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Optimal decay rate for the generalized Oldroyd-B model with only stress tensor diffusion in ℝ2\mathbb{R}^{2}

Zhaonan Luo1\mbox{Luo}^{1} 111E-mail: [email protected], Wei Luo2\mbox{Luo}^{2}222E-mail: [email protected]  and Zhaoyang Yin2,3\mbox{Yin}^{2,3}333E-mail: [email protected]
School1{}^{1}\mbox{School} of Mathematical Sciences, Fudan University, Shanghai 200433, China.
Department2{}^{2}\mbox{Department} of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
Faculty3{}^{3}\mbox{Faculty} of Information Technology,
Macau University of Science and Technology, Macau, China
Abstract

In this paper, we are concerned with optimal decay rate for the 2-D generalized Oldroyd-B model with only stress tensor diffusion (−Δ)β​τ(-\Delta)^{\beta}\tau. In the case β=1\beta=1, we first establish optimal decay rate in H1H^{1} framework and remove the smallness assumption of low frequencies by virtue of the Fourier splitting method and the Littlewood-Paley decomposition theory. Furthermore, we prove optimal decay rate for the highest derivative of the solution by a different method combining time frequency decomposition and the time weighted energy estimate. In the case 12≤β<1\frac{1}{2}\leq\beta<1, we study optimal decay rate for the highest derivative of the solution by the improved Fourier splitting method.
2020 Mathematics Subject Classification: 35Q31, 76A05, 74B20, 42A38.

Keywords: The generalized Oldroyd-B model; Optimal decay rate; The Fourier splitting method.

 

1 Introduction

The classical Oldroyd-B model can be written as follows:

(1.4) {∂tu+u⋅∇u+∇P=d​i​v​τ+ν​Δ​u,d​i​v​u=0,∂tτ+u⋅∇τ+a0​τ+Q​(∇u,τ)=α​D​(u),u|t=0=u0,τ|t=0=τ0.\displaystyle\left\{\begin{array}[]{l}\partial_{t}u+u\cdot\nabla u+\nabla P=div~{}\tau+\nu\Delta u,~{}~{}~{}~{}div~{}u=0,\\[4.30554pt] \partial_{t}\tau+u\cdot\nabla\tau+a_{0}\tau+Q(\nabla u,\tau)=\alpha D(u),\\[4.30554pt] u|_{t=0}=u_{0},~{}~{}\tau|_{t=0}=\tau_{0}.\\[4.30554pt] \end{array}\right.

The Oldroyd-B model derived by J. G. Oldroyd in [24] which was used to describe the dynamics of viscoelastic fluids. In (1.4), u​(t,x)u(t,x) represents the velocity of the liquid, PP denotes the pressure and τ​(t,x)\tau(t,x) is the symmetric stress tensor. The parameters a0a_{0}, ν\nu and α\alpha are nonnegative. Furthermore, QQ is the following bilinear form

Q​(∇u,τ)=τ​Ω−Ω​τ+b​(D​(u)​τ+τ​D​(u)),b∈[−1,1],Q(\nabla u,\tau)=\tau\Omega-\Omega\tau+b(D(u)\tau+\tau D(u)),~{}~{}~{}~{}b\in[-1,1],

with the deformation tensor D​(u)=∇u+(∇u)T2D(u)=\frac{\nabla u+(\nabla u)^{T}}{2} and the vorticity tensor Ω=∇u−(∇u)T2\Omega=\frac{\nabla u-(\nabla u)^{T}}{2}. One can refer to [9] for more explanations of the Oldroyd-B model.

In this paper, we are concerned with the generalized Oldroyd-B model with only stress tensor diffusion and without the damping (a0=0)(a_{0}=0):

(1.8) {∂tu+u⋅∇u+∇P=d​i​v​τ,d​i​v​u=0,∂tτ+u⋅∇τ+Q​(∇u,τ)+(−Δ)β​τ=D​(u),u|t=0=u0,τ|t=0=τ0.\displaystyle\left\{\begin{array}[]{l}\partial_{t}u+u\cdot\nabla u+\nabla P=div~{}\tau,~{}~{}~{}~{}div~{}u=0,\\[4.30554pt] \partial_{t}\tau+u\cdot\nabla\tau+Q(\nabla u,\tau)+(-\Delta)^{\beta}\tau=D(u),\\[4.30554pt] u|_{t=0}=u_{0},~{}~{}\tau|_{t=0}=\tau_{0}.\\[4.30554pt] \end{array}\right.

Taking τ=0\tau=0 in (​1.8​)\eqref{eq1}, then we obtain D​u=0Du=0, which means that u=0u=0 in Sobolev spaces. This observation reveals the essential difference between (1.8) and the well-known Euler equation.

1.1.  The Oldroyd-B model

We first recall some mathematic results for the classical Oldroyd-B model (1.4). In [11], C. Guillopé, and J. C. Saut first proved that the Oldroyd-B model admits a unique global strong solution in Sobolev spaces. The LpL^{p}-setting was showed by E. Fernández-Cara, F.Guillén and R. Ortega [10]. The global week solutions of the Oldroyd-B model were obtained by P. L. Lions and N. Masmoudi [17] for the co-rotation case that is b=0b=0. However, the problem for the general case b≠0b\neq 0 is still open, see [20, 21]. J. Y. Chemin and N. Masmoudi [3] showed the existence and uniqueness of strong solutions in homogenous Besov spaces with critical index of regularity. Optimal decay rates for solutions to the 3-D Oldroyd-B model were given by M. Hieber, H. Wen and R. Zi [13]. An approach based on the deformation tensor can be found in [5, 4, 15, 14, 16, 29, 2].

Some mathematic results for the generalized Oldroyd-B model (1.8) are given as follows. T. M. Elgindi and F. Rousset [9] first showed global regularity for (1.8) with a0>0a_{0}>0 and β=1\beta=1. In [8], T. M. Elgindi and J. Liu proved global strong solutions of the 3-D case under the assumption that initial data is sufficiently small. Recently, W. Deng, Z. Luo and Z. Yin [7] obtained the global solutions in co-rotation case and proved the H1H^{1} decay rate for the global solutions to (1.8). For the case a0=0a_{0}=0 and β∈[12,1]\beta\in[\frac{1}{2},1], P. Constantin, J. Wu, J. Zhao and Y. Zhu [6] established the global well-posedness for (1.8) with small data. In [28], J. Wu and J. Zhao investigated the global well-posedness in Besov spaces for fractional dissipation with small data. Optimal time decay rate in H1H^{1} framework of global solutions to (1.8) was given by [27, 18]. However, they can’t deal with critical case d=2d=2 and β=1\beta=1.

1.2.  Main results

Optimal decay rate for (1.8) has been studied widely. The problem of time decay rate with d=2d=2 is more difficult than the case with d≥3d\geq 3. Since the additional stress tensor τ\tau does not decay fast enough, we failed to use the bootstrap argument as in [25, 19]. For critical case d=2d=2, we can not get any algebraic decay rate by the Fourier splitting method directly as in [12]. We will study this problem by virtue of the refinement to Schonbek’s [26] strategy. Using the Fourier splitting method, we obtain initial logarithmic decay rate

‖(u,τ)‖L2≤C​ln−l⁡(e+t),\displaystyle\|(u,\tau)\|_{L^{2}}\leq C\ln^{-l}(e+t),

for any l∈N+l\in N^{+}. The main difficulty for proving optimal decay rate is lack of damping term and low-frequency estimation of L1L^{1} for τ\tau. However, we have

∫S​(t)∫0t|ℱ​Q​(∇u,τ)⋅τ^¯|​𝑑s′​𝑑ξ≤C​(1+t)−12​∫0t‖τ‖L22​‖∇u‖L2​𝑑s′.\displaystyle\int_{S(t)}\int_{0}^{t}|\mathcal{F}Q(\nabla u,\tau)\cdot\bar{\hat{\tau}}|ds^{\prime}d\xi\leq C(1+t)^{-\frac{1}{2}}\int_{0}^{t}\|\tau\|^{2}_{L^{2}}\|\nabla u\|_{L^{2}}ds^{\prime}.

By virtue of the time weighted energy estimate and the logarithmic decay rate, then we improve the time decay rate to algebraic decay rate (1+t)−12(1+t)^{-\frac{1}{2}}. Notice that this is not the optimal decay rate. Then, we prove optimal decay rate in H1H^{1} framework for the 2-D generalized Oldroyd-B type model by virtue of the Fourier spiltting method and the Littlewood-Paley decomposition theory. Considering the decay rate for the highest derivative of the solution to (1.8), the main difficulty is unclosed energy estimate. The complete dissipation ‖∇Λs​τ‖L22+‖∇Λs−1​u‖L22\|\nabla\Lambda^{s}\tau\|^{2}_{L^{2}}+\|\nabla\Lambda^{s-1}u\|^{2}_{L^{2}} can be obtained by estimating the mixed term ‖Λs​(u,τ)‖L22+⟨Λs−1​τ,−∇Λs−1​u⟩\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\langle\Lambda^{s-1}\tau,-\nabla\Lambda^{s-1}u\rangle. One can see that the decay rate of inner product is slower than the decay rate of energy. To overcome this difficulty, we construct a different energy and dissipation functional for (u,τ)(u,\tau) as follows:

E~s=k​(1+t)​‖Λs​(u,τ)‖L22+⟨Λs−1​τ,−∇Λs−1​u⟩,\widetilde{E}_{s}=k(1+t)\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\langle\Lambda^{s-1}\tau,-\nabla\Lambda^{s-1}u\rangle,

and

D~s=k​(1+t)​‖∇Λs​τ‖L22+14​‖∇Λs−1​u‖L22,\widetilde{D}_{s}=k(1+t)\|\nabla\Lambda^{s}\tau\|^{2}_{L^{2}}+\frac{1}{4}\|\nabla\Lambda^{s-1}u\|^{2}_{L^{2}},

where kk is small enough. Finally, we introduce a new method which flexibly combines the Fourier splitting method and the time weighted energy estimate to prove the decay rate for the highest derivative.

We also present optimal decay rate for the generalized Oldroyd-B model (1.8) in the fractional case 12≤β<1\frac{1}{2}\leq\beta<1. By virtue of the traditional Fourier splitting method, one can not obtain the optimal decay for the fractional case. P. Wang, J. Wu, X. Xu and Y. Zhong [27] use spectral analysis method to prove the optimal decay rate of the lower order energy of the fractional case. The corresponding linearized system of (1.8) is given by

(1.11) {∂tu=ℙ​d​i​v​τ,d​i​v​u=0,∂tℙ​d​i​v​τ+(−Δ)β​ℙ​d​i​v​τ=Δ​u.\displaystyle\left\{\begin{array}[]{l}\partial_{t}u=\mathbb{P}div~{}\tau,~{}~{}~{}~{}div~{}u=0,\\[4.30554pt] \partial_{t}\mathbb{P}div~{}\tau+(-\Delta)^{\beta}\mathbb{P}div~{}\tau=\Delta u.\\[4.30554pt] \end{array}\right.

where ℙ\mathbb{P} denotes the Leray projection onto divergence free vector fields. One can see that (u,ℙ​d​i​v​τ)(u,\mathbb{P}div~{}\tau) satisfy the same system of wave-type equation

(1.12) ∂t​tW+(−Δ)β​∂tW−Δ​W=0.\displaystyle\partial_{tt}W+(-\Delta)^{\beta}\partial_{t}W-\Delta W=0.

The structure in (1.12) reveals that there are both dissipative and dispersive effects on (u,ℙ​d​i​v​τ)(u,\mathbb{P}div~{}\tau). However, we point out that the refinement of Schonbek’s strategy also work in this case with extra lower energy dissipation estimate

(1.13) dd​t​[‖(u,τ)‖Hs2+2​k​(⟨−∇u,τ⟩Hs−β+⟨τ,−Λ−2+2​β​∇u⟩)]+k2​‖Λβ​u‖Hs+1−2​β2+‖Λβ​τ‖Hs2≤0,\displaystyle\frac{d}{dt}[\|(u,\tau)\|^{2}_{H^{s}}+2k(\langle-\nabla u,\tau\rangle_{H^{s-\beta}}+\langle\tau,-\Lambda^{-2+2\beta}\nabla u\rangle)]+\frac{k}{2}\|\Lambda^{\beta}u\|^{2}_{H^{s+1-2\beta}}+\|\Lambda^{\beta}\tau\|^{2}_{H^{s}}\leq 0,

where we use the properties of Calderon-Zygmund type operator. Then, we rediscover optimal decay rates in H1H^{1} [27] by the improved Fourier splitting method. The main difficulty to obtain optimal decay rate for the highest derivative of the solution is unclosed energy estimate. The complete dissipation ‖Λs+β​τ‖L22+‖∇Λs−β​u‖L22\|\Lambda^{s+\beta}\tau\|^{2}_{L^{2}}+\|\nabla\Lambda^{s-\beta}u\|^{2}_{L^{2}} can be obtained by estimating the mixed term ‖Λs​(u,τ)‖L22+⟨Λs−β​τ,−∇Λs−β​u⟩\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\langle\Lambda^{s-\beta}\tau,-\nabla\Lambda^{s-\beta}u\rangle. One can see that the decay rate of inner product is slower than the decay rate of energy. To overcome this difficulty, we construct a different energy and dissipation functional for (u,τ)(u,\tau) as follows:

E¯β=(1+t)a​‖Λs​(u,τ)‖L22+k​⟨Λs−β​τ,−∇Λs−β​u⟩,\overline{E}_{\beta}=(1+t)^{a}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+k\langle\Lambda^{s-\beta}\tau,-\nabla\Lambda^{s-\beta}u\rangle,

and

D¯β=(1+t)a​‖Λs+β​τ‖L22+k4​‖∇Λs−β​u‖L22,\overline{D}_{\beta}=(1+t)^{a}\|\Lambda^{s+\beta}\tau\|^{2}_{L^{2}}+\frac{k}{4}\|\nabla\Lambda^{s-\beta}u\|^{2}_{L^{2}},

where a=2−1β∈[0,1)a=2-\frac{1}{\beta}\in[0,1) and kk is small enough. We prove optimal decay rate for the highest derivative of the solution to 2-D generalized Oldroyd-B type model by the improved Fourier splitting method. Notice that this novel result for (1.8) has not been studied before.

Let’s review the global existence result for (1.8).

Theorem 1.1.

