Obsidian/Zotero/001_artiklid/chenoptimaldecayglobalstrong2025.md

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category: literaturenote citekey: chenoptimaldecayglobalstrong2025 title: Optimal decay of global strong solutions to nematic liquid crystal flows in the half-space authors: "Chen, Haokun; Wang, Yong" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.10255v1" zotero_key: AT35TNDQ zotero_storage: PQ3UBNMH collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Chen, Haokun"

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OPTIMAL DECAY OF GLOBAL STRONG SOLUTIONS TO NEMATIC LIQUID CRYSTAL FLOWS IN THE HALF-SPACE

HAOKUN CHEN AND YONG WANG

Abstract. We study asymptotic behaviors of the higher-order spatial derivatives and the first-order time derivatives for the strong solution to nematic liquid crystal flows in the half-space R 3 + . Furthermore, when the initial data lie in an appropriately weighted Sobolev space, we obtain the decay rates that are faster than the heat kernel. The main tools employed in this paper are the L ^p − L estimates of the Stokes semigroup, the a priori estimates of the steady Stokes system in R 3 + , and the representation formula of the Leray projection operator.

1. Introduction

In this paper, we consider the large time behaviors of global strong solutions to the nematic liquid crystal flows in the half-space R 3 + (see [30, 31]) :

$$\begin{cases} u_t + u \cdot \nabla u + \nabla p = \mu \Delta u - \lambda \nabla \cdot (\nabla d \odot \nabla d), \ dt + u \cdot \nabla d = \theta (\Delta d + |\nabla d|^2 d), \ \nabla \cdot u = 0, \qquad (x, t) \in \mathbb{R}^3+ \times \mathbb{R}^+, \end{cases}$$

$$(1.1)$$

with the initial and boundary conditions

$$\begin{cases} u = \frac{\partial d}{\partial x3} = 0, & \text{on } \partial \mathbb{R}^3+ \times \mathbb{R}^+, \ (u, d) \to (0, e_3), & \text{as } |x| \to \infty, \ (u(x, 0), d(x, 0)) = (u_0(x), d0(x)), & \text{in } \mathbb{R}^3+. \end{cases}$$ (1.2)

Here, R 3 ^{+ =} {^{x =} (*x*1, *^x*2, *^x*3) ^{∈ R} 3 : ^{x3 &}gt; ⁰} with the boundary ^∂R 3 ^{+ =} {^{x =} (*x*1, *^x*2, *^x*3) ^{∈ R} 3 : x³ = 0} and ^{e3 =} (0, ⁰, 1). The unknown variables ^{u =} (*u*1, *^u*2, *^u*3)(x, ^t) and ^{p = p}(x, ^t) denote the velocity field of the fluid and the pressure function, respectively, and ^{d =} (*d*1, *^d*2, *^d*3)(x, ^t) is the unit vector field representing the macroscopic orientation field of liquid crystal molecules. The notation ∇d ⊙ ∇d denotes a 3 × 3 matrix whose (i, ^j)−th entry is given by ∂i^d · ∂j^d for 1 ^{≤ i}, ^{j ≤} 3. The constants µ, λ, θ > 0 represent the viscosity of the fluid, the competition between kinetic and potential energy, and the microscopic elastic relaxation time for the molecular orientation field, respectively. Without loss of generality, we assume µ ⁼ λ ⁼ θ ⁼ 1 in this paper.

If the unknown function d is vanishing in the system (1.1), then one has the half-space problem of the classical incompressible Navier-Stokes equations:

$$\left{ \begin{array}{lll} ut + u \cdot \nabla u + \nabla p = \mu \, \Delta u & \text{in} & \mathbb{R}^n+ \times \mathbb{R}^+, \ \nabla \cdot u = 0 & \text{in} & \mathbb{R}^n+ \times \mathbb{R}^+, \ u(x,t) = 0 & \text{on} & \partial \mathbb{R}^n+ \times \mathbb{R}^+, \ u(x,0) = u0(x) & \text{in} & \mathbb{R}^n+, \end{array} \right.$$

where n ≥ 2 is the spatial dimension. There are lots of literature on the existence, uniqueness and large time behaviors of solutions of the Cauchy problem and the initial-boundary value problem for the incompressible Navier-Stokes equations. For the Cauchy problem, Leray [28] proved the existence of global weak solutions for the incompressible Navier-Stokes equations in dimension three. Significantly, Leray posed the problem of determining whether or not weak solutions decay to zero in the L 2 -norm as time tends to infinity. This problem remains unsolved for a long time until the emergence of Kato's seminal work [24]. Specifically, Kato proved the existence of Leray weak solutions in spatial dimensions n ≤ 4 and established the asymptotic behavior: ∥u(t)∥^{L 2} → 0 as t → ∞. Later, Schonbek [41, 42]

²⁰²⁰ Mathematics Subject Classification. 76A15; 35B40; 35G61.

Key words and phrases. Large-time behavior; Strong solutions; Nematic liquid crystal flows; Half-space.

Corresponding author: Yong Wang (wangyongxmu@163.com).

introduced the Fourier splitting method and obtained the algebraic decay for weak solutions in R 3 . More precisely, under the condition that the initial velocity satisfies u⁰ ∈ L 2 σ(R 3 ) ∩ L r (R 3 ) (1 ^{≤ r} < 2), Schonbek [41, 42] deduced the explicit decay rate:

$$||u(t)||_{L^2(\mathbb{R}^3)} \le C(1+t)^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{2})}, \ \forall \ t>0.$$

Kajikiya-Miyakawa [23] generalized these results in [41, 42] to the case n ≥ 2 by using the spectral theory of self-adjoint operator in Hilbert spaces. Wiegner [53] also studied the L 2 -decay problem in R n (n ≥ 2) and obtained the most general results in this direction. However, most of the techniques in the whole space rely on the Fourier transform theory to deal with the nonlinear term u · ∇u, so the arguments developed there cannot be directly applied to the case of unbounded fields with nonempty boundaries. On the upper half-space R n + , using the estimates of the fractional powers of the Stokes operator, Borchers-Miyakawa [2] obtained the time decay rates in L 2 for the Navier-Stokes equations and established the results similar to the case of the Cauchy problem mentioned before. Later, Bae-Choe [1] improved their results by virtue of Ukai's solution formula (see [50]) for the Stokes equations. Fujigaki-Miyakawa [8] considered the long time behavior of strong solutions for Navier-Stokes flows in L q (R n + ) (1 ^&lt; ^{q &}lt; ^∞). Specifically, Fujigaki-Miyakawa [8] verified that if ^{u0 ∈ L} q σ(R n + ) ∩ L 1 (R n + ) (n ≥ 2) for all 1 ^&lt; ^{q &}lt; ^∞ and ^∥*xnu*0(x)∥^L (R n + ) ^&lt; ^∞, then it holds for any ^{t &}gt; 0,

$$|\nabla^k u(t)|{L^q(\mathbb{R}^n+)} \le C(1+t)^{-\frac{k+1}{2}-\frac{n}{2}(1-\frac{1}{q})},$$

where ^{k = 0}, 1. Han [10–13] derived the decay rates for the first-order and second-order derivatives of the Navier-Stokes flows in L r (R n + ) (1 ≤ r ≤ ∞) by using L ^p − L q estimates and a clever analysis on the fractional powers of the Stokes operator. In particular, the estimates

$$\begin{split} |\nabla u(t)|{L^{1}(\mathbb{R}^{n}{+})} &\leq Ct^{-\frac{1}{2}}, \quad |\nabla u(t)|{L^{\infty}(\mathbb{R}^{n}{+})} \leq Ct^{-\frac{1}{2} - \frac{n}{2}}, \ |\nabla^{2}u(t)|{L^{q}(\mathbb{R}^{n}{+})} + |\partial{t}u(t)|{L^{q}(\mathbb{R}^{n}{+})} + |\nabla p(t)|{L^{q}(\mathbb{R}^{n}_{+})} \leq Ct^{-1 - \frac{n}{2}(1 - \frac{1}{q})}, \quad 1 < q < \infty \end{split}$$

were given in [11] for any ^{t &}gt; 0 provided ^{u0 ∈ L} q σ(R n + ) ∩ L 1 (R n + ) (^{n ≥} 2) for all 1 < ^q < ^∞. Under some constraint conditions on the initial data, Han [12, 14] gave various weighted estimates of ∥∇ku(t)∥^L (R n + for the Navier-Stokes flows, where ^{k = 0}, ¹, 2, 1 ^{≤ r} ≤ ∞, and ^r , 1 if ^{k =} 0. Applying the regularity estimates of the steady Stokes system, Han [15] gave the decay results for the higher-order spatial derivatives of the Navier-Stokes flows, i.e., it was shown in [15] that

$$|\nabla^{k+2}u(t)|{L^{r}(\mathbb{R}^{n}{+})} \leq \begin{cases} C_{\varepsilon}t^{\varepsilon-\frac{2+k}{2}-(1-\frac{1}{r})}, & \text{if } n=2, \ Ct^{-\frac{2+k}{2}-\frac{n}{2}(1-\frac{1}{r})}, & \text{if } n\geq 3 \end{cases}$$

as ^t → ∞ for ^{k ≥} 1 and 1 ^&lt; ^r ≤ ∞, provided that ^u0 also belongs to ^L q σ(R n + ) ∩ L 1 (R n + ) (n ≥ 2) for all ¹ < ^q < ^∞. For further details, the interested readers may refer to [5, 16, 17, 19, 25, 26, 43, 44, 46] and the references cited therein.

