Δ}}_{j} τ = & e^{Δ t} {\dot{Δ}}_{j} τ_{0} + \int_{0}^{t} e^{Δ (t - t^{'})} {\dot{Δ}}_{j} H d t^{'} \\ + \overset{˘}{B} * Λ^{- 1} {\dot{Δ}}_{j} (\nabla u_{0} + {(\nabla u_{0})}^{⊤}) + \int_{0}^{t} \overset{ˇ}{B} (t - t^{'}, \cdot) * (Λ^{- 1} Δ_{j} (\nabla F + {(\nabla F)}^{⊤})) d t^{'} \\ + (e^{Δ t} + \overset{ˇ}{C}) * (Λ^{- 2} \nabla P div {\dot{Δ}}_{j} τ_{0} + {(Λ^{- 2} \nabla P div {\dot{Δ}}_{j} τ_{0})}^{⊤}) \\ + \int_{0}^{t} (e^{Δ (t - t^{'})} + \overset{ˇ}{C} (t - t^{'}, \cdot) *) (Λ^{- 2} \nabla P div {\dot{Δ}}_{j} H + {(Λ^{- 2} \nabla P div {\dot{Δ}}_{j} H)}^{⊤}) d t^{'} . \end{matrix}$ Since $j ⩽ j_{0}$ , it follows from the Proposition 3.1 and the estimate of the Heat Flow that $\begin{matrix} ‖ {\dot{Δ}}_{j} u ‖_{L^{2}} ⩽ & C e^{- c t 2^{2 j}} ‖ {\dot{Δ}}_{j} u_{0} ‖_{L^{2}} + C e^{- c t 2^{2 j}} ‖ {\dot{Δ}}_{j} τ_{0} ‖_{L^{2}} \\ + C \int_{0}^{t} e^{- c (t - t^{'}) 2^{2 j}} (‖ {\dot{Δ}}_{j} F ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} H ‖_{L^{2}}) d t^{'}, \end{matrix}$ and $\begin{matrix} ‖ {\dot{Δ}}_{j} τ ‖_{L^{2}} ⩽ & C e^{- c t 2^{2 j}} ‖ {\dot{Δ}}_{j} u_{0} ‖_{L^{2}} + C e^{- c t 2^{2 j}} ‖ {\dot{Δ}}_{j} τ_{0} ‖_{L^{2}} \\ + C \int_{0}^{t} e^{- c (t - t^{'}) 2^{2 j}} (‖ {\dot{Δ}}_{j} F ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} H ‖_{L^{2}}) d t^{'} . \end{matrix}$ Taking the $L^{\infty}$ and $L^{1}$ norms with respect to time t, by Young’s inequality with convolution, when $j ⩽ j_{0}$ , one has $\begin{matrix} (3.7) & \begin{matrix} ‖ {\dot{Δ}}_{j} u, {\dot{Δ}}_{j} τ ‖_{L_{t}^{\infty} (L^{2})} + 2^{2 j} ‖ {\dot{Δ}}_{j} u, {\dot{Δ}}_{j} τ ‖_{L_{t}^{1} (L^{2})} \\ ⩽ C (‖ {\dot{Δ}}_{j} τ_{0} ‖_{L^{2}} + ‖ {\dot{Δ}}_{j} u_{0} ‖_{L^{2}}) + C (‖ {\dot{Δ}}_{j} F ‖_{L_{t}^{1} (L^{2})} + ‖ {\dot{Δ}}_{j} H ‖_{L_{t}^{1} (L^{2})}) . \end{matrix} \end{matrix}$ Multiplying inequality (3.7) by $2^{j (\frac{d}{2} - 1)}$ , and taking sum with respect to j over $j ⩽ j_{0}$ , one has $\begin{matrix} (3.8) & E^{l} (t) ⩽ ‖ u_{0}, τ_{0} ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + \int_{0}^{t} {‖ (u \cdot \nabla u) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + {‖ (u \cdot \nabla τ) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + {‖ Q (τ, \nabla u) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} d t^{'} . \end{matrix}$ Now, we will estimate the nonlinear term. Firstly, we estimate the typical term $Q (τ, \nabla u)$ , the other terms are similar. Using the Bony decomposition, $\begin{matrix} {\dot{S}}_{j_{0} + 1} (τ \cdot \nabla u) = {\dot{S}}_{j_{0} + 1} {\dot{T}}_{τ} \nabla u + {\dot{S}}_{j_{0} + 1} \dot{R} (τ, \nabla u) + {\dot{T}}_{\nabla u} {\dot{S}}_{j_{0} + 1} τ + [{\dot{S}}_{j_{0} + 1}, {\dot{T}}_{\nabla u}] τ . \end{matrix}$ It’s clear that $\frac{d}{2} - 1 ⩽ \frac{d}{p}$ by (1.4), so if $\frac{d}{2} - 1 < \frac{d}{p}$ , denoting $\frac{1}{p *} = \frac{1}{2} - \frac{1}{p}$ , then for $p ⩽ 4$ (i.e., $p^{*} ⩾ p$ ), one has $\begin{matrix} ‖ {\dot{S}}_{j_{0} + 1} {\dot{T}}_{τ} \nabla u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} & ⩽ ‖ τ ‖_{{\dot{B}}_{p^{*}, 