[6] Let d=2d=2, 12≤β≤1\frac{1}{2}\leq\beta\leq 1 and s>2s>2. Let (u,τ)(u,\tau) be a strong solution of (1.8) with the initial data (u0,τ0)∈Hs(u_{0},\tau_{0})\in H^{s} and τ0\tau_{0} is symmetric. There exists a small constant δ\delta such that if

‖(u0,τ0)‖Hs≤δ,\displaystyle\|(u_{0},\tau_{0})\|_{H^{s}}\leq\delta,

then the system (1.8) admits a unique global strong solution (u,τ)∈C​([0,∞);Hs)(u,\tau)\in C([0,\infty);H^{s}). Moreover, the energy estimate for (u,τ)(u,\tau) implies that

(1.14) dd​t​(‖(u,τ)‖Hs2+2​k​⟨−∇u,τ⟩Hs−β)+k2​‖∇u‖Hs−β2+‖Λβ​τ‖Hs2≤0,\displaystyle\frac{d}{dt}(\|(u,\tau)\|^{2}_{H^{s}}+2k\langle-\nabla u,\tau\rangle_{H^{s-\beta}})+\frac{k}{2}\|\nabla u\|^{2}_{H^{s-\beta}}+\|\Lambda^{\beta}\tau\|^{2}_{H^{s}}\leq 0,

where kk is a sufficiently small constant.

Our main results can be stated as follows:

Theorem 1.2.

Let β=1\beta=1. Let (u,τ)(u,\tau) be a strong solution of (1.4) with the initial data (u0,τ0)(u_{0},\tau_{0}) under the condition in Theorem 1.1. Suppose that (u0,τ0)∈B˙2,∞−1(u_{0},\tau_{0})\in\dot{B}^{-1}_{2,\infty}, then there exists C>0C>0 such that for every t>0t>0 and s1∈[0,s]s_{1}\in[0,s], there holds

(1.15) ‖Λs1​(u,τ)‖L2≤C​(1+t)−1+s12.\displaystyle\|\Lambda^{s_{1}}(u,\tau)\|_{L^{2}}\leq C(1+t)^{-\frac{1+{s_{1}}}{2}}.

In addition, if (u0,τ0)∈Hs+1(u_{0},\tau_{0})\in H^{s+1} and 0<c0=|∫ℝ2(u0,τ0)​𝑑x|0<c_{0}=|\int_{\mathbb{R}^{2}}(u_{0},\tau_{0})dx|, then there exists C0​(s1,c0)≤CC_{0}(s_{1},c_{0})\leq C such that

(1.16) ‖Λs1​(u,τ)‖L2≥C02​(1+t)−s1+12.\displaystyle\|\Lambda^{s_{1}}(u,\tau)\|_{L^{2}}\geq\frac{C_{0}}{2}(1+t)^{-\frac{s_{1}+1}{2}}.
Theorem 1.3.

Let β∈[12,1)\beta\in[\frac{1}{2},1). Let (u,τ)(u,\tau) be a strong solution of (1.4) with the initial data (u0,τ0)(u_{0},\tau_{0}) under the condition in Theorem 1.1. If (u0,τ0)∈B˙2,∞−1(u_{0},\tau_{0})\in\dot{B}^{-1}_{2,\infty}, then there exists C>0C>0 such that for every t>0t>0,

(1.17) ‖Λs​(u,τ)‖L2≤C​(1+t)−s+12​β.\displaystyle\|\Lambda^{s}(u,\tau)\|_{L^{2}}\leq C(1+t)^{-\frac{{s}+1}{2\beta}}.

Furthermore, suppose that (u0,τ0)∈Hs+β(u_{0},\tau_{0})\in H^{s+\beta} and 0<|∫ℝ2(u0,τ0)​𝑑x|0<|\int_{\mathbb{R}^{2}}(u_{0},\tau_{0})dx|, then there exists Cβ≤CC_{\beta}\leq C such that

(1.18) ‖Λs​(u,τ)‖L2≥Cβ2​(1+t)−s+12​β.\displaystyle\|\Lambda^{s}(u,\tau)\|_{L^{2}}\geq\frac{C_{\beta}}{2}(1+t)^{-\frac{s+1}{2\beta}}.
Remark 1.4.

The classical result about large time behaviour often supposed that the initial data belongs to L1L^{1}(See [25]). Since L1↪B˙2,∞−1L^{1}\hookrightarrow\dot{B}^{-1}_{2,\infty}, it follows that the above results still hold true when (u0,τ0)∈L1(u_{0},\tau_{0})\in L^{1}.

The paper is organized as follows. In Section 2 we introduce some lemmas which will be used in the sequel. In Section 3 we prove optimal decay rate for the 2-D generalized Oldroyd-B model in critical case by virtue of a different Fourier splitting method and the time weighted energy estimate. In Section 4 we prove optimal decay rate for the highest derivative of the solution to 2-D generalized Oldroyd-B model in fractional case by the improved Fourier splitting method and the time weighted energy estimate.

2 Preliminaries

In this section we introduce some lemmas which will be used in the sequel.

The Littlewood-Paley decomposition theory and and Besov spaces are given as follows.

Lemma 2.1.

[1] Let 𝒞={ξ∈ℝ2:34≤|ξ|≤83}\mathcal{C}=\{\xi\in\mathbb{R}^{2}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}. There exists radial function φ\varphi, valued in the interval [0,1][0,1], belonging respectively to 𝒟​(𝒞)\mathcal{D}(\mathcal{C}), and

∀ξ∈ℝ2\{0},∑j∈ℤφ​(2−j​ξ)=1,\forall\xi\in\mathbb{R}^{2}\backslash\{0\},\ \sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,
|j−j′|≥2⇒Suppφ(2−j⋅)∩Suppφ(2−j′⋅)=∅.|j-j^{\prime}|\geq 2\Rightarrow\mathrm{Supp}\ \varphi(2^{-j}\cdot)\cap\mathrm{Supp}\ \varphi(2^{-j^{\prime}}\cdot)=\emptyset.

Moreover, there holds

∀ξ∈ℝ2\{0},12≤∑j∈ℤφ2​(2−j​ξ)≤1.\forall\xi\in\mathbb{R}^{2}\backslash\{0\},\ \frac{1}{2}\leq\sum_{j\in\mathbb{Z}}\varphi^{2}(2^{-j}\xi)\leq 1.

ℱ\mathcal{F} denotes the Fourier transform and its inverse is represented by ℱ−1\mathcal{F}^{-1}. Suppose that uu is a tempered distribution in 𝒮h′​(ℝ2)\mathcal{S}^{\prime}_{h}(\mathbb{R}^{2}). For all j∈ℤj\in\mathbb{Z}, define

Δ˙ju=ℱ−1(φ(2−j⋅)ℱu).\dot{\Delta}_{j}u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u).

Then the Littlewood-Paley decomposition is defined by:

u=∑j∈ℤΔ˙j​uin​𝒮′​(ℝ2).u=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}u\quad\text{in}\ \mathcal{S}^{\prime}(\mathbb{R}^{2}).

Let s∈ℝ, 1≤p,r≤∞.s\in\mathbb{R},\ 1\leq p,r\leq\infty. The homogeneous Besov space B˙p,rs\dot{B}^{s}_{p,r} is given as follows

B˙p,rs={u∈𝒮h′:‖u‖B˙p,rs=‖(2j​s​‖Δ˙j​u‖Lp)j‖lr​(ℤ)<∞}.\dot{B}^{s}_{p,r}=\{u\in\mathcal{S}^{\prime}_{h}:\|u\|_{\dot{B}^{s}_{p,r}}=\Big{\|}(2^{js}\|\dot{\Delta}_{j}u\|_{L^{p}})_{j}\Big{\|}_{l^{r}(\mathbb{Z})}<\infty\}.

Take Λ​f=ℱ−1​(|ξ|​ℱ​(f)),\Lambda f=\mathcal{F}^{-1}(|\xi|\mathcal{F}(f)), we introduce the Gagliardo-Nirenberg inequality of Sobolev type with d=2d=2.

Lemma 2.2.

[23] For d=2,p∈[2,+∞)d=2,~{}p\in[2,+\infty) and 0≤s,s1≤s20\leq s,s_{1}\leq s_{2}, there holds

‖Λs​f‖Lp≤C​‖Λs1​f‖L21−θ​‖Λs2​f‖L2θ,\|\Lambda^{s}f\|_{L^{p}}\leq C\|\Lambda^{s_{1}}f\|^{1-\theta}_{L^{2}}\|\Lambda^{s_{2}}f\|^{\theta}_{L^{2}},

where 0≤θ≤10\leq\theta\leq 1 and

s+1−2p=s1​(1−θ)+θ​s2.s+1-\frac{2}{p}=s_{1}(1-\theta)+\theta s_{2}.

Note that we also require that 0<θ<10<\theta<1, 0≤s1≤s0\leq s_{1}\leq s, when p=∞p=\infty.

Lemma 2.3.

[22] Assume that s≥1s\geq 1, p,p1,p4∈(1,∞)p,p_{1},p_{4}\in(1,\infty) and 1p=1p1+1p2=1p3+1p4\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}, then we obtain

‖[Λs,f]​g‖Lp≤C​(‖Λs​f‖Lp1​‖g‖Lp2+‖∇f‖Lp3​‖Λs−1​g‖Lp4).\displaystyle\|[\Lambda^{s},f]g\|_{L^{p}}\leq C(\|\Lambda^{s}f\|_{L^{p_{1}}}\|g\|_{L^{p_{2}}}+\|\nabla f\|_{L^{p_{3}}}\|\Lambda^{s-1}g\|_{L^{p_{4}}}).

3 Optimal decay rate with β=1\beta=1

In this section, we present optimal decay rate for the generalized Oldroyd-B model (1.8) in critical case with β=1\beta=1. Inspired by [12], we consider the coupling effect between (u,τ)(u,\tau). We need to introduce the following energy and dissipation functionals for (u,τ)(u,\tau):

Eθ=‖Λθ​(u,τ)‖Hs−θ2+2​k​⟨∇Λθ​u,Λθ​τ⟩Hs−1−θ,E_{\theta}=\|\Lambda^{\theta}(u,\tau)\|^{2}_{H^{s-\theta}}+2k\langle\nabla\Lambda^{\theta}u,\Lambda^{\theta}\tau\rangle_{H^{s-1-\theta}},

and

Dθ=k2​‖∇Λθ​u‖Hs−1−θ2+‖∇Λθ​τ‖Hs−θ2,D_{\theta}=\frac{k}{2}\|\nabla\Lambda^{\theta}u\|^{2}_{H^{s-1-\theta}}+\|\nabla\Lambda^{\theta}\tau\|^{2}_{H^{s-\theta}},

where θ=0​o​r​1\theta=0~{}or~{}1. The key point is the refinement of Schonbek’s [26] strategy. Using a different Fourier splitting method and taking Fourier transform in (1.8), we obtain the initial L2L^{2} decay rate in following proposition.

Proposition 3.1.

Assume that (u0,τ0)(u_{0},\tau_{0}) satisfy the condition in Theorem 1.1 and (u0,τ0)∈B˙2,∞−1(u_{0},\tau_{0})\in\dot{B}^{-1}_{2,\infty}. For any l∈N+l\in N^{+}, there exists constant ClC_{l} such that

(3.1) E0​(t)≤Cl​ln−l⁡(e+t).\displaystyle E_{0}(t)\leq C_{l}\ln^{-l}(e+t).
Proof.

Taking β=1\beta=1 in Theorem 1.1, we get the following global energy estimate:

(3.2) dd​t​E0​(t)+D0​(t)≤0.\displaystyle\frac{d}{dt}E_{0}(t)+D_{0}(t)\leq 0.

Denote that S0​(t)={ξ:|ξ|2≤2​C2​f′​(t)f​(t)}S_{0}(t)=\{\xi:|\xi|^{2}\leq 2C_{2}\frac{f^{\prime}(t)}{f(t)}\}, where f​(t)=ln3⁡(e+t)f(t)=\ln^{3}(e+t) and C2C_{2} is large enough. By (3.2), we deduce that

(3.3) dd​t​[f​(t)​E0​(t)]+C2​f′​(t)​(k2​‖u‖Hs2+‖τ‖Hs2)≤C​f′​(t)​∫S0​(t)|u^​(ξ)|2+|τ^​(ξ)|2​d​ξ.\displaystyle\frac{d}{dt}[f(t)E_{0}(t)]+C_{2}f^{\prime}(t)(\frac{k}{2}\|u\|^{2}_{H^{s}}+\|\tau\|^{2}_{H^{s}})\leq Cf^{\prime}(t)\int_{S_{0}(t)}|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2}d\xi.

The L2L^{2} estimate to the low frequency part of (u,τ)(u,\tau) is useful for studying time decay rates. Applying Fourier transform to (1.8), we have

(3.6) {u^tj+i​ξj​P^−i​ξk​τ^j​k=F^j,τ^tj​k+|ξ|2​τ^j​k−i2​(ξk​u^j+ξj​u^k)=G^j​k,\displaystyle\left\{\begin{array}[]{ll}\hat{u}^{j}_{t}+i\xi_{j}\hat{P}-i\xi_{k}\hat{\tau}^{jk}=\hat{F}^{j},\\[4.30554pt] \hat{\tau}^{jk}_{t}+|\xi|^{2}\hat{\tau}^{jk}-\frac{i}{2}(\xi_{k}\hat{u}^{j}+\xi_{j}\hat{u}^{k})=\hat{G}^{jk},\\[4.30554pt] \end{array}\right.

where F=−u⋅∇uF=-u\cdot\nabla u and G=−u⋅∇τ−Q​(∇u,τ)G=-u\cdot\nabla\tau-Q(\nabla u,\tau). According to ℛe[iξ⊗u^¯(t,ξ):τ^]+ℛe[iξ⊗u^:τ^¯]=0\mathcal{R}e[i\xi\otimes\bar{\hat{u}}(t,\xi):\hat{\tau}]+\mathcal{R}e[i\xi\otimes\hat{u}:\bar{\hat{\tau}}]=0 and (3.6), we infer that

(3.7) 12​dd​t​(|u^|2+|τ^|2)+|ξ|2​|τ^|2=ℛ​e​[F^⋅u^¯]+ℛ​e​[G^​τ^¯].\displaystyle\frac{1}{2}\frac{d}{dt}(|\hat{u}|^{2}+|\hat{\tau}|^{2})+|\xi|^{2}|\hat{\tau}|^{2}=\mathcal{R}e[\hat{F}\cdot\bar{\hat{u}}]+\mathcal{R}e[\hat{G}\bar{\hat{\tau}}].

Integrating (3.7) in time on [0,t][0,t], we get

(3.8) |u^|2+|τ^|2≤C​(|u^0|2+|τ^0|2)+C​∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s′.\displaystyle|\hat{u}|^{2}+|\hat{\tau}|^{2}\leq C(|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2})+C\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|ds^{\prime}.