There is an extensive literature devoted to nematic liquid crystal flows. The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie in 1960s (see [6, 7, 29]). In 1989, Lin [30] derived a simplified Ericksen-Leslie system (1.1) modeling incompressible viscous liquid crystal flows. From the mathematical point of view, (1.1) is a nonlinear coupling of the Navier-Stokes equation and the transported heat flow of harmonic maps to S 2 (the unit sphere in R 3 ). Lin-Liu [31–33] initiated the mathematical analysis of (1.1) by considering its Ginzburg-Landau approximation, namely, (1.1)² is replaced by

$$d_t + u \cdot \nabla d = \Delta d + \frac{1}{\varepsilon^2} (1 - |d|^2) d.$$

ε More precisely, Lin-Liu [31] proved the existence of global weak solutions to system (1.1) in dimensions two and three with Dirichlet boundary conditions. In [32], the authors proved that the one dimensional space-time Hausdorff measure of the singular set of the suitable weak solutions is zero. For more related topics, readers can refer to [20, 21, 27, 34, 51, 52] and the references cited therein. At the same time, there are many works on decay properties of the liquid crystal flows. Based on the Fourier splitting method introduced by Schonbek [41, 45], Liu [35] studied the temporal decay properties for weak solutions in R 2 . Dai and her collaborators [3, 4] obtained the optimal decay rates of the strong solution to (1.1) in ^{H m}(m ≥ 0) provided ∥*u*0∥H¹ (R³ ) + ∥d⁰ − *e*3∥H² (R³ is sufficiently small. Liu-Xu [36] proved that if the initial data (*u*0, *^d*0) ^{∈ H m}(R 3 ) × H m+1 (R 3 ) (^{m ≥} 3) have sufficiently small ^∥(*u*0, ^∇*d0)∥^L* (R³ )−norm, then system (1.1) admits a unique global-in-time smooth solution (u, ^d) and satisfies

$$|\nabla^{\ell}u|{L^{2}}^{2} + |\nabla^{\ell}\nabla d|{L^{2}}^{2} \le C(1+t)^{-\ell-3/2}, \quad \ell = 0, 1, \dots, m,$$

provided that (*u*0, ^∇*d*0) ^{∈ L} 1 (R 3 ) × L 1 (R 3 ). In the case of the half-space, the Fourier transform method does not work well. In addition, the projection operator P : L r (R n + ) → L r σ(R n + ) is unbounded for ^{r = 1}, ^∞, and ^P∂ⁿ , ^∂*n*P, due to the noncompact boundary ^∂R n + , which leads to many difficulties in dealing with the decay problems. Recently, Huang-Wang-Wen [22] obtained the optimal time decay rates in L r (R 3 + ) for ^{r ≥} 1 of global strong solutions to (1.1), provided that ^∥*u0∥^L* (R + ) + ∥∇*d0∥^L* (R 3 + is sufficiently small. However, in comparison to the heat equation and the Navier-Stokes equations, further research is warranted regarding the decay rates of liquid crystal system.

In this paper, we intend to establish the algebraic decay estimates of the global strong solution, including the L 1 -decay rates of second-order derivatives and the L ^∞-decay rates of higher-order derivatives with respect to spatial variables, to the nematic liquid crystal flows in L r (R 3 + ). To solve this problem, we usually transform the system (1.1) into the following integral forms by Duhamel's formula:

$$\begin{cases} u(t) = e^{-t\mathbb{A}}u_0 - \int_0^t e^{-(t-s)\mathbb{A}} \mathbb{P}\left(u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d(s) \odot \nabla d(s))\right) ds, \ (d-e_3)(t) = e^{t\Delta}(d_0 - e_3) - \int_0^t e^{(t-s)\Delta}\left(u(s) \cdot \nabla d(s) - |\nabla d(s)|^2 d(s)\right) ds, \end{cases}$$

$$(1.3)$$

and

$$\begin{cases} u(t) = e^{-\frac{t}{2}\mathbb{A}}u(\frac{t}{2}) - \int_{\frac{t}{2}}^{t} e^{-(t-s)\mathbb{A}}\mathbb{P}\left(u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d(s) \odot \nabla d(s))\right)ds, \ (d-e_3)(t) = e^{\frac{t}{2}\Delta}(d-e3)(\frac{t}{2}) - \int{\frac{t}{2}}^{t} e^{(t-s)\Delta}\left(u(s) \cdot \nabla d(s) - |\nabla d(s)|^2 d(s)\right)ds, \end{cases}$$

$$(1.4)$$

where A = −P∆ is the Stokes operator. The decay rates of the higher-order derivatives of the strong solution (u, ^d) are not so easy to obtain. The difficulty lies in the strong singularity of the integral when calculating ∥(∇ ^mu, ^{∇ m}d)∥^L (R + ) with m ≥ 2 by applying the L ^p − L q estimates of the Stokes flows. For example,

$$\int{\frac{t}{2}}^{t} (t-s)^{-\frac{m}{2}} |u(s)|{L^{2}(\mathbb{R}^{3}{+})}^{2} ds \ge \int{\frac{t}{2}}^{t} (t-s)^{-\frac{m}{2}} ds |u(t)|{L^{2}(\mathbb{R}^{3}{+})}^{2} = +\infty, \quad \forall \ t>0, \ m \ge 2.$$

To address this issue, the equation of ^d is appropriately modified (see (3.8)), and a continuity argument is employed to derive the L r -decay estimate of ∇ ²d, inspired by the work in [22]. By conducting a sophisticated analysis of the fractional powers of the Stokes operator A, the strong singularity arising from ∥∇2*u∥^L* (R 3 + ) can be effectively avoided, see Lemma 3.4. However, the method employed in Lemma ³.4 is not applicable for ^{m ≥} 3 because the singularity is really too strong. Motivated by the work in [15], we aim to apply the regularity estimates of the steady Stokes system to overcome the difficulty appearing from ∥∇*mu∥^L* (R 3 + ) with m ≥ 3. Besides, the standard elliptic estimates are employed to derive the decay rates of ∥∇*md∥^L r* (R 3 + ) (m ≥ 3). Since ∇d is not divergence free, the required estimates on ∇ · (∇d ⊙ ∇d) is more delicate than the nonlinear term u · ∇u, as stated in [22]. In fact, the third-order and forth-order derivatives of ^d emerge in the estimate of ∥∇P(^u · ∇^{u +} ∇ · (∇^d ⊙ ∇d))∥^L (R + , see (2.12). Consequently, to formally obtain the decay rates of ∥∇*ku∥^L r* (R + ) , it is necessary to first determine the decay rates of ∥∇k+1*d∥^L r* (R 3 ) . To achieve this, we need to jointly solve for the decay rates of the higher derivatives of (u, ^d), see (3.40) and (4.16) below. This approach differs from that used for the Navier-Stokes equations (see [15]) and the magnetohydrodynamic equations (see [37]) due to the complexity of the nonlinear terms with respect to the director field d.