1}^{\frac{d}{2} - \frac{d}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} ⩽ ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} \\ ⩽ (‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) . \end{matrix}$ If $\frac{d}{2} - 1 = \frac{d}{p}$ , by using of ${\dot{B}}_{p, 1}^{\frac{2 d}{p} - \frac{d}{2}} ↪ L^{p^{*}}$ for all $p ⩽ 4$ , we have $\begin{matrix} ‖ {\dot{S}}_{j_{0} + 1} {\dot{T}}_{τ} \nabla u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} & = ‖ {\dot{T}}_{τ} \nabla u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{p}}} ⩽ ‖ τ ‖_{L^{p *}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} ⩽ ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} \\ ⩽ (‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) . \end{matrix}$ Note that $p < 2 d$ by $p ⩽ 4$ and $p \neq 4$ as $d = 2$ , then we have $\begin{matrix} {‖ {\dot{S}}_{j_{0} + 1} \dot{R} (τ, \nabla u) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} & ⩽ {‖ \dot{R} (τ, \nabla u) ‖}_{{\dot{B}}_{\frac{p}{2}, 1}^{\frac{2 d}{p} - 1}} ⩽ ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} \\ ⩽ (‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) . \end{matrix}$ In view of Proposition 2.2, we get $\begin{matrix} ‖ {\dot{T}}_{\nabla u} {\dot{S}}_{j_{0} + 1} τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} & ⩽ ‖ \nabla u ‖_{L^{\infty}} ‖ {\dot{S}}_{j_{0} + 1} τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} ⩽ ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} \\ ⩽ ‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) . \end{matrix}$ By the Lemma 2.2, it holds that $\begin{matrix} {‖ [{\dot{S}}_{j_{0} + 1}, {\dot{T}}_{\nabla u}] τ ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}} & ⩽ ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} - 1}} {‖ \nabla^{2} u ‖}_{{\dot{B}}_{p *, 1}^{\frac{d}{p *} - 1}} ⩽ ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} - 1}} ‖ \nabla u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}} \\ ⩽ (‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) . \end{matrix}$ Consequently, we conclude that $\begin{matrix} (3.9) & \begin{matrix} \int_{0}^{t} {‖ Q (τ, \nabla u) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} d t^{'} & ⩽ \int_{0}^{t} (‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) d t^{'} \\ ⩽ C E^{2} (t) . \end{matrix} \end{matrix}$ Similar to the above estimate, we get the following estimates $\begin{matrix} (3.10) & \begin{matrix} \int_{0}^{t} {‖ (u \cdot \nabla u) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} d t^{'} & ⩽ \int_{0}^{t} (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) d t^{'} \\ ⩽ C E^{2} (t), \end{matrix} \end{matrix}$ and $\begin{matrix} (3.11) & \begin{matrix} \int_{0}^{t} {‖ (u \cdot \nabla τ) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} d t^{'} & ⩽ \int_{0}^{t} (‖ τ ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} + 1}}^{l} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 2}}^{h}) (‖ u ‖_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + ‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h}) d t^{'} \\ ⩽ C E^{2} (t) . \end{matrix} \end{matrix}$