Integrating (3.8) over S0​(t)S_{0}(t) with ξ\xi, then we obtain the following estimate for (3.6):

(3.9) ∫S0​(t)|u^|2+|τ^|2​d​ξ≤C​∫S0​(t)|u^0|2+|τ^0|2​d​ξ+C​∫S0​(t)∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s′​d​ξ.\displaystyle\int_{S_{0}(t)}|\hat{u}|^{2}+|\hat{\tau}|^{2}d\xi\leq C\int_{S_{0}(t)}|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2}d\xi+C\int_{S_{0}(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|ds^{\prime}d\xi.

According to E0​(0)<∞E_{0}(0)<\infty, (u0,τ0)∈B˙2,∞−1(u_{0},\tau_{0})\in\dot{B}^{-1}_{2,\infty} and applying Lemma 2.1, we infer that

(3.10) ∫S0​(t)(|u^0|2+|τ^0|2)​𝑑ξ\displaystyle\int_{S_{0}(t)}(|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2})d\xi ≤∑j≤log2⁡[43​C212​f′​(t)f​(t)]∫ℝ22​φ2​(2−j​ξ)​(|u^0|2+|τ^0|2)​𝑑ξ\displaystyle\leq\sum_{j\leq\log_{2}[\frac{4}{3}C_{2}^{\frac{1}{2}}\sqrt{\frac{f^{\prime}(t)}{f(t)}}]}\int_{\mathbb{R}^{2}}2\varphi^{2}(2^{-j}\xi)(|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2})d\xi
≤∑j≤log2⁡[43​C212​f′​(t)f​(t)]C​(‖Δ˙j​u0‖L22+‖Δ˙j​τ0‖L22)\displaystyle\leq\sum_{j\leq\log_{2}[\frac{4}{3}C_{2}^{\frac{1}{2}}\sqrt{\frac{f^{\prime}(t)}{f(t)}}]}C(\|\dot{\Delta}_{j}u_{0}\|^{2}_{L^{2}}+\|\dot{\Delta}_{j}\tau_{0}\|^{2}_{L^{2}})
≤C​f′​(t)f​(t)​‖(u0,τ0)‖B˙2,∞−12.\displaystyle\leq C\frac{f^{\prime}(t)}{f(t)}\|(u_{0},\tau_{0})\|^{2}_{\dot{B}^{-1}_{2,\infty}}.

By (3.2) and Minkowski’s inequality, we obtain

(3.11) ∫S0​(t)∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s′​d​ξ\displaystyle\int_{S_{0}(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|ds^{\prime}d\xi =∫0t∫S0​(t)|F^⋅u^¯|+|G^⋅τ^¯|​d​ξ​d​s′\displaystyle=\int_{0}^{t}\int_{S_{0}(t)}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|d\xi ds^{\prime}
≤C​f′​(t)f​(t)​∫0t(‖u‖L22+‖τ‖L22)​D0​(s′)12​𝑑s′\displaystyle\leq C\sqrt{\frac{f^{\prime}(t)}{f(t)}}\int_{0}^{t}(\|u\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{2}})D_{0}(s^{\prime})^{\frac{1}{2}}ds^{\prime}
≤C​f′​(t)f​(t)​(1+t)12.\displaystyle\leq C\sqrt{\frac{f^{\prime}(t)}{f(t)}}(1+t)^{\frac{1}{2}}.

It follows from (3.9)-(3.11) that

(3.12) ∫S0​(t)|u^|2+|τ^|2​d​ξ≤C​ln−12⁡(e+t).\displaystyle\int_{S_{0}(t)}|\hat{u}|^{2}+|\hat{\tau}|^{2}d\xi\leq C\ln^{-\frac{1}{2}}(e+t).

Combining (3.3) and (3.12), we have

(3.13) dd​t​[f​(t)​E0​(t)]≤C​f′​(t)​ln−12⁡(e+t).\displaystyle\frac{d}{dt}[f(t)E_{0}(t)]\leq Cf^{\prime}(t)\ln^{-\frac{1}{2}}(e+t).

Consequently, we get the initial time decay rate:

(3.14) E0​(t)≤C​ln−12⁡(e+t).\displaystyle E_{0}(t)\leq C\ln^{-\frac{1}{2}}(e+t).

By virtue of the bootstrap argument, for any l∈N+l\in N^{+}, we infer that E0≤Cl​ln−l⁡(e+t).E_{0}\leq C_{l}\ln^{-l}(e+t). ∎

We now consider the the initial time decay rate of E1​(t)E_{1}(t).

Proposition 3.2.

Let (u0,τ0)∈B˙2,∞−1(u_{0},\tau_{0})\in\dot{B}^{-1}_{2,\infty}. Under the condition in Theorem 1.1, for any l∈N+l\in N^{+}, then there exists a constant ClC_{l} such that

(3.15) E1​(t)≤Cl​(1+t)−1​ln−l⁡(e+t).\displaystyle E_{1}(t)\leq C_{l}(1+t)^{-1}\ln^{-l}(e+t).
Proof.

Since ⟨u⋅∇u,Δ​u⟩=0\langle u\cdot\nabla u,\Delta u\rangle=0 with d=2d=2, then we have

(3.16) 12​dd​t​‖∇(u,τ)‖L22+‖∇2τ‖L22=⟨u⋅∇τ+Q​(u,τ),Δ​τ⟩≤C​δ​D1,\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla(u,\tau)\|^{2}_{L^{2}}+\|\nabla^{2}\tau\|^{2}_{L^{2}}=\langle u\cdot\nabla\tau+Q(u,\tau),\Delta\tau\rangle\leq C\delta D_{1},

and

(3.17) dd​t​⟨Δ​τ,∇u⟩+12​‖∇2u‖L22\displaystyle\frac{d}{dt}\langle\Delta\tau,\nabla u\rangle+\frac{1}{2}\|\nabla^{2}u\|^{2}_{L^{2}} =⟨d​i​v​(u⋅∇τ+Q​(u,τ)−Δ​τ),Δ​u⟩\displaystyle=\langle div~{}(u\cdot\nabla\tau+Q(u,\tau)-\Delta\tau),\Delta u\rangle
−⟨∇ℙ​(u⋅∇u−d​i​v​τ),Δ​τ⟩\displaystyle-\langle\nabla\mathbb{P}(u\cdot\nabla u-div~{}\tau),\Delta\tau\rangle
≤C​δ​D1+C​‖∇2τ‖L22.\displaystyle\leq C\delta D_{1}+C\|\nabla^{2}\tau\|^{2}_{L^{2}}.

Using Lemma 2.3, we obtain

(3.18) 12​dd​t​‖Λs​(u,τ)‖L22+‖∇Λs​τ‖L22\displaystyle\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\|\nabla\Lambda^{s}\tau\|^{2}_{L^{2}} =−⟨[Λs,u⋅∇]​u,Λs​u⟩−⟨Λs​(u⋅∇τ+Q​(u,τ)),Λs​τ⟩\displaystyle=-\langle[\Lambda^{s},u\cdot\nabla]u,\Lambda^{s}u\rangle-\langle\Lambda^{s}(u\cdot\nabla\tau+Q(u,\tau)),\Lambda^{s}\tau\rangle
≤C(∥∇u∥L∞∥Λsu∥L22+∥τ∥L∞∥Λsu∥L2∥Λs+1τ∥L2\displaystyle\leq C(\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}u\|^{2}_{L^{2}}+\|\tau\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}
+∥u∥L∞∥Λsτ∥L2∥Λs+1τ∥L2+∥∇u∥L∞∥Λs−1τ∥L2∥Λs+1τ∥L2).\displaystyle+\|u\|_{L^{\infty}}\|\Lambda^{s}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}+\|\nabla u\|_{L^{\infty}}\|\Lambda^{s-1}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}).

By Lemma 2.2, we have ‖∇u‖L∞​‖Λs−1​τ‖L2≤C​‖u‖L2s−2s​‖∇su‖L22s​‖τ‖L21s​‖Λs​τ‖L2s−1s\|\nabla u\|_{L^{\infty}}\|\Lambda^{s-1}\tau\|_{L^{2}}\leq C\|u\|^{\frac{s-2}{s}}_{L^{2}}\|\nabla^{s}u\|^{\frac{2}{s}}_{L^{2}}\|\tau\|^{\frac{1}{s}}_{L^{2}}\|\Lambda^{s}\tau\|^{\frac{s-1}{s}}_{L^{2}}. This together with (3.18) and Theorem 1.1 ensure that

(3.19) dd​t​‖Λs​(u,τ)‖L22+‖∇Λs​τ‖L22≤C​δ​D1.\displaystyle\frac{d}{dt}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\|\nabla\Lambda^{s}\tau\|^{2}_{L^{2}}\leq C\delta D_{1}.

Using Lemmas 2.2 and 2.3, we deduce that

(3.20) dd​t​⟨Λs−1​τ,−∇Λs−1​u⟩+12​‖∇Λs−1​u‖L22\displaystyle\frac{d}{dt}\langle\Lambda^{s-1}\tau,-\nabla\Lambda^{s-1}u\rangle+\frac{1}{2}\|\nabla\Lambda^{s-1}u\|^{2}_{L^{2}}
=⟨Λs−1​(u⋅∇τ+Q​(u,τ)−Δ​τ),∇Λs−1​u⟩\displaystyle=\langle\Lambda^{s-1}(u\cdot\nabla\tau+Q(u,\tau)-\Delta\tau),\nabla\Lambda^{s-1}u\rangle
−⟨Λs−1​ℙ​(u⋅∇u−d​i​v​τ),d​i​v​Λs−1​τ⟩\displaystyle-\langle\Lambda^{s-1}\mathbb{P}(u\cdot\nabla u-div~{}\tau),div~{}\Lambda^{s-1}\tau\rangle
≤C​‖Λs​u‖L2​(‖u‖L∞​‖Λs​τ‖L2+‖τ‖L∞​‖Λs​u‖L2+‖Λs+1​τ‖L2)\displaystyle\leq C\|\Lambda^{s}u\|_{L^{2}}(\|u\|_{L^{\infty}}\|\Lambda^{s}\tau\|_{L^{2}}+\|\tau\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}+\|\Lambda^{s+1}\tau\|_{L^{2}})
+C​‖Λs​τ‖L2​(‖u‖L∞​‖Λs​u‖L2+‖Λs​τ‖L2)\displaystyle+C\|\Lambda^{s}\tau\|_{L^{2}}(\|u\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}+\|\Lambda^{s}\tau\|_{L^{2}})
≤C​δ​D1+C​‖Λs​τ‖H12.\displaystyle\leq C\delta D_{1}+C\|\Lambda^{s}\tau\|^{2}_{H^{1}}.

Combining (3.16)-(3.20), we obtain

(3.21) dd​t​E1+D1≤0,\displaystyle\frac{d}{dt}E_{1}+D_{1}\leq 0,

which implies that

(3.22) dd​t​[f​(t)​E1]+C2​f′​(t)​(k2​‖∇u‖Hs−12+‖∇τ‖Hs−12)≤C​f′​(t)​∫S0​(t)|ξ|2​(|u^​(ξ)|2+|τ^​(ξ)|2)​𝑑ξ.\displaystyle\frac{d}{dt}[f(t)E_{1}]+C_{2}f^{\prime}(t)(\frac{k}{2}\|\nabla u\|^{2}_{H^{s-1}}+\|\nabla\tau\|^{2}_{H^{s-1}})\leq Cf^{\prime}(t)\int_{S_{0}(t)}|\xi|^{2}(|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2})d\xi.

By Proposition 3.1, we obtain

(3.23) f′​(t)​∫S0​(t)|ξ|2​(|u^​(ξ)|2+|τ^​(ξ)|2)​𝑑ξ≤C​(1+t)−2​ln−l+1⁡(e+t).\displaystyle f^{\prime}(t)\int_{S_{0}(t)}|\xi|^{2}(|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2})d\xi\leq C(1+t)^{-2}\ln^{-l+1}(e+t).

According to (3.22)-(3.23), we infer that

(3.24) E1≤C​(1+t)−1​ln−l⁡(e+t).\displaystyle E_{1}\leq C(1+t)^{-1}\ln^{-l}(e+t).

Therefore, we complete the proof of Proposition 3.2. ∎

By virtue of the standard method, we can’t immediately obtain the optimal decay rate. However, we obtain a weak result as follows.

Proposition 3.3.

Assume that (u0,τ0)(u_{0},\tau_{0}) satisfy the condition in Proposition 3.1, there holds

(3.25) E0​(t)≤C​(1+t)−12,\displaystyle E_{0}(t)\leq C(1+t)^{-\frac{1}{2}},

and

(3.26) E1​(t)≤C​(1+t)−32.\displaystyle E_{1}(t)\leq C(1+t)^{-\frac{3}{2}}.
Proof.

Define S​(t)={ξ:|ξ|2≤C2​(1+t)−1}S(t)=\{\xi:|\xi|^{2}\leq C_{2}(1+t)^{-1}\} with C2C_{2} large enough. By (3.2), we infer that

(3.27) dd​t​E0​(t)+k​C22​(1+t)​‖u‖Hs2+C21+t​‖τ‖Hs2≤C1+t​∫S​(t)|u^​(ξ)|2+|τ^​(ξ)|2​d​ξ.\displaystyle\frac{d}{dt}E_{0}(t)+\frac{kC_{2}}{2(1+t)}\|u\|^{2}_{H^{s}}+\frac{C_{2}}{1+t}\|\tau\|^{2}_{H^{s}}\leq\frac{C}{1+t}\int_{S(t)}|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2}d\xi.

Integrating (3.8) over S​(t)S(t) with ξ\xi, then we get

(3.28) ∫S​(t)|u^|2+|τ^|2​d​ξ≤C​∫S​(t)|u^0|2+|τ^0|2​d​ξ+C​∫S​(t)∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s′​d​ξ.\displaystyle\int_{S(t)}|\hat{u}|^{2}+|\hat{\tau}|^{2}d\xi\leq C\int_{S(t)}|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2}d\xi+C\int_{S(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|ds^{\prime}d\xi.