Notations. We denote the usual L p spaces by

$$L^{p}(\mathbb{R}^{3}{+}) = { f(x) \mid ||f(x)||{L^{p}(\mathbb{R}^{3}_{+})} < \infty }, \quad 1 \le p \le \infty,$$

where

$$||f(x)||{L^p(\mathbb{R}^3+)} := \left{ \begin{array}{ll} \Big( \int{\mathbb{R}^3+} |f(x)|^p \, dx \Big)^{\frac{1}{p}}, & 1 \leq p < \infty, \ \inf{|E| = 0} { \sup{E \subset \mathbb{R}^3+} \mathbb{R}^3+ \backslash E }, & p = \infty. \end{array} \right.$$

We denote the usual Sobolev spaces by

$$W^{k,p}(\mathbb{R}^3+) = { f(x) \mid ||f(x)||{W^{k,p}(\mathbb{R}^3_+)} < \infty }, \quad k = 0, 1, \dots, \quad 1 \le p \le \infty,$$

where

$$|f(x)|{W^{k,p}(\mathbb{R}^3+)} := \left{ \begin{array}{ll} \left( \sum{0 \leq l \leq k} |\nabla^l f(x)|{L^p(\mathbb{R}^3+)}^p \right)^{\frac{1}{p}}, & 1 \leq p < \infty, \ \sum{0 \leq l \leq k} |\nabla^l f(x)|{L^\infty(\mathbb{R}^3+)}, & p = \infty. \end{array} \right.$$

For simplicity, we denote H k (R 3 + ) = Wk,² (R 3 + ) (^{k = 1}, ², . . .). And we denote by ^C ∞ 0,σ (R 3 + ) the set of all C ^∞ divergence-free vector fields with compact support in R 3 + . The space L r σ(R 3 + ) (1 < ^r < ^∞) is the closure of C ∞ 0,σ (R 3 + ) with respect to ∥ · ∥^L (R 3 + , namely,

$$L{\sigma}^{r}(\mathbb{R}^{3}{+}) = \left{ u = (u{1}, u{2}, u{3}) \in L^{r}(\mathbb{R}^{3}{+}) \mid \nabla \cdot u = 0, \ u{3}|{\partial \mathbb{R}^{3}_{+}} = 0 \right}.$$

Set

$$L^r\pi(\mathbb{R}^3+) = \left{ \nabla p \in L^r(\mathbb{R}^3+) \mid p \in L^r{loc}(\overline{\mathbb{R}^3_+}) \right}.$$

Then the Helmholtz decomposition holds (see [2]):

$$L^{r}(\mathbb{R}^{3}{+}) = L^{r}{\sigma}(\mathbb{R}^{3}{+}) \oplus L^{r}{\pi}(\mathbb{R}^{3}_{+}), \quad 1 < r < \infty.$$

In addition, we denote

$$D^{k,r}(\mathbb{R}^3+) = \left{ v \in L^1{\text{loc}}(\mathbb{R}^3+) \mid ||\nabla^k v||{L^r(\mathbb{R}^3_+)} < \infty \right},\,$$

and $$D^k(\mathbb{R}^3+) = D^{k,2}(\mathbb{R}^3+)$$ .

For completeness, we first recall the following proposition on the global existence of strong solutions to the system (1.1)-(1.2), see Theorem 1.1 in [22].

Proposition 1.1. There exists an ^{ε0 &}gt; ⁰ such that if u^{0 ∈ L} r σ(R 3 + ) and d⁰ ∈ D 1,r (R 3 + ) for r ^{= 2}, ³ satisfies

$$||u0||{L^3(\mathbb{R}^3_+)} + ||\nabla d0||{L^3(\mathbb{R}^3_+)} \le \varepsilon_0,$$

then the half-space problem (1.1)-(1.2) admits a unique global strong solution (u,d) such that for any τ > ⁰, the following properties hold:

$$\begin{cases} u \in C([0,\infty), L^2(\mathbb{R}^3+) \cap L^2([0,\infty), D^1(\mathbb{R}^3+)), \ u \in C(\mathbb{R}^+, D^2(\mathbb{R}^3+)) \cap L^2([\tau,+\infty), D^3(\mathbb{R}^3+)), \ ut \in C(\mathbb{R}^+, L^2(\mathbb{R}^3+)) \cap L^2([\tau,+\infty), W^{1,2}(\mathbb{R}^3_+)), \ ut \in C'([0,\infty), L^3(\mathbb{R}^3+)), \nabla d \in L^\infty([0,\infty), L^3(\mathbb{R}^3+)), \ (|u|^{\frac{3}{2}}, |\nabla d|^{\frac{3}{2}}) \in L^2([0,\infty), D^1(\mathbb{R}^3+)), \ \nabla d \in C([0,\infty), L^2(\mathbb{R}^3+)) \cap L^2([0,\infty), D^1(\mathbb{R}^3+)), \ \nabla d \in C(\mathbb{R}^+, D^2(\mathbb{R}^3+)) \cap L^2([\tau,+\infty), D^3(\mathbb{R}^3+)), \ dt \in L^2([0,\infty), L^2(\mathbb{R}^3+)), \ dt \in C(\mathbb{R}^+, W^{1,2}(\mathbb{R}^3+)) \cap L^2([\tau,+\infty), D^2(\mathbb{R}^3_+)). \end{cases}$$

Our first theorem considers the decay rates of ∥(∇ ^ku, ^{∇ k}d)(t)∥^L (R + (^{k = 1}, 2) with 1 ^{≤ p} ≤ ∞, where (u, ^d) is the strong solution obtained in Proposition 1.1.

Theorem 1.1. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ), if (u, ^d, ^p) is the global strong solution obtained by Proposition 1.1, then the following estimates

$$\begin{cases} |\nabla u(t)|{L^{p}(\mathbb{R}^{3}{+})} \leq Ct^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{p})}, \ |\nabla^{2} d(t)|{L^{p}(\mathbb{R}^{3}{+})} \leq Ct^{-1 - \frac{3}{2}(1 - \frac{1}{p})}, \ |\nabla^{2} u(t)|{L^{1}(\mathbb{R}^{3}{+})} \leq Ct^{-1}, \ |\nabla^{2} u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} \leq Ct^{-\frac{5}{2}}, \ |\nabla^{2} u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} + |\partial{t} u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla p(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\partial{t} d(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq Ct^{-1 - \frac{3}{2}(1 - \frac{1}{r})}. \end{cases}$$

hold for any t > ⁰, where p ^∈ [1, ^∞], and r ^∈ (1, ^∞).

Remark 1.1. Huang-Wang-Wen [22] obtained the decay rates of ∥∇u(t)∥^L (R + and ∥∇2d(t)∥^L (R + for ^{1 ≤ p ≤ 6}. Theorem 1.1 further generalizes these results to the extended range ⁶ < ^p ≤ ∞.

Remark 1.2. The algebraic decay rates are similar to that of incompressible Navier-Stokes equations in the half-space obtained in [10–12].

Remark 1.3. In order to establish the L¹ -decay of the strong solution, we need to avoid the unboundedness of the projection operator P in L¹ (R 3 + ). To overcome this problem, we use a decomposition for ^P(*u*·∇u+∇·(∇*d*⊙∇d)), see (2.10), and translate the unboundedness of the operator ^P into the treatment of a carefully designed elliptic Neumann problem (2.8).

Remark 1.4. Compared with Theorem 1.2 in [22] (i.e., Lemma 2.6 in this paper), the additional conditions "u⁰ ∈ D 1 (R 3 + ) and ∇d⁰ ∈ D 1 (R 3 + )"are not required to establish the decay estimate ∥∇u(t)∥^L (R 3 + ) ≤ Ct^{− 2} . This is attributed to our ability to derive the decay rate of ∇ ³d in L^r (R 3 + ), as detailed in Lemma 3.7.

Our second result is to establish the decay estimates for the higher-order spatial derivatives of the strong solution.

Theorem 1.2. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ). Then for every integer k ^{≥ 1}, there exists t(k) > ⁰ such that for ¹ < ^r ≤ ∞ and t ^{≥ t}(k), the strong solution (u, ^d, ^p) obtained in Proposition 1.¹ satisfies

$$|\nabla^{2+k}u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{1+k}p(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{2+k}d(t)|{L^r(\mathbb{R}^3+)} \leq Ct^{-\frac{2+k}{2}-\frac{3}{2}(1-\frac{1}{r})}$$

Remark 1.5. The algebraic decay rate is similar to that of the incompressible Navier-Stokes equations in the half-space obtained in [15], which can be considered the optimal compared to the heat kernel.