In the view of the inequalities (3.8), (3.9), (3.10) and (3.11), we have $\begin{matrix} (3.12) & E^{l} (t) ⩽ {‖ (u_{0}, τ_{0}) ‖}_{{\dot{B}}_{2, 1}^{\frac{d}{2} - 1}}^{l} + E^{2} (t) . \end{matrix}$

Step 2 (The high frequency estimates of the solutions). By [8], we have the following estimate for the high frequency part: $\begin{matrix} E^{h} (t) ⩽ & C e^{C V (t)} (2^{- N_{0}} + 2^{3 N_{0}} (e^{C V (t)} - 1)) E^{h} (t) + C 2^{3 N_{0}} e^{C V (t)} E^{h} (0) \\ + C 2^{3 N_{0}} e^{C V (t)} ({‖ {\dot{T}}_{\partial_{k} u}^{'} u^{k} ‖}_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{d}{p} + 1})}^{h} + {‖ {\dot{T}}_{\partial_{k} τ}^{'} u^{k} ‖}_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{d}{p}})}^{h} + {‖ Q (τ, \nabla u) ‖}_{L_{t}^{1} ({\dot{B}}_{p, 1}^{\frac{d}{p}})}^{h}) \\ + C 2^{3 N_{0}} e^{C V (t)} \sum_{| q^{'} - q | ⩽ 4} \int_{0}^{t} V^{'} (t^{'}) (‖ u ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p} + 1}}^{h} + ‖ τ ‖_{{\dot{B}}_{p, 1}^{\frac{d}{p}}}^{h}) d t^{'} \\ + C 2^{3 N_{0}} e^{C V (t)} E^{l} (t) \int_{0}^{t} V^{'} (t^{'}) d t^{'} . \end{matrix}$ Now we complete the proof of the inequality (3.1) similar to the inequality (3.46) in [8].

Proof of Theorem 1.2.

As well known, the bootstrap argument starts with the assumptions that $E (t)$ is bounded and $V (t)$ is small, $\begin{matrix} E (t) ⩽ δ, V (t) ⩽ ε . \end{matrix}$ Choosing $δ, ε > 0$ small enough, there exists a suitable constant $C > 0$ such that $\begin{matrix} (3.13) & \begin{matrix} C E (t) + C 2^{3 N_{0}} e^{C V (t)} E (t) + C e^{C V (t)} (2^{- N_{0}} + 2^{3 N_{0}} (e^{C V (t)} - 1) \\ + C 2^{3 N_{0}} e^{C V (t)} (e^{C V} - 1) + C 2^{3 N_{0}} e^{C V (t)} (e^{C V} - 1) ⩽ \frac{1}{2} . \end{matrix} \end{matrix}$ Combining the inequalities (3.1) and (3.13), we have $\begin{matrix} E (t) ⩽ C E (0) + C 2^{3 N_{0}} e^{C V (t)} E^{h} (0) . \end{matrix}$ If $E (0)$ is sufficiently small, then one obtains $\begin{matrix} E (t) ⩽ \frac{1}{2} δ . \end{matrix}$ It follows from $V (t) ⩽ C E (t)$ that $\begin{matrix} V (t) ⩽ \frac{1}{2} ε . \end{matrix}$ Thus we accomplish the proof of Theorem 1.2. □

Footnotes

Acknowledgements

This work is partially supported by China Scholarship Council (Nos. 202006370255), the National Natural Science Foundation of China (Nos. 11971485, 12171486), Natural Science Foundation for Excellent Young Scholars of Hunan Province (No. 2023JJ20057) and Fundamental Research Funds for the Central Universities of Central South University, China (No. 2021zzts0041).

Conflict of interest

The authors declare that they have no competing interests.

Data availability

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

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