According to (u0,τ0)∈Hs∩B˙2,∞−1(u_{0},\tau_{0})\in H^{s}\cap\dot{B}^{-1}_{2,\infty} and applying Lemma 2.1, we deduce that

(3.29) ∫S​(t)|u^0|2+|τ^0|2​d​ξ\displaystyle\int_{S(t)}|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2}d\xi ≤∑j≤log2⁡[43​C212​(1+t)−12]∫ℝ22​φ2​(2−j​ξ)​(|u^0|2+|τ^0|2)​𝑑ξ\displaystyle\leq\sum_{j\leq\log_{2}[\frac{4}{3}C_{2}^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}]}\int_{\mathbb{R}^{2}}2\varphi^{2}(2^{-j}\xi)(|\hat{u}_{0}|^{2}+|\hat{\tau}_{0}|^{2})d\xi
≤∑j≤log2⁡[43​C212​(1+t)−12]2​(‖Δ˙j​u0‖L22+‖Δ˙j​τ0‖L22)\displaystyle\leq\sum_{j\leq\log_{2}[\frac{4}{3}C_{2}^{\frac{1}{2}}(1+t)^{-\frac{1}{2}}]}2(\|\dot{\Delta}_{j}u_{0}\|^{2}_{L^{2}}+\|\dot{\Delta}_{j}\tau_{0}\|^{2}_{L^{2}})
≤C​(1+t)−1​‖(u0,τ0)‖B˙2,∞−12,\displaystyle\leq C(1+t)^{-1}\|(u_{0},\tau_{0})\|^{2}_{\dot{B}^{-1}_{2,\infty}},

and

(3.30) ∫S​(t)∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s′​d​ξ≤C​(1+t)−12​∫0t(‖u‖L22+‖τ‖L22)​(‖∇u‖L2+‖∇τ‖L2)​𝑑s′.\displaystyle\int_{S(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|ds^{\prime}d\xi\leq C(1+t)^{-\frac{1}{2}}\int_{0}^{t}(\|u\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{2}})(\|\nabla u\|_{L^{2}}+\|\nabla\tau\|_{L^{2}})ds^{\prime}.

Combining (3.27)-(3.30), we have

(3.31) dd​t​E0​(t)+k​C22​(1+t)​‖u‖Hs2+C21+t​‖τ‖Hs2\displaystyle\frac{d}{dt}E_{0}(t)+\frac{kC_{2}}{2(1+t)}\|u\|^{2}_{H^{s}}+\frac{C_{2}}{1+t}\|\tau\|^{2}_{H^{s}}
≤C1+t​[(1+t)−1+(1+t)−12​∫0t(‖u‖L22+‖τ‖L22)​(‖∇u‖L2+‖∇τ‖L2)​𝑑s′],\displaystyle\leq\frac{C}{1+t}[(1+t)^{-1}+(1+t)^{-\frac{1}{2}}\int_{0}^{t}(\|u\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{2}})(\|\nabla u\|_{L^{2}}+\|\nabla\tau\|_{L^{2}})ds^{\prime}],

which implies that

(3.32) (1+t)32​E0​(t)≤C​(1+t)12+C​(1+t)​∫0t(‖u‖L22+‖τ‖L22)​(‖∇u‖L2+‖∇τ‖L2)​𝑑s′.\displaystyle(1+t)^{\frac{3}{2}}E_{0}(t)\leq C(1+t)^{\frac{1}{2}}+C(1+t)\int_{0}^{t}(\|u\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{2}})(\|\nabla u\|_{L^{2}}+\|\nabla\tau\|_{L^{2}})ds^{\prime}.

Let N​(t)=sup0≤s≤t(1+s)12​E0​(s)N(t)=\sup_{0\leq s\leq t}(1+s)^{\frac{1}{2}}E_{0}(s). By (3.32), we obtain

(3.33) N​(t)≤C+C​∫0tN​(s)​(1+s)−12​(‖∇u‖L2+‖∇τ‖L2)​𝑑s′.\displaystyle N(t)\leq C+C\int_{0}^{t}N(s)(1+s)^{-\frac{1}{2}}(\|\nabla u\|_{L^{2}}+\|\nabla\tau\|_{L^{2}})ds^{\prime}.

Applying Gronwall’s inequality and Proposition 3.2, we deduce that N​(t)≤CN(t)\leq C, which implies that

(3.34) E0≤C​(1+t)−12.\displaystyle E_{0}\leq C(1+t)^{-\frac{1}{2}}.

According to (3.21), we infer that

(3.35) dd​t​E1+C21+t​(k2​‖∇u‖Hs−12+‖∇τ‖Hs−12)≤C1+t​∫S​(t)|ξ|2​(|u^​(ξ)|2+|τ^​(ξ)|2)​𝑑ξ.\displaystyle\frac{d}{dt}E_{1}+\frac{C_{2}}{1+t}(\frac{k}{2}\|\nabla u\|^{2}_{H^{s-1}}+\|\nabla\tau\|^{2}_{H^{s-1}})\leq\frac{C}{1+t}\int_{S(t)}|\xi|^{2}(|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2})d\xi.

By (3.34), we get

(3.36) C1+t​∫S​(t)|ξ|2​(|u^​(ξ)|2+|τ^​(ξ)|2)​𝑑ξ≤C​(1+t)−2​(‖u‖L22+‖τ‖L22)≤C​(1+t)−52.\displaystyle\frac{C}{1+t}\int_{S(t)}|\xi|^{2}(|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2})d\xi\leq C(1+t)^{-2}(\|u\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{2}})\leq C(1+t)^{-\frac{5}{2}}.

This together with (3.2), (3.34) and (3.35) ensure that

(3.37) E1≤C​(1+t)−32.\displaystyle E_{1}\leq C(1+t)^{-\frac{3}{2}}.

We thus complete the proof of Proposition 3.3. ∎

Using Proposition 3.3, we can prove that the solution of (1.8) belongs to some Besov space with negative index.

Lemma 3.4.

Let 0<α,σ≤10<\alpha,\sigma\leq 1 and σ<2​α\sigma<2\alpha. Under the condition in Proposition 3.1. If

(3.38) E0​(t)≤C​(1+t)−α,E1​(t)≤C​(1+t)−α−1,\displaystyle E_{0}(t)\leq C(1+t)^{-\alpha},~{}~{}~{}~{}E_{1}(t)\leq C(1+t)^{-\alpha-1},

then we have

(3.39) (u,τ)∈L∞​(0,∞;B˙2,∞−σ).\displaystyle(u,\tau)\in L^{\infty}(0,\infty;\dot{B}^{-\sigma}_{2,\infty}).
Proof.

Applying Δ˙j\dot{\Delta}_{j} to (1.8), we have

(3.42) {Δ˙j​ut+∇Δ˙j​P−d​i​v​Δ˙j​τ=Δ˙j​F,Δ˙j​τt−Δ​Δ˙j​τ−Δ˙j​D​(u)=Δ˙j​G.\displaystyle\left\{\begin{array}[]{ll}\dot{\Delta}_{j}u_{t}+\nabla\dot{\Delta}_{j}P-div~{}\dot{\Delta}_{j}\tau=\dot{\Delta}_{j}F,\\[4.30554pt] \dot{\Delta}_{j}\tau_{t}-\Delta\dot{\Delta}_{j}\tau-\dot{\Delta}_{j}D(u)=\dot{\Delta}_{j}G.\\[4.30554pt] \end{array}\right.

We first deduce from (3.42) that

(3.43) dd​t​(‖Δ˙j​u‖L22+‖Δ˙j​τ‖L22)+2​‖∇Δ˙j​τ‖L22≤C​(‖Δ˙j​F‖L2​‖Δ˙j​u‖L2+‖Δ˙j​G‖L2​‖Δ˙j​τ‖L2).\displaystyle\frac{d}{dt}(\|\dot{\Delta}_{j}u\|^{2}_{L^{2}}+\|\dot{\Delta}_{j}\tau\|^{2}_{L^{2}})+2\|\nabla\dot{\Delta}_{j}\tau\|^{2}_{L^{2}}\leq C(\|\dot{\Delta}_{j}F\|_{L^{2}}\|\dot{\Delta}_{j}u\|_{L^{2}}+\|\dot{\Delta}_{j}G\|_{L^{2}}\|\dot{\Delta}_{j}\tau\|_{L^{2}}).

Applying 2−2​j​σ2^{-2j\sigma} to (3.43) and taking l∞l^{\infty} norm, we obtain

(3.44) dd​t​(‖u‖B˙2,∞−σ2+‖τ‖B˙2,∞−σ2)≤C​(‖F‖B˙2,∞−σ​‖u‖B˙2,∞−σ+‖G‖B˙2,∞−σ​‖τ‖B˙2,∞−σ).\displaystyle\frac{d}{dt}(\|u\|^{2}_{\dot{B}^{-\sigma}_{2,\infty}}+\|\tau\|^{2}_{\dot{B}^{-\sigma}_{2,\infty}})\leq C(\|F\|_{\dot{B}^{-\sigma}_{2,\infty}}\|u\|_{\dot{B}^{-\sigma}_{2,\infty}}+\|G\|_{\dot{B}^{-\sigma}_{2,\infty}}\|\tau\|_{\dot{B}^{-\sigma}_{2,\infty}}).

Let M​(t)=∑0≤s≤t‖u‖B˙2,∞−σ+‖τ‖B˙2,∞−σM(t)=\sum_{0\leq s\leq t}\|u\|_{\dot{B}^{-\sigma}_{2,\infty}}+\|\tau\|_{\dot{B}^{-\sigma}_{2,\infty}}. According to (3.44), we infer that

(3.45) M2​(t)\displaystyle M^{2}(t) ≤C​M2​(0)+C​M​(t)​∫0t‖F‖B˙2,∞−σ+‖G‖B˙2,∞−σ​d​s′.\displaystyle\leq CM^{2}(0)+CM(t)\int_{0}^{t}\|F\|_{\dot{B}^{-\sigma}_{2,\infty}}+\|G\|_{\dot{B}^{-\sigma}_{2,\infty}}ds^{\prime}.

Using (3.38) and inclusion between Lesbesgue and Besov space, we deduce that

(3.46) ∫0t‖(F,G)‖B˙2,∞−σ​𝑑s′\displaystyle\int_{0}^{t}\|(F,G)\|_{\dot{B}^{-\sigma}_{2,\infty}}ds^{\prime} ≤C​∫0t‖(F,G)‖L2σ+1​𝑑s′\displaystyle\leq C\int_{0}^{t}\|(F,G)\|_{L^{\frac{2}{\sigma+1}}}ds^{\prime}
≤C​∫0t(‖u‖L2σ+‖τ‖L2σ)​(‖∇u‖L2+‖∇τ‖L2)​𝑑s′\displaystyle\leq C\int_{0}^{t}(\|u\|_{L^{\frac{2}{\sigma}}}+\|\tau\|_{L^{\frac{2}{\sigma}}})(\|\nabla u\|_{L^{2}}+\|\nabla\tau\|_{L^{2}})ds^{\prime}
≤C​∫0t‖(u,τ)‖L2σ​‖∇(u,τ)‖L22−σ​𝑑s′\displaystyle\leq C\int_{0}^{t}\|(u,\tau)\|^{\sigma}_{L^{2}}\|\nabla(u,\tau)\|^{2-\sigma}_{L^{2}}ds^{\prime}
≤C​∫0t(1+s)−(1+α−σ2)​𝑑s′≤C.\displaystyle\leq C\int_{0}^{t}(1+s)^{-(1+\alpha-\frac{\sigma}{2})}ds^{\prime}\leq C.

Combining (3.45) and (3.46), we get M​(t)≤CM(t)\leq C. ∎

Lemma 3.5.

Let 0<β,σ≤10<\beta,\sigma\leq 1 and 12≤α\frac{1}{2}\leq\alpha. Under the condition in Proposition 3.1. For any t∈[0,+∞)t\in[0,+\infty), if

(3.47) E0​(t)≤C​(1+t)−α,E1​(t)≤C​(1+t)−α−1,\displaystyle E_{0}(t)\leq C(1+t)^{-\alpha},~{}~{}~{}~{}E_{1}(t)\leq C(1+t)^{-\alpha-1},

and

(3.48) (u,τ)∈L∞​(0,∞;B˙2,∞−σ),\displaystyle(u,\tau)\in L^{\infty}(0,\infty;\dot{B}^{-\sigma}_{2,\infty}),

then there exists a constant CC such that

(3.49) E0​(t)≤C​(1+t)−βa​n​dE1​(t)≤C​(1+t)−β−1,\displaystyle E_{0}(t)\leq C(1+t)^{-\beta}~{}~{}~{}~{}and~{}~{}~{}~{}E_{1}(t)\leq C(1+t)^{-\beta-1},

where β<σ+12\beta<\frac{\sigma+1}{2} for α=12\alpha=\frac{1}{2} and β=σ+12\beta=\frac{\sigma+1}{2} for α>12\alpha>\frac{1}{2}.

Proof.

According to the proof of Proposition 3.2, we have

(3.50) dd​t​E0​(t)+k​C22​(1+t)​‖u‖Hs2+C21+t​‖τ‖Hs2\displaystyle\frac{d}{dt}E_{0}(t)+\frac{kC_{2}}{2(1+t)}\|u\|^{2}_{H^{s}}+\frac{C_{2}}{1+t}\|\tau\|^{2}_{H^{s}}
≤C​C21+t​((1+t)−1+∫S​(t)∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s​d​ξ).\displaystyle\leq\frac{CC_{2}}{1+t}((1+t)^{-1}+\int_{S(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|dsd\xi).

Using (3.47) and (3.48), we infer that

(3.51) ∫S​(t)∫0t|F^⋅u^¯|+|G^⋅τ^¯|​d​s​d​ξ\displaystyle\int_{S(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}|+|\hat{G}\cdot\bar{\hat{\tau}}|dsd\xi ≤C​∫0t(‖F‖L1​∫S​(t)|u^|​𝑑ξ+‖G‖L1​∫S​(t)|τ^|​𝑑ξ)​𝑑s\displaystyle\leq C\int_{0}^{t}(\|F\|_{L^{1}}\int_{S(t)}|\hat{u}|d\xi+\|G\|_{L^{1}}\int_{S(t)}|\hat{\tau}|d\xi)ds
≤C​(1+t)−12​∫0t(‖F‖L1+‖G‖L1)​(∫S​(t)|u^|2+|τ^|2​d​ξ)12​𝑑s\displaystyle\leq C(1+t)^{-\frac{1}{2}}\int_{0}^{t}(\|F\|_{L^{1}}+\|G\|_{L^{1}})(\int_{S(t)}|\hat{u}|^{2}+|\hat{\tau}|^{2}d\xi)^{\frac{1}{2}}ds
≤C​(1+t)−σ+12​M​(t)​∫0t‖F‖L1+‖G‖L1​d​s\displaystyle\leq C(1+t)^{-\frac{\sigma+1}{2}}M(t)\int_{0}^{t}\|F\|_{L^{1}}+\|G\|_{L^{1}}ds
≤C​(1+t)−σ+12​∫0t(1+s)−(α+12)​𝑑s\displaystyle\leq C(1+t)^{-\frac{\sigma+1}{2}}\int_{0}^{t}(1+s)^{-(\alpha+\frac{1}{2})}ds
≤C​(1+t)−β.\displaystyle\leq C(1+t)^{-\beta}.

By virtue of (3.50) and (3.51), we get

(3.52) E0​(t)≤C​(1+t)−β.\displaystyle E_{0}(t)\leq C(1+t)^{-\beta}.