Our third main result states that: the more rapid decay rates of the strong solution are obtained if the given initial data lie in a suitable weighted space.

Theorem 1.3. Under the same assumptions of Theorem 1.1, if, in addition,

$$\int{\mathbb{R}^3+} x_3 |u_0(x)| \, dx < \infty,\tag{1.5}$$

then the decay estimates on u, p can be improved as

$$\left{ \begin{array}{l} |\nabla u(t)|{L^p(\mathbb{R}^3+)} \leq Ct^{-1-\frac{3}{2}(1-\frac{1}{p})}, \ |\nabla^2 u(t)|{L^\infty(\mathbb{R}^3+)} \leq Ct^{-3}, \ |\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} + |\partialt u(t)|{L^r(\mathbb{R}^3+)} + |\nabla p(t)|{L^r(\mathbb{R}^3+)} \leq Ct^{-\frac{3}{2}-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^{2+k} u(t)|{L^p(\mathbb{R}^3+)} + |\nabla^{1+k} p(t)|{L^p(\mathbb{R}^3_+)} \leq Ct^{-\frac{12}{2}-\frac{2+k}{2}-\frac{3}{2}(1-\frac{1}{p})}, \end{array} \right.$$

for t ^&gt; ^tk with some suitable number t^{k &}gt; ⁰, p ^∈ (1, ^∞], and r ^∈ (1, ^∞).

Remark 1.6. It should be pointed out that the faster decay rates of ∥u(t)∥^L (R + and ∥∇u(t)∥^L (R + are obtained by Huang-Wang-Wen [22] for ¹ < ^r ≤ ∞ and ¹ < ^{p ≤ 6}.

Remark 1.7. When the condition (1.5) is satisfied, the decay rates of u will be accelerated. At this point, the estimate of ∥u(t)∥^L (R + is equivalent to the estimate of ∥∇d(t)∥^L (R + for all ¹ < ^r ≤ ∞, so that the L r -decay rates of u · ∇u aligns with that of ∇ · (∇^d ⊙ ∇d). In terms of (1.4)1, we are able to obtain the faster decay rates of the higher-order derivatives of u.

Remark 1.8. As stated in [22], it remains an interesting question whether the director field d satisfies improved estimates on ^∇d, analogously to those presented in Theorem 1.3, under the condition that ∥*x*3∇*d0∥^L* (R 3 ) ^&lt; ^∞.

The remaining part of this paper is organized as follows. In Section 2, we introduce some basic known results about the Stokes operator and some important inequalities which will be used throughout the paper. In Section 3, we firstly give the decay estimates of ∥∇2*d*∥L^∞ and ∥∇*u*∥L^∞ for the strong solution (u, ^d) through a continuity argument. Then, we will make full use of the estimates of the fractional powers of the Stokes operator to prove the decay estimates of ∥∇2*u∥^{L p}* in the case 1 < ^p < ^∞. In the end, we give the decay estimates of ∥∇*iu∥^{L j}* for (i, ^j) ⁼ (1, 1), (2, 1) and (2, ^∞). The unboundedness of the projector P is overcome by employing a decomposition for the nonlinear terms. In Section 4, the asymptotic behavior for the higher-order spacial derivatives of the strong solution is given in R 3 + . In Section 5, we establish further decay estimates of the strong solution u if the initial data lie in a suitable weighted space.

2. Preliminaries

In section 2, we list some useful lemmas which are frequently used in remaining sections. First, we recall some facts about Stokes operator A that can be founded in [47]. The Stokes operator A is defined by A = −P∆: D(A) → L 2 σ(R 3 + ) with

$$D(\mathbb{A}) := H^2(\mathbb{R}^3+) \cap H^1{0,\sigma}(\mathbb{R}^3_+),$$

where

$$\mathbb{P}: L^r(\mathbb{R}^3+) \mapsto L^r{\sigma}(\mathbb{R}^3_+)$$

is the associated projection operator, which is bounded for any 1 < ^r < ^∞ (see [38]). Since ^A is a positive self-adjoint operator, there exists a uniquely determined resolution {*E*λ ^{| λ ≥ 0}} of identity in L 2 σ(R 3 + ) such that A has the spectral representation

$$\mathbb{A} = \int0^\infty \lambda \, dE\lambda$$

with the domain

$$D(\mathbb{A}) = \Big{ v \in L^2\sigma(\mathbb{R}^3+) \mid \big| \mathbb{A} v \big|{L^2(\mathbb{R}^3+)}^2 = \int0^\infty \lambda^2 \, d \big| E\lambda v \big|{L^2(\mathbb{R}^3+)}^2 < \infty \Big}.$$

More generally, for any α ^∈ (0, 1), we define the fractional order of Stokes operator ^A as:

$$\mathbb{A}^\alpha = \int0^\infty \lambda^\alpha \, dE\lambda$$

with the domain

$$D(\mathbb{A}^\alpha) = \Big{ v \in L^2\sigma(\mathbb{R}^3+) \mid |\mathbb{A}^\alpha v|^2{L^2(\mathbb{R}^3+)} = \int0^\infty \lambda^{2\alpha} \, d|E\lambda v|^2{L^2(\mathbb{R}^3+)} < \infty \Big}.$$

The operator

$$-\Delta: D(-\Delta) \to L^2(\mathbb{R}^3_+)$$

is defined by

$$-\Delta = \int0^\infty \lambda \, d\tilde{E}\lambda$$

with the domain

$$D(-\Delta) = D(\Delta) = { u \in W0^{1,2}(\mathbb{R}^3+) \mid \Delta u \in L^2(\mathbb{R}^3_+) },$$

where {*E*˜ λ ^{| λ ≥ 0}} denotes the resolution of identity for ^−∆ (see [47, p.101]). The fractional order of Laplace operator

$$(-\Delta)^{\alpha} = \int0^{\infty} \lambda^{\alpha} \, d\tilde{E}{\lambda}$$

with the domain

$$D((-\Delta)^{\alpha}) = \left{ v \in L^{2}(\mathbb{R}^{3}{+}) \mid |(-\Delta)^{\alpha}v|{L^{2}(\mathbb{R}^{3}{+})}^{2} = \int{0}^{\infty} \lambda^{2\alpha} \, d|\tilde{E}{\lambda}v|{L^{2}(\mathbb{R}^{3}_{+})}^{2} < \infty \right}$$

is well defined for ^{α ∈} (0, 1). We conclude that *^E*λ ⁼ *^E*˜ λ^P for all ^{λ ≥} 0, since the resolution of identity is uniquely determined by A. In this case we see that

$$\mathbb{A}^{\alpha}u=\int{0}^{\infty}\lambda^{\alpha}\,dE{\lambda}u=\int{0}^{\infty}\lambda^{\alpha}\,d\tilde{E}{\lambda}\mathbb{P}u=\int{0}^{\infty}\lambda^{\alpha}\,d\tilde{E}{\lambda}u=(-\Delta)^{\alpha}u$$

holds for all u ∈ D(A ) (see [47, p.140] for details).

It should be noted that the operator A on R 3 ⁺ generates a bounded analytic *C*0−semigroup {e ⁻*t*A}*t*≥⁰ on L r σ(R 3 + ), so the solution u of Stokes problem

$$\left{ \begin{array}{ll} ut - \Delta u + \nabla p = 0 & \text{in} \quad \mathbb{R}^3+ \times \mathbb{R}^+, \ \nabla \cdot u = 0 & \text{in} \quad \mathbb{R}^3+ \times \mathbb{R}^+, \ u(x,t) = 0 & \text{on} \quad \partial \mathbb{R}^3+ \times \mathbb{R}^+, \ u(x,0) = u0(x) & \text{in} \quad \mathbb{R}^3+ \end{array} \right.$$

can be expressed by u(t) = e ⁻*t*A*u*0. Moreover, the Stokes flow e ⁻*t*Au⁰ satisfies the following L ^p − L q estimates.

Lemma 2.1. ([8]) Let a ∈ L q σ(R 3 + ), then for any t > ⁰, the following estimate hold

$$|\nabla^k e^{-t\mathbb{A}} a|{L^p(\mathbb{R}^3+)} \le C t^{-\frac{k}{2} - \frac{3}{2}(\frac{1}{q} - \frac{1}{p})} |a|{L^q(\mathbb{R}^3+)} \tag{2.1}$$

with k ^{= 0}, ¹, . . . , provided that ^{1 ≤ q} < ^p ≤ ∞ or ¹ < ^{q ≤ p} < ^∞. Furthermore,

$$|\nabla e^{-t\mathbb{A}}a|{L^{1}(\mathbb{R}^{3}{+})} \le Ct^{-\frac{1}{2}}|a|{L^{1}(\mathbb{R}^{3}{+})}, \ \forall \ t>0,$$ (2.2)

and (2.1) and (2.2) *still hold, if we replace the operator e*−t^A by et^∆ with a ^{∈ L} q (R 3 + ).

Lemma 2.2. ([12]) Let ¹ < ^q < ^∞. Assume that a ^∈ *^W1,^q* (R 3 + ) satisfies ∇ · a = 0 in R 3 ⁺ *and a3|∂^R 3 = 0. Then for any* ⁰ < θ < ¹ and t > ⁰,

$$|\nabla^2 e^{-t\mathbb{A}} a|{L^{\infty}(\mathbb{R}^3+)} \leq C(t^{-\frac{1}{2} - \frac{3}{2q}} |\nabla a|{L^q(\mathbb{R}^3+)} + t^{-1 + \frac{\theta}{2} - \frac{3}{2q}} |y3^{-\theta} a|{L^q(\mathbb{R}^3_+)}).$$

Lemma 2.3. ([13]) Let a ∈ *W*1,¹ (R 3 + ) satisfies ∇ · a = 0 in R 3 ⁺ *and a3|∂^R + ^{= 0}. Then for any* ⁰ < η < ¹ and t > ⁰,

$$|\nabla^2 e^{-t\mathbb{A}}a|{L^1(\mathbb{R}^3+)} \le C(t^{-\frac{1}{2}}|\nabla a|{L^1(\mathbb{R}^3+)} + t^{-1+\frac{\eta}{2}}\int{\mathbb{R}^3+} y_3^{-\eta}|a(y)|\,dy).$$

and

$$|\nabla^2 e^{-t\mathbb{A}} a|{L^1(\mathbb{R}^3+)} \le Ct^{-1} |a|{L^1(\mathbb{R}^3+)}.$$

We will also use the following elementary inequalities.

Lemma 2.4. ([22]) For any ^{α1 ∈} (0, 1) and ^α2, α^{3 &}gt; ⁰, it holds that

$$\begin{cases} \int_0^t (t-s)^{-\alpha_1} s^{-\alpha_2} \, ds \leq C t^{1-\alpha_1-\alpha_2}, \text{ for } 0 < \alpha_2 < 1, \ \int_0^t (t-s)^{-\alpha_1} s^{-\alpha_2} \, ds \leq C t^{1-\alpha_1-\alpha_2}, \text{ for } \alpha_2 > 0; \end{cases}$$

$$\int_0^t (t-s)^{-\alpha_1} (1+s)^{-\alpha_3} \, ds \leq \begin{cases} C t^{-\alpha_1}, \text{ for } \alpha_3 > 1, \ C t^{1-\alpha_1-\alpha_3}, \text{ for } 0 < \alpha_3 < 1, \ C t^{-\alpha_1} \ln(1+t), \text{ for } \alpha_3 = 1; \end{cases}$$

and

$$\int_{\frac{t}{2}}^{t} (t-s)^{-\alpha_1} (1+s)^{-\alpha_3} ds \le Ct^{1-\alpha_1-\alpha_3} \text{ for } \alpha_3 > 0.$$

To establish further decay estimates, we will use the following results obtained in [22].

Lemma 2.5. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ). Then the strong solution (u, ^d) of (1.1)-(1.2) obtained in Proposition 1.¹ satisfies the following decay estimates:

$$|u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |(d - e{3})(t)|{L^{r}(\mathbb{R}^{3}_{+})} \le Ct^{-\frac{3}{2}(1 - \frac{1}{r})},$$

$$|\nabla d(t)|{L^{s}(\mathbb{R}^{3}{+})} \le Ct^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{s})},$$ (2.3)

$$|\nabla u(t)|{L^p(\mathbb{R}^3+)} \le Ct^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{p})},\tag{2.4}$$

$$|\nabla^2 d(t)|{L^q(\mathbb{R}^3+)} \le Ct^{-1-\frac{3}{2}(1-\frac{1}{q})},\tag{2.5}$$

for any t > ⁰, r ^∈ (1, ^∞], s ^∈ [1, ^∞], p ^∈ (1, 6], and q ^∈ [1, 6].

Further, if u⁰ satisfies ∥*x*3*u0∥^L* (R + ) ^&lt; ^∞, then the estimates on u can be improved as

$$\begin{cases} |u(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq Ct^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{r})}, \ |\nabla u(t)|{L^{p}(\mathbb{R}^{3}{+})} \leq Ct^{-1 - \frac{3}{2}(1 - \frac{1}{p})} \end{cases}$$

for any t > ⁰, r ^∈ (1, ^∞], and p ^∈ (1, 6].

The following result on the global solution u is from Theorem 1.2 in [22].

Lemma 2.6. Under the same assumptions of Theorem 1.1, if, in addition, u⁰ ∈ D 1 (R 3 + ) and ∇d⁰ ∈ D 1 (R 3 + ), then

$$|\nabla u(t)|{L^1(\mathbb{R}^3+)} \le Ct^{-\frac{1}{2}}$$

holds for any t > ⁰.

Next we recall the classical Gagliardo-Nirenberg-Sobolev inequality in the half-space.

Lemma 2.7. ([39]) Let m be a positive integer, p, ^q,^{r ∈} [1, ^∞]. If u ^{∈ L} q (R 3 + ) ∩ D m,p (R 3 + ), then for any integer k ^∈ [0, ^m),

$$|\nabla^k u|{L^r(\mathbb{R}^3+)} \leq C |\nabla^m u|{L^p(\mathbb{R}^3+)}^{\alpha} |u|{L^q(\mathbb{R}^3+)}^{1-\alpha}.$$

where

$$\frac{1}{r} - \frac{k}{3} = \alpha (\frac{1}{p} - \frac{m}{3}) + (1 - \alpha) \frac{1}{q},$$

with

$$\left{ \begin{array}{l} \alpha \in [\frac{k}{m},1), \quad if \ p \in (1,+\infty) \ and \ m-k-\frac{3}{p} \in \mathbb{N}, \ \alpha \in [\frac{k}{m},1], \quad otherwise. \end{array} \right.$$

However, when q = +∞, k ^{= 0} and mp < ³, the additional condition is needed: u(x) ^{→ 0}, as x → ∞ or u ∈ L γ (R 3 + ) for some finite γ ^{≥ 1}. The above positive constant C depends only on m, ^k, ^p, q and α.

As a special case often used in this paper, it holds that

$$||f||{L^{\infty}(\mathbb{R}^{3}{+})} \le C||f||{L^{s}(\mathbb{R}^{3}{+})}^{1-\frac{3}{s}} ||\nabla f||{L^{s}(\mathbb{R}^{3}{+})}^{\frac{3}{s}}, \ \forall \ f \in W^{1,s}(\mathbb{R}^{3}_{+}), \ 3 < s < \infty.$$

$$(2.6)$$

Now we recall the regularity estimates of the steady Stokes equations in the half-space (see [9, p.254]).