Moreover, we have

(3.53) dd​t​E1+C21+t​(k2​‖∇u‖Hs−12+‖∇τ‖Hs−12)\displaystyle\frac{d}{dt}E_{1}+\frac{C_{2}}{1+t}(\frac{k}{2}\|\nabla u\|^{2}_{H^{s-1}}+\|\nabla\tau\|^{2}_{H^{s-1}}) ≤C1+t​∫S​(t)|ξ|2​(|u^​(ξ)|2+|τ^​(ξ)|2)​𝑑ξ.\displaystyle\leq\frac{C}{1+t}\int_{S(t)}|\xi|^{2}(|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2})d\xi.
≤C​(1+t)−2​(‖u‖L22+‖τ‖L22)\displaystyle\leq C(1+t)^{-2}(\|u\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{2}})
≤C​(1+t)−2−β,\displaystyle\leq C(1+t)^{-2-\beta},

which implies that E1≤C​(1+t)−1−βE_{1}\leq C(1+t)^{-1-\beta}. ∎

We improve the decay rates in H1H^{1} by Lemmas 3.4 and 3.5. However, considering the decay rate for the highest derivative of the solution to (1.8), the main difficulty is unclosed energy estimate. To overcome this difficulty, we introduce a new method which flexibly combines the Fourier splitting method and the time weighted energy estimate.

Proposition 3.6.

Assume that (u0,τ0)(u_{0},\tau_{0}) satisfy the condition in Proposition 3.1, then there exists a constant CC such that

(3.54) ‖Λs1​(u,τ)‖L2≤C​(1+t)−1+s12,\displaystyle\|\Lambda^{s_{1}}(u,\tau)\|_{L^{2}}\leq C(1+t)^{-\frac{1+{s_{1}}}{2}},

where s1∈[0,s]s_{1}\in[0,s].

Proof.

We first improve the decay rate in Proposition 3.3. According to Proposition 3.3 and Lemma 3.4 with σ=α=12\sigma=\alpha=\frac{1}{2}, we have

(u,τ)∈L∞​(0,∞;B˙2,∞−12).\displaystyle(u,\tau)\in L^{\infty}(0,\infty;\dot{B}^{-\frac{1}{2}}_{2,\infty}).

By virtue of Lemma 3.5 with α=σ=12\alpha=\sigma=\frac{1}{2} and β=58\beta=\frac{5}{8}, we infer that

E0​(t)≤C​(1+t)−58a​n​dE1​(t)≤C​(1+t)−58−1.\displaystyle E_{0}(t)\leq C(1+t)^{-\frac{5}{8}}~{}~{}~{}~{}and~{}~{}~{}~{}E_{1}(t)\leq C(1+t)^{-\frac{5}{8}-1}.

Taking σ=1\sigma=1 and α=58\alpha=\frac{5}{8} in Lemma 3.4, we get

(3.55) (u,τ)∈L∞​(0,∞;B˙2,∞−1).\displaystyle(u,\tau)\in L^{\infty}(0,\infty;\dot{B}^{-1}_{2,\infty}).

Taking advantage of Lemma 3.5 again with α=58\alpha=\frac{5}{8} and σ=β=1\sigma=\beta=1, we deduce that

(3.56) E0​(t)≤C​(1+t)−1a​n​dE1​(t)≤C​(1+t)−2.\displaystyle E_{0}(t)\leq C(1+t)^{-1}~{}~{}~{}~{}and~{}~{}~{}~{}E_{1}(t)\leq C(1+t)^{-2}.

Then, we introduce some new energy and dissipation functionals for (u,τ)(u,\tau) as follows:

E~s=k​(1+t)​‖Λs​(u,τ)‖L22+⟨Λs−1​τ,−∇Λs−1​u⟩,\widetilde{E}_{s}=k(1+t)\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\langle\Lambda^{s-1}\tau,-\nabla\Lambda^{s-1}u\rangle,

and

D~s=k​(1+t)​‖∇Λs​τ‖L22+14​‖∇Λs−1​u‖L22,\widetilde{D}_{s}=k(1+t)\|\nabla\Lambda^{s}\tau\|^{2}_{L^{2}}+\frac{1}{4}\|\nabla\Lambda^{s-1}u\|^{2}_{L^{2}},

where kk is small enough. Using (3.18), (3.20), (3.56) and Lemmas 2.2-2.3, we deduce that

(3.57) dd​t​E~s+2​D~s\displaystyle\frac{d}{dt}\widetilde{E}_{s}+2\widetilde{D}_{s} ≤k∥Λs(u,τ)∥L22+Ck(1+t)(∥∇u∥L∞∥Λsu∥L22+∥τ∥L∞∥Λsu∥L2∥Λs+1τ∥L2\displaystyle\leq k\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+Ck(1+t)(\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}u\|^{2}_{L^{2}}+\|\tau\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}
+∥u∥L∞∥Λsτ∥L2∥Λs+1τ∥L2+∥∇u∥L∞∥Λs−1τ∥L2∥Λs+1τ∥L2),\displaystyle+\|u\|_{L^{\infty}}\|\Lambda^{s}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}+\|\nabla u\|_{L^{\infty}}\|\Lambda^{s-1}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}),
+C​‖Λs​u‖L2​(‖u‖L∞​‖Λs​τ‖L2+‖τ‖L∞​‖Λs​u‖L2+‖Λs+1​τ‖L2)\displaystyle+C\|\Lambda^{s}u\|_{L^{2}}(\|u\|_{L^{\infty}}\|\Lambda^{s}\tau\|_{L^{2}}+\|\tau\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}+\|\Lambda^{s+1}\tau\|_{L^{2}})
+C​‖Λs​τ‖L2​(‖u‖L∞​‖Λs​u‖L2+‖Λs​τ‖L2)\displaystyle+C\|\Lambda^{s}\tau\|_{L^{2}}(\|u\|_{L^{\infty}}\|\Lambda^{s}u\|_{L^{2}}+\|\Lambda^{s}\tau\|_{L^{2}})
≤(C​k+12)​D~s+C​k​(1+t)​‖u‖L∞2​‖Λs​τ‖L22+C​‖Λs​τ‖L22\displaystyle\leq(Ck+\frac{1}{2})\widetilde{D}_{s}+Ck(1+t)\|u\|^{2}_{L^{\infty}}\|\Lambda^{s}\tau\|^{2}_{L^{2}}+C\|\Lambda^{s}\tau\|^{2}_{L^{2}}
+C​k​(1+t)​‖∇u‖L∞​‖Λs−1​τ‖L2​‖Λs+1​τ‖L2\displaystyle+Ck(1+t)\|\nabla u\|_{L^{\infty}}\|\Lambda^{s-1}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}
≤(C​k+12)​D~s+C​‖Λs−1​τ‖L2​‖Λs+1​τ‖L2,\displaystyle\leq(Ck+\frac{1}{2})\widetilde{D}_{s}+C\|\Lambda^{s-1}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}},

where we use ‖Λs​τ‖L22≤C​‖Λs−1​τ‖L2​‖Λs+1​τ‖L2\|\Lambda^{s}\tau\|^{2}_{L^{2}}\leq C\|\Lambda^{s-1}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}}. By (3.56) and Lemma 2.2, we have

(3.58) C​‖Λs−1​τ‖L2​‖Λs+1​τ‖L2\displaystyle C\|\Lambda^{s-1}\tau\|_{L^{2}}\|\Lambda^{s+1}\tau\|_{L^{2}} ≤C​‖τ‖L22s+1​‖Λs+1​τ‖L22​ss+1\displaystyle\leq C\|\tau\|^{\frac{2}{s+1}}_{L^{2}}\|\Lambda^{s+1}\tau\|^{\frac{2s}{s+1}}_{L^{2}}
≤k4​(1+t)​‖Λs+1​τ‖L22+C​(1+t)−s​‖τ‖L22\displaystyle\leq\frac{k}{4}(1+t)\|\Lambda^{s+1}\tau\|^{2}_{L^{2}}+C(1+t)^{-s}\|\tau\|^{2}_{L^{2}}
≤14​D~s+C​(1+t)−s−1.\displaystyle\leq\frac{1}{4}\widetilde{D}_{s}+C(1+t)^{-s-1}.

Combining (3.57) and (3.58), we infer that

(3.59) dd​t​E~s+D~s≤C​(1+t)−s−1,\displaystyle\frac{d}{dt}\widetilde{E}_{s}+\widetilde{D}_{s}\leq C(1+t)^{-s-1},

which implies that

(3.60) dd​t​E~s+k​C2​‖Λs​τ‖L22+14​‖∇Λs−1​u‖L22\displaystyle\frac{d}{dt}\widetilde{E}_{s}+kC_{2}\|\Lambda^{s}\tau\|^{2}_{L^{2}}+\frac{1}{4}\|\nabla\Lambda^{s-1}u\|^{2}_{L^{2}} ≤C​(1+t)−s−1+k​C2​∫S​(t)|ξ|2​s​|τ^​(ξ)|2​𝑑ξ\displaystyle\leq C(1+t)^{-s-1}+kC_{2}\int_{S(t)}|\xi|^{2s}|\hat{\tau}(\xi)|^{2}d\xi
≤C​(1+t)−s−1.\displaystyle\leq C(1+t)^{-s-1}.

According to (3.60), we infer that

(3.61) (1+t)s+1​E~s+∫0tk​(1+s′)s+1​‖Λs​(τ,u)‖L22​𝑑s′\displaystyle(1+t)^{s+1}\widetilde{E}_{s}+\int_{0}^{t}k(1+s^{\prime})^{s+1}\|\Lambda^{s}(\tau,u)\|^{2}_{L^{2}}ds^{\prime}
≤C​(1+t)+C​∫0t(1+s′)s​⟨Λs−1​τ,−∇Λs−1​u⟩​𝑑s′\displaystyle\leq C(1+t)+C\int_{0}^{t}(1+s^{\prime})^{s}\langle\Lambda^{s-1}\tau,-\nabla\Lambda^{s-1}u\rangle ds^{\prime}
≤C​(1+t)+k2​∫0t(1+s′)s+1​‖Λs​u‖L22​𝑑s′+C​∫0t(1+s′)s−1​‖Λs−1​τ‖L22​𝑑s′\displaystyle\leq C(1+t)+\frac{k}{2}\int_{0}^{t}(1+s^{\prime})^{s+1}\|\Lambda^{s}u\|^{2}_{L^{2}}ds^{\prime}+C\int_{0}^{t}(1+s^{\prime})^{s-1}\|\Lambda^{s-1}\tau\|^{2}_{L^{2}}ds^{\prime}
≤C​(1+t)+k2​∫0t(1+s′)s+1​‖Λs​(τ,u)‖L22​𝑑s′+C​∫0t(1+s′)1−s​‖τ‖L22​𝑑s′\displaystyle\leq C(1+t)+\frac{k}{2}\int_{0}^{t}(1+s^{\prime})^{s+1}\|\Lambda^{s}(\tau,u)\|^{2}_{L^{2}}ds^{\prime}+C\int_{0}^{t}(1+s^{\prime})^{1-s}\|\tau\|^{2}_{L^{2}}ds^{\prime}
≤C​(1+t)+k2​∫0t(1+s′)s+1​‖Λs​(τ,u)‖L22​𝑑s′.\displaystyle\leq C(1+t)+\frac{k}{2}\int_{0}^{t}(1+s^{\prime})^{s+1}\|\Lambda^{s}(\tau,u)\|^{2}_{L^{2}}ds^{\prime}.

Using Lemma 2.2 again, we obtain

(3.62) (1+t)s+2​‖Λs​(u,τ)‖L22\displaystyle(1+t)^{s+2}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}} ≤C​(1+t)+C​(1+t)s+1​⟨Λs−1​τ,−∇Λs−1​u⟩\displaystyle\leq C(1+t)+C(1+t)^{s+1}\langle\Lambda^{s-1}\tau,-\nabla\Lambda^{s-1}u\rangle
≤C​(1+t)+k​(1+t)s+2​‖Λs​u‖L22+C​(1+t)s​‖Λs−1​τ‖L22\displaystyle\leq C(1+t)+k(1+t)^{s+2}\|\Lambda^{s}u\|^{2}_{L^{2}}+C(1+t)^{s}\|\Lambda^{s-1}\tau\|^{2}_{L^{2}}
≤C​(1+t)+k​(1+t)s+2​‖Λs​(u,τ)‖L22+C​(1+t)2−s​‖τ‖L22,\displaystyle\leq C(1+t)+k(1+t)^{s+2}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+C(1+t)^{2-s}\|\tau\|^{2}_{L^{2}},

which implies that

(3.63) ‖Λs​(u,τ)‖L22≤C​(1+t)−s−1.\displaystyle\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}\leq C(1+t)^{-s-1}.

We thus complete the proof of Proposition 3.6. ∎

The proof of Theorem 1.2: By Proposition 3.6, we only need to prove the lower bound of the decay rate. We first consider the linear system of (1.8) with β=1\beta=1:

(3.67) {∂tuL+∇PL−d​i​v​τL=0,d​i​v​uL=0,∂tτL−D​(uL)−Δ​τL=0,uL|t=0=u0,τL|t=0=τ0.\displaystyle\left\{\begin{array}[]{l}\partial_{t}u_{L}+\nabla P_{L}-div~{}\tau_{L}=0,~{}~{}~{}~{}div~{}u_{L}=0,\\[4.30554pt] \partial_{t}\tau_{L}-D(u_{L})-\Delta\tau_{L}=0,\\[4.30554pt] u_{L}|_{t=0}=u_{0},~{}~{}\tau_{L}|_{t=0}=\tau_{0}.\\[4.30554pt] \end{array}\right.

According to Proposition 3.6 and Lemma 3.4, one can deduce that ‖Λs1​(uL,τL)‖L22≤C​(1+t)−s1−1\|\Lambda^{s_{1}}(u_{L},\tau_{L})\|^{2}_{L^{2}}\leq C(1+t)^{-s_{1}-1} and (uL,τL)∈L∞​(0,∞;B˙2,∞−1)(u_{L},\tau_{L})\in L^{\infty}(0,\infty;\dot{B}^{-1}_{2,\infty}). Applying Fourier transform to (3.67), we get

(3.70) {∂tu^Lj+i​ξj​P^L−i​ξk​τ^Lj​k=0,∂tτ^Lj​k+|ξ|2​τ^Lj​k−i2​(ξk​u^Lj+ξj​u^Lk)=0.\displaystyle\left\{\begin{array}[]{ll}\partial_{t}\hat{u}_{L}^{j}+i\xi_{j}\hat{P}_{L}-i\xi_{k}\hat{\tau}_{L}^{jk}=0,\\[4.30554pt] \partial_{t}\hat{\tau}_{L}^{jk}+|\xi|^{2}\hat{\tau}_{L}^{jk}-\frac{i}{2}(\xi_{k}\hat{u}_{L}^{j}+\xi_{j}\hat{u}_{L}^{k})=0.\\[4.30554pt] \end{array}\right.