Lemma 2.8. Let f, ^{g ∈ C} ∞ 0 (R 3 + ), m ^{≥ 0}, q ^∈ (1, ^∞). Then any solution (ϕ, π) to the nonhomogeneous Stokes system

$$\begin{cases} -\Delta \phi + \nabla \pi = f & in & \mathbb{R}^3+, \ \nabla \cdot \phi = g & in & \mathbb{R}^3+, \ \phi(x) = 0 & on & \partial \mathbb{R}^3_+, \ \phi(x) \to 0 & as & |x| \to \infty \end{cases}$$

*is C∞, and belongs to Wm+2,^q* (R 3 + )×Wm+1,^q (R 3 + ). Moreover, for each integer ℓ ^{≥ 0}, the following estimate holds

$$|\nabla^{\ell+2}\phi|{L^{q}(\mathbb{R}^{3}{+})} + |\nabla^{\ell+1}\pi|{L^{q}(\mathbb{R}^{3}{+})} \le C(\ell, q)(|\nabla^{\ell}f|{L^{q}(\mathbb{R}^{3}{+})} + |\nabla^{\ell+1}g|{L^{q}(\mathbb{R}^{3}{+})}). \tag{2.7}$$

Let g = N f denote the solution of the Neumann problem

$$\begin{cases} -\Delta g = f & \text{in } \mathbb{R}^3_+, \ g(x) \to 0 & as |x| \to \infty, \ \partialv g|{\partial \mathbb{R}^3_+} = 0. \end{cases} (2.8)$$

Then (see [10])

$$\mathcal{N} = \int_0^\infty F(\tau) \, d\tau,\tag{2.9}$$

where the operator F(t) is defined as follows:

$$(F(t)f)(x) = \int{\mathbb{R}^3+} [G_t(x'-y',x_3-y_3) + G_t(x'-y',x_3+y_3)]f(y) \, dy,$$

and ^Gt(x) ⁼ (4πt) − 2 e − ^4t is the Gaussian kernel in R 3 . In addition, it holds that (see [22])

$$\mathbb{P}(u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d))$$

$$= (u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d)) + \sum{i,j=1}^{3} \nabla \mathcal{N} \partial{i} \partial{j} (u{i} u{j} + \langle \partial{i} d, \partial_{j} d \rangle). \tag{2.10}$$

Moreover, we have the following estimates.

Lemma 2.9. ([37]) Let ⁰ < θ < ¹. Then for any u, ^{v ∈ C} ∞ 0,σ (R 3 + )

$$\begin{split} |x3^{-\theta}\mathbb{P}(u\cdot\nabla v)|{L^q(\mathbb{R}^3_+)} = & |x3^{-\theta}(u\cdot\nabla v + \sum{i,j=1}^3 \nabla \mathcal{N}\partial_i\partial_j(u_ivj))|{L^q(\mathbb{R}^3+)} \ \leq & C(|u|{L^{q1}(\mathbb{R}^3+)}|v|_{L^{q2}(\mathbb{R}^3+)} + |\nabla u|_{L^{q1}(\mathbb{R}^3+)}|\nabla v|_{L^{q2}(\mathbb{R}^3+)} \ & + |u|_{L^{q1}(\mathbb{R}^3+)}|\nabla^2 v|_{L^{q2}(\mathbb{R}^3+)} + |u|_{L^{q1}(\mathbb{R}^3+)}|\nabla v|_{L^{q2}(\mathbb{R}^3+)}. \end{split}$$

where ^{1 ≤ q &}lt; ^∞, q ^≤ *^q*1, ^q2 ≤ ∞, 1 q = 1 *q*1 + 1 *q*2 .

We refer to [22, p.1774] and [11, p.3945] for the following lemma.

Lemma 2.10. Let ^{1 ≤ k}, ^{m ≤ 3}. Then for any u, ^{v ∈ C} ∞ 0,σ (R 3 + ) and d, ^{g ∈ C ∞}(R 3 + ), we have the following estimates: for any t > ⁰,

$$\begin{split} &| \sum{i,j=1}^{3} \partial{k} \mathcal{N} \partial{i} \partial{j} (u{i}v{j} + \langle \partial{i}d, \partial{j}g \rangle)(t) |{L^{q}(\mathbb{R}^{3}{+})} \ \leq & C(|u(t)|{L^{q{1}}(\mathbb{R}^{3}{+})} |v(t)|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla u(t)|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla v(t)|{L^{q{2}}(\mathbb{R}^{3}{+})} \ &+ |\nabla d(t)|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla g(t)|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla^{3}d(t)|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla g(t)|{L^{q{2}}(\mathbb{R}^{3}{+})} \ &+ |\nabla^{2}d(t)|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla^{2}g(t)|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla d(t)|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla^{3}g(t)|{L^{q{2}}(\mathbb{R}^{3}{+})}); \end{split} \tag{2.11}$$

and

$$\begin{split} &| \sum{i,j=1}^{3} \partial{k} \partial{m} N \partial{i} \partial{j} (u{i} v{j} + \langle \partial{i} d, \partial{j} g \rangle)(t) |{L^{q}(\mathbb{R}^{3}{+})} \ \leq & C (| \nabla^{2} u(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla v(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} + | \nabla u(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla^{2} v(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} \ &+ | \nabla u(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | v(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} + | u(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla v(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} \ &+ | \nabla d(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla^{4} g(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} + | \nabla^{2} d(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla^{3} g(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} \ &+ | \nabla^{3} d(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla^{2} g(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} + | \nabla^{4} d(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla g(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} \ &+ | \nabla^{2} d(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla g(t) |{L^{q{2}}(\mathbb{R}^{3}{+})} + | \nabla d(t) |{L^{q{1}}(\mathbb{R}^{3}{+})} | \nabla^{2} g(t) |{L^{q{2}}(\mathbb{R}^{3}_{+})} ), \end{split} \tag{2.12}$$

where 1 ≤ q ≤ ∞, q = *q*1 + *q*2 , ^{1 ≤} *^q*1, ^q2 ≤ ∞. Especially, we have

$$| \sum{i,j=1}^{3} \nabla N \partial{i} \partial{j} (u{i}u{j} + \langle \partial{i}d, \partial{j}d \rangle)(t) |{L^{1}(\mathbb{R}^{3}_{+})}$$

$$\leq C(|u(t)|{H^{1}(\mathbb{R}^{3}{+})}^{2} + |\nabla d(t)|{H^{1}(\mathbb{R}^{3}{+})}^{2} + ||\nabla d(t)||\nabla^{3}d(t)|{L^{1}(\mathbb{R}^{3}{+})}).$$ (2.13)

Proof. We only prove (2.12), and (2.11) can be proved in the same way. Let 1 ^{≤ k}, ^{m ≤} 3. Using (2.9), we deduce for 1 ^{≤ q}, *^q*1, ^q2 ≤ ∞,

$$\begin{split} & \left| \sum{i,j=1}^{3} \partial{k} \partial{m} \mathcal{N} \partial{i} \partial{j} (u{i}v{j} + \langle \partial{i}d, \partial{j}g \rangle)(t) \right|{L^{q}(\mathbb{R}^{3}{+})} \ & = \left| \sum{i,j=1}^{3} \partial{k} \partial{m} \int{0}^{\infty} F(\tau) \partial{i} \partial{j} (u{i}v{j} + \langle \partial{i}d, \partial{j}g \rangle)(t) \, d\tau \right|{L^{q}(\mathbb{R}^{3}{+})} \ & \leq \left| \sum{i,j=1}^{3} \partial{k} \partial{m} (\int{0}^{1} + \int{1}^{\infty}) G{\tau} * \left( \partial{i} \partial{j} (u{i}v{j} + \langle \partial{i}d, \partial{j}g \rangle) \right){*}(t) \, d\tau \right|{L^{q}(\mathbb{R}^{3}{+})} \ & \leq \sum{i,j=1}^{3} (\int{0}^{1} \tau^{-\frac{1}{2}} \, d\tau) |\nabla G{1}|{L^{1}(\mathbb{R}^{3})} |\partial{m} \left( \partial{i} \partial{j} (u{i}v{j} + \langle \partial{i}d, \partial{j}g \rangle) \right)(t) |{L^{q}(\mathbb{R}^{3}{+})} \ & + (\int{1}^{\infty} \tau^{-\frac{3}{2}} \, d\tau) |\nabla^{3} G{1}|{L^{1}(\mathbb{R}^{3})} (|\nabla u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |v(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla v(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & + |\nabla^{2} d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |\nabla d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{2} g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & \leq C(|\nabla^{2} u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla v(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |\nabla u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{2} v(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & + |\nabla d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{2} g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |\nabla^{2} d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{3} g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & + |\nabla u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{2} g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla v(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & + |\nabla u(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |\nabla d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla v(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & + |\nabla^{2} d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |\nabla d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{2} g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} \ & + |\nabla^{2} d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} + |\nabla d(t)|{L^{q{1}(\mathbb{R}^{3}{+})}} |\nabla^{2} g(t)|{L^{q{2}(\mathbb{R}^{3}{+})}} ), \end{split}$$

where f ∗ g represents the convolution of f and g, and f^∗ represents the even extension of function f with respect to x³ from R 3 + to R 3 :

$$f_*(x', x_3) = \begin{cases} f(x', x_3), & \text{if } x_3 \ge 0, \ f(x', -x_3), & \text{if } x_3 < 0. \end{cases}$$

Next we recall the classical Young inequality in the half-space.