We introduce a new weighted energy estimate instead of complex spectral analysis to prove the lower bound of the decay rate. From (3.70), we have

(3.71) 12​dd​t​[e2​|ξ|2​t​|(u^L,τ^L)|2]−|ξ|2​e2​|ξ|2​t​|u^L|2=0,\displaystyle\frac{1}{2}\frac{d}{dt}[e^{2|\xi|^{2}t}|(\hat{u}_{L},\hat{\tau}_{L})|^{2}]-|\xi|^{2}e^{2|\xi|^{2}t}|\hat{u}_{L}|^{2}=0,

which implies that

(3.72) |ξ|2​s1​|(u^L,τ^L)|2=|ξ|2​s1​e−2​|ξ|2​t​|(u^0,τ^0)|2+∫0t2​|ξ|2​s1+2​e−2​|ξ|2​(t−s′)​|u^L|2​𝑑s′.\displaystyle|\xi|^{2s_{1}}|(\hat{u}_{L},\hat{\tau}_{L})|^{2}=|\xi|^{2s_{1}}e^{-2|\xi|^{2}t}|(\hat{u}_{0},\hat{\tau}_{0})|^{2}+\int_{0}^{t}2|\xi|^{2s_{1}+2}e^{-2|\xi|^{2}(t-s^{\prime})}|\hat{u}_{L}|^{2}ds^{\prime}.

According to 0<c0=|∫ℝ2(u0,τ0)​𝑑x|=|(u^0​(0),τ^0​(0))|0<c_{0}=|\int_{\mathbb{R}^{2}}(u_{0},\tau_{0})dx|=|(\hat{u}_{0}(0),\hat{\tau}_{0}(0))|, we deduce that there exists η>0\eta>0 such that |(u^0​(ξ),τ^0​(ξ))|≥c02|(\hat{u}_{0}(\xi),\hat{\tau}_{0}(\xi))|\geq\frac{c_{0}}{2} if ξ∈B​(0,η)\xi\in B(0,\eta). From (3.72), we have

(3.73) ‖(uL,τL)‖H˙s12\displaystyle\|(u_{L},\tau_{L})\|_{\dot{H}^{s_{1}}}^{2} ≥∫|ξ|≤η|ξ|2​s1​e−2​|ξ|2​t​|(u^0,τ^0)|2​𝑑ξ\displaystyle\geq\int_{|\xi|\leq\eta}|\xi|^{2s_{1}}e^{-2|\xi|^{2}t}|(\hat{u}_{0},\hat{\tau}_{0})|^{2}d\xi
≥c024​∫|ξ|≤η|ξ|2​s1​e−2​|ξ|2​t​𝑑ξ\displaystyle\geq\frac{c^{2}_{0}}{4}\int_{|\xi|\leq\eta}|\xi|^{2s_{1}}e^{-2|\xi|^{2}t}d\xi
≥C02​(1+t)−1−s1,\displaystyle\geq C^{2}_{0}(1+t)^{-1-s_{1}},

where C02=c024​∫|y|≤η|y|2​s1​e−2​|y|2​𝑑yC^{2}_{0}=\frac{c^{2}_{0}}{4}\int_{|y|\leq\eta}|y|^{2s_{1}}e^{-2|y|^{2}}dy. Taking uN=u−uLu_{N}=u-u_{L}, τN=τ−τL\tau_{N}=\tau-\tau_{L} and PN=P−PLP_{N}=P-P_{L}, then we immediately obtain ‖Λs1​(uN,τN)‖L22≤C​(1+t)−s1−1\|\Lambda^{s_{1}}(u_{N},\tau_{N})\|^{2}_{L^{2}}\leq C(1+t)^{-s_{1}-1} and (uN,τN)∈L∞​(0,∞;B˙2,∞−1)(u_{N},\tau_{N})\in L^{\infty}(0,\infty;\dot{B}^{-1}_{2,\infty}). Moreover, we have

(3.77) {∂tuN+∇PN−d​i​v​τN=F,d​i​v​uN=0,∂tτN−D​(uN)−Δ​τN=G,uN|t=0=τN|t=0=0.\displaystyle\left\{\begin{array}[]{l}\partial_{t}u_{N}+\nabla P_{N}-div~{}\tau_{N}=F,~{}~{}~{}~{}div~{}u_{N}=0,\\[4.30554pt] \partial_{t}\tau_{N}-D(u_{N})-\Delta\tau_{N}=G,\\[4.30554pt] u_{N}|_{t=0}=\tau_{N}|_{t=0}=0.\\[4.30554pt] \end{array}\right.

According to (3.77) and time decay rates for (uN,τN)(u_{N},\tau_{N}) and (u,τ)(u,\tau), we deduce that

(3.78) 12​dd​t​‖(uN,τN)‖L22+‖∇τN‖L22\displaystyle\frac{1}{2}\frac{d}{dt}\|(u_{N},\tau_{N})\|^{2}_{L^{2}}+\|\nabla\tau_{N}\|^{2}_{L^{2}} =⟨F,uN⟩+⟨G,τN⟩\displaystyle=\langle F,u_{N}\rangle+\langle G,\tau_{N}\rangle
≤C​‖∇u‖L2​‖u‖L4​‖uN‖L4+C​‖τN‖L4​‖∇(τ,u)‖L2​‖(τ,u)‖L4\displaystyle\leq C\|\nabla u\|_{L^{2}}\|u\|_{L^{4}}\|u_{N}\|_{L^{4}}+C\|\tau_{N}\|_{L^{4}}\|\nabla(\tau,u)\|_{L^{2}}\|(\tau,u)\|_{L^{4}}
≤C​δ​(1+t)−2,\displaystyle\leq C\delta(1+t)^{-2},

and

(3.79) dd​t​⟨τN,−∇uN⟩+12​‖∇uN‖L22\displaystyle\frac{d}{dt}\langle\tau_{N},-\nabla u_{N}\rangle+\frac{1}{2}\|\nabla u_{N}\|^{2}_{L^{2}} =⟨G+Δ​τN,−∇uN⟩−⟨ℙ​(F+d​i​v​τN),d​i​v​τN⟩\displaystyle=\langle G+\Delta\tau_{N},-\nabla u_{N}\rangle-\langle\mathbb{P}(F+div~{}\tau_{N}),div~{}\tau_{N}\rangle
≤C​(1+t)−2.\displaystyle\leq C(1+t)^{-2}.

Using (3.77) again, we obtain

(3.80) 12​dd​t​‖∇uN‖L22\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla u_{N}\|^{2}_{L^{2}} =⟨F,−Δ​uN⟩−⟨d​i​v​τN,−Δ​uN⟩\displaystyle=\langle F,-\Delta u_{N}\rangle-\langle div~{}\tau_{N},-\Delta u_{N}\rangle
≤C​δ​(1+t)−2.\displaystyle\leq C\delta(1+t)^{-2}.

By (3.78)-(3.80), we have

(3.81) dd​t​(‖(uN,τN)‖L22+‖∇uN‖L22+2​k0​⟨τN,−∇uN⟩)\displaystyle\frac{d}{dt}(\|(u_{N},\tau_{N})\|^{2}_{L^{2}}+\|\nabla u_{N}\|^{2}_{L^{2}}+2k_{0}\langle\tau_{N},-\nabla u_{N}\rangle)
+2​‖∇τN‖L22+k0​‖∇uN‖L22\displaystyle+2\|\nabla\tau_{N}\|^{2}_{L^{2}}+k_{0}\|\nabla u_{N}\|^{2}_{L^{2}}
≤C​(δ+k0)​(1+t)−2,\displaystyle\leq C(\delta+k_{0})(1+t)^{-2},

which implies that

(3.82) dd​t​(‖(uN,τN)‖L22+‖∇uN‖L22+2​k0​⟨τN,−∇uN⟩)+k0​C12​(1+t)​‖uN‖H12+C11+t​‖τN‖L22\displaystyle\frac{d}{dt}(\|(u_{N},\tau_{N})\|^{2}_{L^{2}}+\|\nabla u_{N}\|^{2}_{L^{2}}+2k_{0}\langle\tau_{N},-\nabla u_{N}\rangle)+\frac{k_{0}C_{1}}{2(1+t)}\|u_{N}\|^{2}_{H^{1}}+\frac{C_{1}}{1+t}\|\tau_{N}\|^{2}_{L^{2}}
≤C​C11+t​∫S​(t)|u^N​(ξ)|2+|τ^N​(ξ)|2​d​ξ+C​(δ+k0)​(1+t)−2.\displaystyle\leq\frac{CC_{1}}{1+t}\int_{S(t)}|\hat{u}_{N}(\xi)|^{2}+|\hat{\tau}_{N}(\xi)|^{2}d\xi+C(\delta+k_{0})(1+t)^{-2}.

Similar to (3.28) and (3.51), we infer that

(3.83) ∫S​(t)|u^N|2+|τ^N|2​d​ξ\displaystyle\int_{S(t)}|\hat{u}_{N}|^{2}+|\hat{\tau}_{N}|^{2}d\xi ≤C​∫S​(t)∫0t|F^⋅u^¯N|+|G^⋅τ^¯N|​d​s′​d​ξ\displaystyle\leq C\int_{S(t)}\int_{0}^{t}|\hat{F}\cdot\bar{\hat{u}}_{N}|+|\hat{G}\cdot\bar{\hat{\tau}}_{N}|ds^{\prime}d\xi
≤C​(1+t)−1​∫0t‖(u,τ)‖L2​‖∇(u,τ)‖L2​‖(uN,τN)‖B˙2,∞−1​𝑑s′\displaystyle\leq C(1+t)^{-1}\int_{0}^{t}\|(u,\tau)\|_{L^{2}}\|\nabla(u,\tau)\|_{L^{2}}\|(u_{N},\tau_{N})\|_{\dot{B}^{-1}_{2,\infty}}ds^{\prime}
≤C​δ​(1+t)−1.\displaystyle\leq C\delta(1+t)^{-1}.

According to (3.82) and (3.83), we obtain

‖(uN,τN)‖L22+‖∇uN‖L22≤C​(δ​C1+k0)​(1+t)−1.\displaystyle\|(u_{N},\tau_{N})\|^{2}_{L^{2}}+\|\nabla u_{N}\|^{2}_{L^{2}}\leq C(\delta C_{1}+k_{0})(1+t)^{-1}.

Applying Λs1\Lambda^{s_{1}} to (3.77), 0≤s1≤s0\leq s_{1}\leq s, we get

(3.86) {∂tΛs1​uN+∇Λs1​PN−d​i​v​Λs1​τN=Λs1​F,∂tΛs1​τN−D​(Λs1​uN)−Δ​Λs1​τN=Λs1​G.\displaystyle\left\{\begin{array}[]{l}\partial_{t}\Lambda^{s_{1}}u_{N}+\nabla\Lambda^{s_{1}}P_{N}-div~{}\Lambda^{s_{1}}\tau_{N}=\Lambda^{s_{1}}F,\\[4.30554pt] \partial_{t}\Lambda^{s_{1}}\tau_{N}-D(\Lambda^{s_{1}}u_{N})-\Delta\Lambda^{s_{1}}\tau_{N}=\Lambda^{s_{1}}G.\\[4.30554pt] \end{array}\right.

Using (3.86), Lemmas 2.2-2.3 and time decay rates, we infer that

(3.87) 12​dd​t​‖Λs​(uN,τN)‖L22+‖∇Λs​τN‖L22\displaystyle\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}}+\|\nabla\Lambda^{s}\tau_{N}\|^{2}_{L^{2}}
=−⟨[Λs,u⋅∇]​u,Λs​uN⟩−⟨u⋅∇Λs​u,Λs​uN⟩+⟨Λs​G,Λs​τN⟩\displaystyle=-\langle[\Lambda^{s},u\cdot\nabla]u,\Lambda^{s}u_{N}\rangle-\langle u\cdot\nabla\Lambda^{s}u,\Lambda^{s}u_{N}\rangle+\langle\Lambda^{s}G,\Lambda^{s}\tau_{N}\rangle
=−⟨[Λs,u⋅∇]​u,Λs​uN⟩−⟨u⋅∇Λs​uL,Λs​uN⟩+⟨Λs−1​G,Λs+1​τN⟩\displaystyle=-\langle[\Lambda^{s},u\cdot\nabla]u,\Lambda^{s}u_{N}\rangle-\langle u\cdot\nabla\Lambda^{s}u_{L},\Lambda^{s}u_{N}\rangle+\langle\Lambda^{s-1}G,\Lambda^{s+1}\tau_{N}\rangle
≤C​δ​((1+t)−2−s+(1+t)−1−s2​‖Λs+1​τN‖L2)+C​‖u‖L∞​‖∇Λs​uL‖L2​‖Λs​uN‖L2\displaystyle\leq C\delta((1+t)^{-2-s}+(1+t)^{-1-\frac{s}{2}}\|\Lambda^{s+1}\tau_{N}\|_{L^{2}})+C\|u\|_{L^{\infty}}\|\nabla\Lambda^{s}u_{L}\|_{L^{2}}\|\Lambda^{s}u_{N}\|_{L^{2}}
≤C​δ​(1+t)−2−s+C​δ​‖Λs+1​τN‖L22,\displaystyle\leq C\delta(1+t)^{-2-s}+C\delta\|\Lambda^{s+1}\tau_{N}\|^{2}_{L^{2}},

and

(3.88) dd​t​⟨Λs−1​τN,−∇Λs−1​uN⟩+12​‖∇Λs−1​uN‖L22\displaystyle\frac{d}{dt}\langle\Lambda^{s-1}\tau_{N},-\nabla\Lambda^{s-1}u_{N}\rangle+\frac{1}{2}\|\nabla\Lambda^{s-1}u_{N}\|^{2}_{L^{2}}
=⟨Λs−1​(G+Δ​τN),−∇Λs−1​uN⟩+⟨Λs−1​ℙ​(F+d​i​v​τN),d​i​v​Λs−1​τN⟩\displaystyle=\langle\Lambda^{s-1}(G+\Delta\tau_{N}),-\nabla\Lambda^{s-1}u_{N}\rangle+\langle\Lambda^{s-1}\mathbb{P}(F+div~{}\tau_{N}),div~{}\Lambda^{s-1}\tau_{N}\rangle
≤14​‖∇Λs−1​uN‖L22+C​‖Λs​τN‖H12+C​δ​(1+t)−1−s.\displaystyle\leq\frac{1}{4}\|\nabla\Lambda^{s-1}u_{N}\|^{2}_{L^{2}}+C\|\Lambda^{s}\tau_{N}\|^{2}_{H^{1}}+C\delta(1+t)^{-1-s}.