Lemma 2.11. ([15]) Let K(x, ^y), f(y) be defined in ^R 3 ⁺ × R 3 + , R 3 + , respectively, and let ^{1 ≤ p}, ^{q ≤ r} ≤ ∞ satisfy 1 + r = 1 p + 1 q . Then

$$|\int{\mathbb{R}^3+} K(\cdot, y) f(y) \, dy |{L^r(\mathbb{R}^3+)} \leq \sup{x \in \mathbb{R}^3+} |K(x, \cdot)|{L^p(\mathbb{R}^3+)}^{1 - \frac{p}{r}} \sup{y \in \mathbb{R}^3+} |K(\cdot, y)|{L^p(\mathbb{R}^3+)}^{\frac{p}{r}} |f|{L^q(\mathbb{R}^3+)}$$

Especially there holds

$$| \int{\mathbb{R}^3+} K(\cdot, y) f(y) \, dy |{L^1(\mathbb{R}^3+)} \le \sup{y \in \mathbb{R}^3+} | K(\cdot, y) |{L^1(\mathbb{R}^3+)} | f |{L^1(\mathbb{R}^3+)};$$

and

$$|\int{\mathbb{R}^3+} K(\cdot, y) f(y) \, dy|{L^r(\mathbb{R}^3+)} \leq (\sup{x \in \mathbb{R}^3+} |K(x, \cdot)|{L^p(\mathbb{R}^3+)} + \sup{y \in \mathbb{R}^3+} |K(\cdot, y)|{L^p(\mathbb{R}^3+)}) |f|{L^q(\mathbb{R}^3+)}.$$

□

Lemma 2.12. Let ⁰ < θ < ¹. Then for any d, ^{g ∈ C ∞}(R 3 + ), the following estimate

$$\begin{split} &|x3^{-\theta}\mathbb{P}\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^3+)}\ &\leq C(|\nabla^3d|{L^{q1}(\mathbb{R}^3+)}|\nabla g|_{L^{q2}(\mathbb{R}^3+)}+|\nabla^2d|_{L^{q1}(\mathbb{R}^3+)}|\nabla^2g|_{L^{q2}(\mathbb{R}^3+)}\ &+|\nabla d|_{L^{q1}(\mathbb{R}^3+)}|\nabla^3g|_{L^{q2}(\mathbb{R}^3+)}+|\nabla d|_{L^{q1}(\mathbb{R}^3+)}|\nabla g|_{L^{q2}(\mathbb{R}^3+)}\ &+|\nabla^2d|_{L^{q1}(\mathbb{R}^3+)}|\nabla g|_{L^{q2}(\mathbb{R}^3+)}+|\nabla d|_{L^{q1}(\mathbb{R}^3+)}|\nabla^2g|_{L^{q2}(\mathbb{R}^3+)}) \end{split}$$

holds for any t ^&gt; ⁰, where ^{1 ≤ q &}lt; ^∞, q ^≤ *^q*1, ^q2 ≤ ∞, 1 q = 1 *q*1 + 1 *q*2 .

Proof. We closely follow the proof of Lemma 2.2 in [37]. From (2.9), (2.11) and Lemma 2.11, one has for any 1 ≤ k ≤ 3 and 1 ≤ q ≤ ∞ that

$$\begin{split} & \left| \sum{i,j=1}^{3} x{3}^{-\theta} \partial{k} \mathcal{N} \partial{i} \partial{j} (\langle \partial{i} d, \partial{j} g \rangle)(t) \right|{L^{q}(\mathbb{R}^{3}{+})} \ \leq & \left| \sum{i,j=1}^{3} x{3}^{-\theta} \partial{k} \int{0}^{1} G{\tau} * \left( \partial{i} \partial{j} (\langle \partial{i} d, \partial{j} g \rangle) \right){*}(t) \, d\tau \right|{L^{q}(\mathbb{R}^{2} \times (0,1))} \ & + \left| \sum{i,j=1}^{3} x{3}^{-\theta} \partial{k} \int{1}^{\infty} G{\tau} * \left( \partial{i} \partial{j} (\langle \partial{i} d, \partial{j} g \rangle) \right){*}(t) \, d\tau \right|{L^{q}(\mathbb{R}^{2} \times (0,1))} \ & + \left| \sum{i,j=1}^{3} \partial{k} \mathcal{N} \partial{i} \partial{j} (\langle \partial{i} d, \partial{j} g \rangle)(t) \right|{L^{q}(\mathbb{R}^{3}{+})} \ \leq & C \sup{y \in \mathbb{R}^{3}} \left| \int{0}^{1} x{3}^{-\theta} \partial{k} G{\tau}(x - y) \, d\tau \right|{L^{1}(\mathbb{R}^{2} \times (0,1))} \left| \sum{i,j=1}^{3} \partial{i} \partial{j} (\langle \partial{i} d, \partial{j} g \rangle) \right|{L^{q}(\mathbb{R}^{3}{+})} \ & + C \sup{y \in \mathbb{R}^{3}} \sum{i,j=1}^{3} \left| \int{1}^{\infty} x{3}^{-\theta} \partial{k} \partial{i} \partial{j} G{\tau}(x - y) \, d\tau \right|{L^{1}(\mathbb{R}^{2} \times (0,1))} \left| |\nabla d| |\nabla g| \right|{L^{q}(\mathbb{R}^{3}{+})} \ & + \left| \sum{i,j=1}^{3} \partial{k} \mathcal{N} \partial{i} \partial{j} (\langle \partial{i} d, \partial{j} g \rangle)(t) \right|{L^{q}(\mathbb{R}^{3}{+})} \ \leq & C(|\nabla^{3} d|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla g|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla d|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla g|{L^{q{2}}(\mathbb{R}^{3}{+})} \ & + |\nabla d|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla^{3} g|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla d|{L^{q{1}}(\mathbb{R}^{3}{+})} |\nabla g|{L^{q{2}}(\mathbb{R}^{3}_{+})}, \tag{2.14} \end{split}$$

where we use the fact that (see [12, p.1758])

$$| \int_0^1 x_3^{-\theta} \partialk G{\tau}(x - y) \, d\tau |_{L^1(\mathbb{R}^2 \times (0, 1))} \le C_1,$$

and

$$|\int_1^\infty x_3^{-\theta} \partial_k \partial_i \partialj G\tau(x-y) \, d\tau|_{L^1(\mathbb{R}^2 \times (0,1))} \le C_2.$$

By one-dimensional Hardy inequality (see [18]), we get for 0 < θ < 1 that

$$\begin{split} &|x3^{-\theta}\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^3_+)}^q\ \leq &|x3^{-\theta}\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^3_+)}^q\ \leq &|x3^{-\theta}\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^2\times(0,1))}^q + |x3^{-\theta}\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^2\times[1,\infty))}^q\ \leq &\int_{\mathbb{R}^2}\int_0^1 x_3^{q-q\theta}\frac{|(\nabla\cdot(\nabla d\odot\nabla g))(x',x_3)|^q}{x_3^q}\,dx3dx' + |\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^2\times[1,+\infty))}^q\ \leq &\int_{\mathbb{R}^2}\int_0^\infty\frac{|(\nabla\cdot(\nabla d\odot\nabla g))(x',x_3)|^q}{x_3^q}\,dx3dx' + |\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^3+)}^q\ \leq &C\int{\mathbb{R}^2}\int_0^\infty|\partial_3(\nabla\cdot(\nabla d\odot\nabla g))(x',x_3)|^q\,dx3dx' + |\nabla\cdot(\nabla d\odot\nabla g)|{L^q(\mathbb{R}^3_+)}^q \end{split}$$

$$\leq C(|\nabla^{3}d|{L^{q{1}}(\mathbb{R}^{3}{+})}|\nabla g|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla^{2}d|{L^{q{1}}(\mathbb{R}^{3}{+})}|\nabla^{2}g|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla d|{L^{q{1}}(\mathbb{R}^{3}{+})}|\nabla^{2}g|{L^{q{2}}(\mathbb{R}^{3}{+})} + |\nabla d|{L^{q{1}}(\mathbb{R}^{3}{+})}|\nabla^{2}g|{L^{q{2}}(\mathbb{R}^{3}{+})},$$

$$(2.15)$$

where 1 ^{≤ q &}lt; ^∞, ^{q ≤} *^q*1, ^q2 ≤ ∞, q = *q*1 + *q*2 . Hence, Lemma 2.12 directly follows from (2.14), (2.15) and the decomposition

$$\mathbb{P}\nabla\cdot(\nabla d\odot\nabla g)=\nabla\cdot(\nabla d\odot\nabla g)+\sum{i,j=1}^{3}\nabla\mathcal{N}\partial{i}\partial{j}(\langle\partial{i}d,\partial_{j}g\rangle).$$

3. Proof of ^Theorem 1.¹

In this section, we will provide the proof for Theorem 1.1. We first recall a lemma on the estimates for the L 3 -norm of (u, ^∇d), which will play an important role in the sequel.