Combining (3.87) and (3.88), we get

(3.89) dd​t​[(1+t)​‖Λs​(uN,τN)‖L22+4​⟨Λs−1​τN,−∇Λs−1​uN⟩]+(1+t)​‖∇Λs​τN‖L22+‖∇Λs−1​uN‖L22\displaystyle\frac{d}{dt}[(1+t)\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}}+4\langle\Lambda^{s-1}\tau_{N},-\nabla\Lambda^{s-1}u_{N}\rangle]+(1+t)\|\nabla\Lambda^{s}\tau_{N}\|^{2}_{L^{2}}+\|\nabla\Lambda^{s-1}u_{N}\|^{2}_{L^{2}}
≤‖Λs​(uN,τN)‖L22+C​‖Λs​τN‖L22+C​δ​(1+t)−1−s,\displaystyle\leq\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}}+C\|\Lambda^{s}\tau_{N}\|^{2}_{L^{2}}+C\delta(1+t)^{-1-s},

which implies that

(3.90) dd​t​[(1+t)​‖Λs​(uN,τN)‖L22+4​⟨Λs−1​τN,−∇Λs−1​uN⟩]+C22​‖Λs​τN‖L22+‖∇Λs−1​uN‖L22\displaystyle\frac{d}{dt}[(1+t)\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}}+4\langle\Lambda^{s-1}\tau_{N},-\nabla\Lambda^{s-1}u_{N}\rangle]+\frac{C_{2}}{2}\|\Lambda^{s}\tau_{N}\|^{2}_{L^{2}}+\|\nabla\Lambda^{s-1}u_{N}\|^{2}_{L^{2}}
≤C2​∫S0​(t)|ξ|2​s​|τ^N​(ξ)|2​𝑑ξ+C​δ​(1+t)−1−s\displaystyle\leq C_{2}\int_{S_{0}(t)}|\xi|^{2s}|\hat{\tau}_{N}(\xi)|^{2}d\xi+C\delta(1+t)^{-1-s}
≤C​[δ+(δ​C1+k0)​C2]​(1+t)−1−s.\displaystyle\leq C[\delta+(\delta C_{1}+k_{0})C_{2}](1+t)^{-1-s}.

Note that k0​C1k_{0}C_{1} and C2C_{2} large enough. We need to take (δ​C1+k0)​C2(\delta C_{1}+k_{0})C_{2} small enough. According to (3.90), we obtain

(3.91) (1+t)s+2​‖Λs​(uN,τN)‖L22\displaystyle(1+t)^{s+2}\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}} ≤4​(1+t)s+1​⟨Λs−1​τN,∇Λs−1​uN⟩+C​[δ+(δ+k0)​k​C2]​(1+t)\displaystyle\leq 4(1+t)^{s+1}\langle\Lambda^{s-1}\tau_{N},\nabla\Lambda^{s-1}u_{N}\rangle+C[\delta+(\delta+k_{0})kC_{2}](1+t)
≤12​(1+t)s+2​‖Λs​(uN,τN)‖L22+C​[δ+(δ​C1+k0)​C2]​(1+t).\displaystyle\leq\frac{1}{2}(1+t)^{s+2}\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}}+C[\delta+(\delta C_{1}+k_{0})C_{2}](1+t).

Take suitable constants in (3.91), we have

(3.92) ‖Λs​(uN,τN)‖L22≤C024​(1+t)−1−s,\displaystyle\|\Lambda^{s}(u_{N},\tau_{N})\|^{2}_{L^{2}}\leq\frac{C^{2}_{0}}{4}(1+t)^{-1-s},

which implies that

(3.93) ‖Λs1​(uN,τN)‖L22≤C024​(1+t)−1−s1.\displaystyle\|\Lambda^{s_{1}}(u_{N},\tau_{N})\|^{2}_{L^{2}}\leq\frac{C^{2}_{0}}{4}(1+t)^{-1-s_{1}}.

According to (3.73) and (3.93), we infer that

(3.94) ‖(u,τ)‖H˙s12≥C024​(1+t)−1−s1.\displaystyle\|(u,\tau)\|_{\dot{H}^{s_{1}}}^{2}\geq\frac{C^{2}_{0}}{4}(1+t)^{-1-s_{1}}.

Therefore, we complete the proof of Theorem 1.2. □\Box

4 Optimal decay rate with 12≤β<1\frac{1}{2}\leq\beta<1

In this section, we present optimal decay rate for the generalized Oldroyd-B model (1.8) in fractional case 12≤β<1\frac{1}{2}\leq\beta<1. We point out that the refinement of Schonbek’s strategy also work in this case with a extra lower energy dissipation estimate. Similarly, we obtain the decay rates in H1H^{1}. Moreover, considering the decay rate for the highest derivative of the solution to (1.8), the main difficulty is unclosed energy estimate. To overcome this difficulty, we introduce another method which flexibly combines the Fourier spiltting method and the time weighted energy estimate. We rewrite (1.8) as follows.

(4.3) {∂tu+∇P−d​i​v​τ=F,d​i​v​u=0,∂tτ+(−Δ)β​τ−D​(u)=G.\displaystyle\left\{\begin{array}[]{l}\partial_{t}u+\nabla P-div~{}\tau=F,~{}~{}~{}~{}div~{}u=0,\\[4.30554pt] \partial_{t}\tau+(-\Delta)^{\beta}\tau-D(u)=G.\\[4.30554pt] \end{array}\right.

For convenience of explanation, we assume that β=12\beta=\frac{1}{2}. By Theorem 1.1, then we have

(4.4) dd​t​(‖(u,τ)‖Hs2+2​k​⟨−∇u,τ⟩Hs−12)+k2​‖∇u‖Hs−122+‖Λ12​τ‖Hs2≤0.\displaystyle\frac{d}{dt}(\|(u,\tau)\|^{2}_{H^{s}}+2k\langle-\nabla u,\tau\rangle_{H^{s-\frac{1}{2}}})+\frac{k}{2}\|\nabla u\|^{2}_{H^{s-\frac{1}{2}}}+\|\Lambda^{\frac{1}{2}}\tau\|^{2}_{H^{s}}\leq 0.

To present optimal decay rate for (4.3), we need additional lower energy dissipation estimate for uu. According to properties of Calderon-Zygmund operator and Lemma 2.2, we infer that

(4.5) dd​t​⟨τ,Λ−1​(−∇u)⟩+12​‖Λ12​u‖L22\displaystyle\frac{d}{dt}\langle\tau,\Lambda^{-1}(-\nabla u)\rangle+\frac{1}{2}\|\Lambda^{\frac{1}{2}}u\|^{2}_{L^{2}} =⟨G−Λ​τ,−Λ−1​∇u⟩−⟨ℙ​(F+d​i​v​τ),Λ−1​d​i​v​τ⟩\displaystyle=\langle G-\Lambda\tau,-\Lambda^{-1}\nabla u\rangle-\langle\mathbb{P}(F+div~{}\tau),\Lambda^{-1}div~{}\tau\rangle
≤C(∥Λ12τ∥L2∥Λ12u∥L2+∥Λ12τ∥L22\displaystyle\leq C(\|\Lambda^{\frac{1}{2}}\tau\|_{L^{2}}\|\Lambda^{\frac{1}{2}}u\|_{L^{2}}+\|\Lambda^{\frac{1}{2}}\tau\|^{2}_{L^{2}}
+∥u∥L42∥∇τ∥L2+∥∇u∥L2∥u∥L4∥τ∥L4)\displaystyle+\|u\|^{2}_{L^{4}}\|\nabla\tau\|_{L^{2}}+\|\nabla u\|_{L^{2}}\|u\|_{L^{4}}\|\tau\|_{L^{4}})
≤C​δ​‖Λ12​u‖L22+C​‖Λ12​τ‖L22.\displaystyle\leq C\delta\|\Lambda^{\frac{1}{2}}u\|^{2}_{L^{2}}+C\|\Lambda^{\frac{1}{2}}\tau\|^{2}_{L^{2}}.

Combining (4.4) and (4.5), we obtain

(4.6) dd​t​[‖(u,τ)‖Hs2+2​k​(⟨−∇u,τ⟩Hs−12+⟨τ,−Λ−1​∇u⟩)]+k2​‖Λ12​u‖Hs2+‖Λ12​τ‖Hs2≤0.\displaystyle\frac{d}{dt}[\|(u,\tau)\|^{2}_{H^{s}}+2k(\langle-\nabla u,\tau\rangle_{H^{s-\frac{1}{2}}}+\langle\tau,-\Lambda^{-1}\nabla u\rangle)]+\frac{k}{2}\|\Lambda^{\frac{1}{2}}u\|^{2}_{H^{s}}+\|\Lambda^{\frac{1}{2}}\tau\|^{2}_{H^{s}}\leq 0.

Define S1​(t)={ξ:|ξ|≤C2​(1+t)−1}S_{1}(t)=\{\xi:|\xi|\leq C_{2}(1+t)^{-1}\} for β=12\beta=\frac{1}{2}. By (4.6), we get

(4.7) dd​t​[‖(u,τ)‖Hs2+2​k​(⟨−∇u,τ⟩Hs−12+⟨τ,−Λ−1​∇u⟩)]+k​C22​(1+t)​‖u‖Hs2+C21+t​‖τ‖Hs2\displaystyle\frac{d}{dt}[\|(u,\tau)\|^{2}_{H^{s}}+2k(\langle-\nabla u,\tau\rangle_{H^{s-\frac{1}{2}}}+\langle\tau,-\Lambda^{-1}\nabla u\rangle)]+\frac{kC_{2}}{2(1+t)}\|u\|^{2}_{H^{s}}+\frac{C_{2}}{1+t}\|\tau\|^{2}_{H^{s}}
≤C1+t​∫S1​(t)|u^​(ξ)|2+|τ^​(ξ)|2​d​ξ.\displaystyle\leq\frac{C}{1+t}\int_{S_{1}(t)}|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2}d\xi.

According to the refinement of the Fourier spiltting method in Theorem 1.2, one can rediscover the following proposition with β=12\beta=\frac{1}{2} , which contain optimal decay for the solution in H1H^{1} [27].

Proposition 4.1.

Let d=2d=2 and 12≤β<1\frac{1}{2}\leq\beta<1. Let (u,τ)(u,\tau) be a strong solution of (1.4) with the initial data (u0,τ0)(u_{0},\tau_{0}) under the condition in Theorem 1.1. Then we have

(4.8) dd​t​[‖(u,τ)‖Hs2+2​k​(⟨−∇u,τ⟩Hs−β+⟨τ,−Λ−2+2​β​∇u⟩)]+k2​‖Λβ​u‖Hs+1−2​β2+‖Λβ​τ‖Hs2≤0.\displaystyle\frac{d}{dt}[\|(u,\tau)\|^{2}_{H^{s}}+2k(\langle-\nabla u,\tau\rangle_{H^{s-\beta}}+\langle\tau,-\Lambda^{-2+2\beta}\nabla u\rangle)]+\frac{k}{2}\|\Lambda^{\beta}u\|^{2}_{H^{s+1-2\beta}}+\|\Lambda^{\beta}\tau\|^{2}_{H^{s}}\leq 0.

In addition, if (u0,τ0)∈B˙2,∞−1(u_{0},\tau_{0})\in\dot{B}^{-1}_{2,\infty}, then there exists C>0C>0 such that for every t>0t>0, there holds

(4.9) ‖Λs2​(u,τ)‖Hs−s2≤C​(1+t)−12​β−s22​β,\displaystyle\|\Lambda^{s_{2}}(u,\tau)\|_{H^{s-s_{2}}}\leq C(1+t)^{-\frac{1}{2\beta}-\frac{s_{2}}{2\beta}},

where 0≤s2≤10\leq s_{2}\leq 1.

We omit the proof of Proposition 4.1, which is similar to the proof of (4.7) and Theorem 1.2.
The proof of Theorem 1.3: Applying Λs1\Lambda^{s_{1}} to (4.3), we obtain

(4.12) {∂tΛs1​u+∇Λs1​P−d​i​v​Λs1​τ=Λs1​F,∂tΛs1​τ−D​(Λs1​u)+(−Δ)β​Λs1​τ=Λs1​G.\displaystyle\left\{\begin{array}[]{l}\partial_{t}\Lambda^{s_{1}}u+\nabla\Lambda^{s_{1}}P-div~{}\Lambda^{s_{1}}\tau=\Lambda^{s_{1}}F,\\[4.30554pt] \partial_{t}\Lambda^{s_{1}}\tau-D(\Lambda^{s_{1}}u)+(-\Delta)^{\beta}\Lambda^{s_{1}}\tau=\Lambda^{s_{1}}G.\\[4.30554pt] \end{array}\right.

We first introduce the energy and dissipation functionals for (u,τ)(u,\tau) as follows:

E¯β=(1+t)a​‖Λs​(u,τ)‖L22+k​⟨Λs−β​τ,−∇Λs−β​u⟩,\overline{E}_{\beta}=(1+t)^{a}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+k\langle\Lambda^{s-\beta}\tau,-\nabla\Lambda^{s-\beta}u\rangle,

and

D¯β=(1+t)a​‖Λs+β​τ‖L22+k4​‖∇Λs−β​u‖L22,\overline{D}_{\beta}=(1+t)^{a}\|\Lambda^{s+\beta}\tau\|^{2}_{L^{2}}+\frac{k}{4}\|\nabla\Lambda^{s-\beta}u\|^{2}_{L^{2}},

where a=2−1β∈[0,1)a=2-\frac{1}{\beta}\in[0,1) and kk is small enough to overcome the difficulty about inner product estimate of linear term d​i​v​τdiv\tau in (4.12)1(\ref{eq10})_{1} with β=12\beta=\frac{1}{2}. By (4.12), we have

(4.13) 12​dd​t​‖Λs​(u,τ)‖L22+‖Λs+β​τ‖L22=⟨Λs​F,Λs​u⟩+⟨Λs​G,Λs​τ⟩,\displaystyle\frac{1}{2}\frac{d}{dt}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+\|\Lambda^{s+\beta}\tau\|^{2}_{L^{2}}=\langle\Lambda^{s}F,\Lambda^{s}u\rangle+\langle\Lambda^{s}G,\Lambda^{s}\tau\rangle,

and

(4.14) dd​t​⟨Λs−β​τ,−∇Λs−β​u⟩+12​‖∇Λs−β​u‖L22\displaystyle\frac{d}{dt}\langle\Lambda^{s-\beta}\tau,-\nabla\Lambda^{s-\beta}u\rangle+\frac{1}{2}\|\nabla\Lambda^{s-\beta}u\|^{2}_{L^{2}}
=⟨(−Λs−β​G−Λs+β​τ),∇Λs−β​u⟩+⟨Λs−β​ℙ​(F+d​i​v​τ),d​i​v​Λs−β​τ⟩.\displaystyle=\langle(-\Lambda^{s-\beta}G-\Lambda^{s+\beta}\tau),\nabla\Lambda^{s-\beta}u\rangle+\langle\Lambda^{s-\beta}\mathbb{P}(F+div~{}\tau),div~{}\Lambda^{s-\beta}\tau\rangle.

Combining (4.13) and (4.14), we get

(4.15) dd​t​E¯β+2​D¯β\displaystyle\frac{d}{dt}\overline{E}_{\beta}+2\overline{D}_{\beta} =a​(1+t)a−1​‖Λs​(u,τ)‖L22+2​(1+t)a​(⟨Λs​F,Λs​u⟩+⟨Λs​G,Λs​τ⟩)\displaystyle=a(1+t)^{a-1}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+2(1+t)^{a}(\langle\Lambda^{s}F,\Lambda^{s}u\rangle+\langle\Lambda^{s}G,\Lambda^{s}\tau\rangle)
+k​⟨(−Λs−β​G−Λs+β​τ),∇Λs−β​u⟩+k​⟨Λs−β​ℙ​(F+d​i​v​τ),d​i​v​Λs−β​τ⟩.\displaystyle+k\langle(-\Lambda^{s-\beta}G-\Lambda^{s+\beta}\tau),\nabla\Lambda^{s-\beta}u\rangle+k\langle\Lambda^{s-\beta}\mathbb{P}(F+div~{}\tau),div~{}\Lambda^{s-\beta}\tau\rangle.

By Lemma 2.2, we obtain ‖Λs​u‖L2≤C​‖u‖L21−θ​‖Λs+1−β​u‖L2θ\|\Lambda^{s}u\|_{L^{2}}\leq C\|u\|^{1-\theta}_{L^{2}}\|\Lambda^{s+1-\beta}u\|^{\theta}_{L^{2}} and ‖∇u‖L∞≤C​‖u‖L21−θ1​‖Λs+1−β​u‖L2θ1\|\nabla u\|_{L^{\infty}}\leq C\|u\|^{1-\theta_{1}}_{L^{2}}\|\Lambda^{s+1-\beta}u\|^{\theta_{1}}_{L^{2}} with (θ,θ1)=(ss+1−β,2s+1−β)(\theta,\theta_{1})=(\frac{s}{s+1-\beta},\frac{2}{s+1-\beta}). By Lemmas 2.2-2.3 and Proposition 4.1, we have

(4.16) (1+t)a​⟨Λs​F,Λs​u⟩\displaystyle(1+t)^{a}\langle\Lambda^{s}F,\Lambda^{s}u\rangle ≤C​(1+t)a​‖∇u‖L∞​‖Λs​u‖L22\displaystyle\leq C(1+t)^{a}\|\nabla u\|_{L^{\infty}}\|\Lambda^{s}u\|^{2}_{L^{2}}
≤C​(1+t)a​‖∇u‖L∞β​‖∇u‖L21−β​‖Λs+1−β​u‖L22\displaystyle\leq C(1+t)^{a}\|\nabla u\|^{\beta}_{L^{\infty}}\|\nabla u\|^{1-\beta}_{L^{2}}\|\Lambda^{s+1-\beta}u\|^{2}_{L^{2}}
≤C​(1+t)a​(1+t)−12−12​β​‖Λs+1−β​u‖L22\displaystyle\leq C(1+t)^{a}(1+t)^{-\frac{1}{2}-\frac{1}{2\beta}}\|\Lambda^{s+1-\beta}u\|^{2}_{L^{2}}
≤C​(1+t)32−32​β​‖Λs+1−β​u‖L22.\displaystyle\leq C(1+t)^{\frac{3}{2}-\frac{3}{2\beta}}\|\Lambda^{s+1-\beta}u\|^{2}_{L^{2}}.

When 12≤β<1\frac{1}{2}\leq\beta<1, we have (1+t)a​⟨Λs​F,Λs​u⟩≤δ​D¯β(1+t)^{a}\langle\Lambda^{s}F,\Lambda^{s}u\rangle\leq\delta\overline{D}_{\beta}. Similarly, we deduce that

(4.17) (1+t)a​⟨Λs​G,Λs​τ⟩\displaystyle(1+t)^{a}\langle\Lambda^{s}G,\Lambda^{s}\tau\rangle ≤C​(1+t)a​‖Λs−β​G‖L2​‖Λs+β​τ‖L2\displaystyle\leq C(1+t)^{a}\|\Lambda^{s-\beta}G\|_{L^{2}}\|\Lambda^{s+\beta}\tau\|_{L^{2}}
≤C​(1+t)a​‖Λs+β​τ‖L2​(‖u‖L∞​‖Λs+1−β​τ‖L2+‖τ‖L∞​‖∇Λs−β​u‖L2)\displaystyle\leq C(1+t)^{a}\|\Lambda^{s+\beta}\tau\|_{L^{2}}(\|u\|_{L^{\infty}}\|\Lambda^{s+1-\beta}\tau\|_{L^{2}}+\|\tau\|_{L^{\infty}}\|\nabla\Lambda^{s-\beta}u\|_{L^{2}})
≤δ​D¯β+C​(1+t)a​(‖u‖L∞2​‖Λs+1−β​τ‖L22+‖τ‖L∞2​‖∇Λs−β​u‖L22)\displaystyle\leq\delta\overline{D}_{\beta}+C(1+t)^{a}(\|u\|^{2}_{L^{\infty}}\|\Lambda^{s+1-\beta}\tau\|^{2}_{L^{2}}+\|\tau\|^{2}_{L^{\infty}}\|\nabla\Lambda^{s-\beta}u\|^{2}_{L^{2}})
≤2​δ​D¯β+(1+t)a−1β​‖Λs+1−β​τ‖L22.\displaystyle\leq 2\delta\overline{D}_{\beta}+(1+t)^{a-\frac{1}{\beta}}\|\Lambda^{s+1-\beta}\tau\|^{2}_{L^{2}}.

Applying Lemma 2.2 again, we get ‖Λs+1−β​τ‖L2≤C​‖τ‖L21−θ2​‖Λs+β​τ‖L2θ2\|\Lambda^{s+1-\beta}\tau\|_{L^{2}}\leq C\|\tau\|^{1-\theta_{2}}_{L^{2}}\|\Lambda^{s+\beta}\tau\|^{\theta_{2}}_{L^{2}} with θ2=s+1−βs+β\theta_{2}=\frac{s+1-\beta}{s+\beta}, which implies that

(4.18) ‖Λs+1−β​τ‖L22\displaystyle\|\Lambda^{s+1-\beta}\tau\|^{2}_{L^{2}} ≤C​(1+t)−s+1β+1​‖τ‖L22+C​(1+t)a​‖Λs+β​τ‖L22\displaystyle\leq C(1+t)^{-\frac{s+1}{\beta}+1}\|\tau\|^{2}_{L^{2}}+C(1+t)^{a}\|\Lambda^{s+\beta}\tau\|^{2}_{L^{2}}
≤C​(1+t)−s+1β+a−1+C​(1+t)a​‖Λs+β​τ‖L22.\displaystyle\leq C(1+t)^{-\frac{s+1}{\beta}+a-1}+C(1+t)^{a}\|\Lambda^{s+\beta}\tau\|^{2}_{L^{2}}.

Combining (4.17) and (4.18), we infer that

(4.19) (1+t)a​⟨Λs​G,Λs​τ⟩≤C​δ​D¯β+C​(1+t)−s+1β+a−1.\displaystyle(1+t)^{a}\langle\Lambda^{s}G,\Lambda^{s}\tau\rangle\leq C\delta\overline{D}_{\beta}+C(1+t)^{-\frac{s+1}{\beta}+a-1}.

Moreover, we can easily deduce from (4.17) and (4.18) that

(4.20) k​⟨(−Λs−β​G−Λs+β​τ),∇Λs−β​u⟩+k​⟨Λs−β​ℙ​(F+d​i​v​τ),d​i​v​Λs−β​τ⟩\displaystyle k\langle(-\Lambda^{s-\beta}G-\Lambda^{s+\beta}\tau),\nabla\Lambda^{s-\beta}u\rangle+k\langle\Lambda^{s-\beta}\mathbb{P}(F+div\tau),div\Lambda^{s-\beta}\tau\rangle
≤C​(δ+k)​D¯β+C​(1+t)−s+1β+a−1.\displaystyle\leq C(\delta+k)\overline{D}_{\beta}+C(1+t)^{-\frac{s+1}{\beta}+a-1}.

According (4.15)-(4.20), we infer that

(4.21) dd​t​E¯β+2​D¯β\displaystyle\frac{d}{dt}\overline{E}_{\beta}+2\overline{D}_{\beta} ≤a(1+t)a−1∥Λs(u,τ)∥L22+C(δ+k)D¯β++C(1+t)−s+1β+a−1.\displaystyle\leq a(1+t)^{a-1}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+C(\delta+k)\overline{D}_{\beta}++C(1+t)^{-\frac{s+1}{\beta}+a-1}.

Define Sβ​(t)={ξ:|ξ|2​β≤4​C2​(1+t)−1}S^{\beta}(t)=\{\xi:|\xi|^{2\beta}\leq 4C_{2}(1+t)^{-1}\}. By (4.21), we get

(4.22) dd​t​E¯β+k​C2​(1+t)a−1​‖Λs​(u,τ)‖L22\displaystyle\frac{d}{dt}\overline{E}_{\beta}+kC_{2}(1+t)^{a-1}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}
≤C​(1+t)−s+1β+a−1+C​(1+t)a−1​∫Sβ​(t)|ξ|2​s​(|u^​(ξ)|2+|τ^​(ξ)|2)​𝑑ξ\displaystyle\leq C(1+t)^{-\frac{s+1}{\beta}+a-1}+C(1+t)^{a-1}\int_{S^{\beta}(t)}|\xi|^{2s}(|\hat{u}(\xi)|^{2}+|\hat{\tau}(\xi)|^{2})d\xi
≤C​(1+t)−s+1β+a−1,\displaystyle\leq C(1+t)^{-\frac{s+1}{\beta}+a-1},

which implies that

(4.23) (1+t)s+1β−a+1​E¯β+∫0tk​C2​(1+s′)s+1β​‖Λs​(τ,u)‖L22​𝑑s′\displaystyle(1+t)^{\frac{s+1}{\beta}-a+1}\overline{E}_{\beta}+\int_{0}^{t}kC_{2}(1+s^{\prime})^{\frac{s+1}{\beta}}\|\Lambda^{s}(\tau,u)\|^{2}_{L^{2}}ds^{\prime}
≤C​(1+t)+Cs​∫0t(1+s′)s+1β−a​⟨Λs−β​τ,−∇Λs−β​u⟩​𝑑s′\displaystyle\leq C(1+t)+C_{s}\int_{0}^{t}(1+s^{\prime})^{\frac{s+1}{\beta}-a}\langle\Lambda^{s-\beta}\tau,-\nabla\Lambda^{s-\beta}u\rangle ds^{\prime}
≤C​(1+t)+Cs​∫0t(1+s′)s+1β​‖Λs​u‖L22​𝑑s′+Cs​∫0t(1+s′)s+1β−2​a​‖Λs+1−2​β​τ‖L22​𝑑s′\displaystyle\leq C(1+t)+C_{s}\int_{0}^{t}(1+s^{\prime})^{\frac{s+1}{\beta}}\|\Lambda^{s}u\|^{2}_{L^{2}}ds^{\prime}+C_{s}\int_{0}^{t}(1+s^{\prime})^{\frac{s+1}{\beta}-2a}\|\Lambda^{s+1-2\beta}\tau\|^{2}_{L^{2}}ds^{\prime}
≤C​(1+t)+Cs​∫0t(1+s′)s+1β​‖Λs​(τ,u)‖L22​𝑑s′+Cs​∫0t(1+s′)1−sβ​‖τ‖L22​𝑑s′\displaystyle\leq C(1+t)+C_{s}\int_{0}^{t}(1+s^{\prime})^{\frac{s+1}{\beta}}\|\Lambda^{s}(\tau,u)\|^{2}_{L^{2}}ds^{\prime}+C_{s}\int_{0}^{t}(1+s^{\prime})^{\frac{1-s}{\beta}}\|\tau\|^{2}_{L^{2}}ds^{\prime}
≤C​(1+t)+Cs​∫0t(1+s′)s+1β​‖Λs​(τ,u)‖L22​𝑑s′.\displaystyle\leq C(1+t)+C_{s}\int_{0}^{t}(1+s^{\prime})^{\frac{s+1}{\beta}}\|\Lambda^{s}(\tau,u)\|^{2}_{L^{2}}ds^{\prime}.

Taking k​C2≥CskC_{2}\geq C_{s} and using Lemma 2.2, we infer that

(4.24) (1+t)s+1β+1​‖Λs​(u,τ)‖L22\displaystyle(1+t)^{\frac{s+1}{\beta}+1}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}} ≤C​(1+t)+C​k​(1+t)s+1β−a+1​⟨Λs−β​τ,−∇Λs−β​u⟩\displaystyle\leq C(1+t)+Ck(1+t)^{\frac{s+1}{\beta}-a+1}\langle\Lambda^{s-\beta}\tau,-\nabla\Lambda^{s-\beta}u\rangle
≤C​[(1+t)+k​(1+t)s+1+ββ​‖Λs​u‖L22+k​(1+t)s+3−3​ββ​‖Λs+1−2​β​τ‖L22]\displaystyle\leq C[(1+t)+k(1+t)^{\frac{s+1+\beta}{\beta}}\|\Lambda^{s}u\|^{2}_{L^{2}}+k(1+t)^{\frac{s+3-3\beta}{\beta}}\|\Lambda^{s+1-2\beta}\tau\|^{2}_{L^{2}}]
≤C​(1+t)+C​k​(1+t)s+1β+1​‖Λs​(u,τ)‖L22+C​k​(1+t)1−sβ+1​‖τ‖L22,\displaystyle\leq C(1+t)+Ck(1+t)^{\frac{s+1}{\beta}+1}\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}+Ck(1+t)^{\frac{1-s}{\beta}+1}\|\tau\|^{2}_{L^{2}},

which implies that

(4.25) ‖Λs​(u,τ)‖L22≤C​(1+t)−s+1β.\displaystyle\|\Lambda^{s}(u,\tau)\|^{2}_{L^{2}}\leq C(1+t)^{-\frac{s+1}{\beta}}.

The lower bound of time decay rate can be obtained by the similar method in Theorem 1.2. We omit the details here. □\Box

Acknowledgments This work was partially supported by the National Natural Science Foundation of China (No.12171493 and No.11671407), the Macao Science and Technology Development Fund (No. 0091/2018/A3), Guangdong Province of China Special Support Program (No. 8-2015), the key project of the Natural Science Foundation of Guangdong province (No. 2016A030311004), and National Key R&\&D Program of China (No. 2021YFA1002100).

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