Lemma 3.1. ([22]) There exists ^{ε0 &}gt; ⁰ such that if

$$||u0||{L^3(\mathbb{R}^3_+)}^3 + ||\nabla d0||{L^3(\mathbb{R}^3_+)}^3 \le \varepsilon_0^3$$

then for any ⁰ < ^t < ^∞,

$$|u(t)|{L^3(\mathbb{R}^3+)}^3 + |\nabla d(t)|{L^3(\mathbb{R}^3+)}^3 \leq |u0|{L^3(\mathbb{R}^3_+)}^3 + |\nabla d0|{L^3(\mathbb{R}^3_+)}^3.$$

Then we have the following lemma.

Lemma 3.2. *Assume there exist C*1,C^{2 &}gt; ⁰ such that if the global solution (u, ^d) given by Proposition 1.¹ satisfies

$$|\nabla^2 d(t)|{L^{\infty}(\mathbb{R}^3+)} \le 2C_1 t^{-\frac{5}{2}},\tag{3.1}$$

$$|\nabla u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} \le 2C_{2}t^{-2} \tag{3.2}$$

for any t ^∈ (0, ^∞). Then the following estimates

$$|\nabla u(t)|{L^{p}(\mathbb{R}^{3}{+})} \le Ct^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{p})},\tag{3.3}$$

$$|\nabla^2 d(t)|{L^r(\mathbb{R}^3+)} \le C t^{-1-\frac{3}{2}(1-\frac{1}{r})},\tag{3.4}$$

$$|\nabla^2 d(t)|{L^{\infty}(\mathbb{R}^3+)} \le \frac{3}{2} C_1 t^{-\frac{5}{2}},\tag{3.5}$$

$$|\nabla u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} \le \frac{3}{2}C_{2}t^{-2} \tag{3.6}$$

hold for any t ^∈ (0, ^∞), where p ^∈ (1, ^∞) and r ^∈ [1, ^∞).

Proof. Step 1: Proof of (3.5).

From (2.5), (3.1) and the interpolation inequality, we obtain that for any 2 < ^r < ^∞,

$$|\nabla^2 d(t)|{L^r(\mathbb{R}^3+)} \leq |\nabla^2 d(t)|{L^2(\mathbb{R}^3+)}^{\frac{r}{r}} |\nabla^2 d(t)|{L^\infty(\mathbb{R}^3+)}^{\frac{r-2}{r}} \leq C t^{-1-\frac{3}{2}(1-\frac{1}{r})}.$$

Similarly, by (2.4) and (3.2), we obtain that for any 2 < ^p < ^∞,

$$|\nabla u(t)|{L^p(\mathbb{R}^3+)} \le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}.$$

Applying ^∇ to (1.1)2, it gives

$$(\nabla d)_t - \Delta(\nabla d) = -\nabla(u \cdot \nabla d) + \nabla(|\nabla d|^2 d),$$

in terms of Duhamel's principle, we have

$$\nabla d(t) = e^{\frac{t}{2}\Delta} \nabla d(\frac{t}{2}) + \int_{\frac{t}{2}}^{t} e^{(t-s)\Delta} \Big( -\nabla (u \cdot \nabla d) + \nabla (|\nabla d|^2 d) \Big)(s) \, ds, \tag{3.7}$$

and

$$\nabla^2 d(t) = \nabla e^{\frac{t}{2}\Delta} \nabla d(\frac{t}{2}) + \int_{\frac{t}{2}}^t \nabla e^{(t-s)\Delta} \Big( -\nabla (u \cdot \nabla d) + \nabla (|\nabla d|^2 d) \Big) (s) \, ds. \tag{3.8}$$

□

This, together with (2.1) and (2.3), gives rise to

$$\begin{split} &|\nabla^2 d(t)|{L^{\infty}(\mathbb{R}^3+)} \ &\leq Ct^{-\frac{5}{4}} |\nabla d(t)|{L^2(\mathbb{R}^3+)} + \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} |\nabla (u \cdot \nabla d) - \nabla (|\nabla d|^2 d)|{L^q(\mathbb{R}^3_+)} ds \ &\leq C1 t^{-\frac{5}{2}} + \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} ||\nabla u||\nabla d| + |u||\nabla^2 d| + |\nabla d||\nabla^2 d| + |\nabla d|^3|{L^q(\mathbb{R}^3+)} ds \ &\leq C_1 t^{-\frac{5}{2}} + K_1 + K_2 + K_3 + K_4, \end{split}$$

where we choose ^{q ∈} (3, ^∞) so that ⁻ 2 − 2q ^&gt; ⁻1. We first estimate ^K1 by using (2.3), (3.3), the interpolation inequality, Lemma 2.4 and Lemma 3.1,

$$\begin{split} K1 &= \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} \left| |\nabla u||\nabla d| \right|{L^q(\mathbb{R}^3+)} ds \ &\leq C \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} |\nabla u(s)|{L^{2q}(\mathbb{R}^3+)} |\nabla d(s)|{L^3(\mathbb{R}^3+)}^{\frac{3}{2q}} |\nabla d(s)|{L^\infty(\mathbb{R}^3_+)}^{\frac{2q-3}{2q}} ds \ &\leq C \varepsilon0^{\frac{3}{2q}} \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} s^{-4+\frac{15}{4q}} ds \ &\leq C \varepsilon_0^{\frac{3}{2q}} t^{-\frac{5}{2}}. \end{split}$$

Similarly, using (3.3), (3.4), the interpolation inequality, Lemma 2.4, Lemma 2.5 and Lemma 3.1, we can establish the estimates as follows:

$$\begin{split} K2 &= \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} \left| |u| |\nabla^2 d| \right|{L^q(\mathbb{R}^3+)} ds \ &\leq C \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} |u(s)|{L^3(\mathbb{R}^3+)}^{\frac{3q}{2q}} |u(s)|{L^\infty(\mathbb{R}^3+)}^{\frac{2q-3}{2q}} |\nabla^2 d(s)|{L^{2q}(\mathbb{R}^3_+)} ds \ &\leq C \varepsilon0^{\frac{3q}{2q}} \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} s^{-4+\frac{3}{q}} ds \ &\leq C \varepsilon_0^{\frac{3q}{2q}} t^{-\frac{5}{2}}, \ K3 &= \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} \left| |\nabla d| |\nabla^2 d| \right|{L^q(\mathbb{R}^3+)} ds \ &\leq C \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} |\nabla^2 d(s)|{L^{2q}(\mathbb{R}^3+)} |\nabla d(s)|{L^3(\mathbb{R}^3+)}^{\frac{3q}{2q}} |\nabla d(s)|{L^\infty(\mathbb{R}^3_+)}^{\frac{2q-3}{2q}} ds \ &\leq C \varepsilon0^{\frac{3}{2q}} \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} s^{-\frac{9}{2}+\frac{15}{4q}} ds \ &\leq C \varepsilon_0^{\frac{3}{2q}} t^{-\frac{5}{2}}, \ K4 &= \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} |\nabla d(s)|{L^3(\mathbb{R}^3+)}^{\frac{3q}{2q}} |\nabla d(s)|{L^3(\mathbb{R}^3+)}^{\frac{3q}{2q}} ds \ &\leq C \varepsilon0^{\frac{3}{q}} \int{\frac{t}{2}}^t (t-s)^{-\frac{1}{2}-\frac{3}{2q}} s^{-6+\frac{6}{q}} ds \ &\leq C \varepsilon_0^{\frac{3}{q}} t^{-\frac{5}{2}}. \end{split}$$

Thus (3.5) follows by choosing ^ε0 sufficiently small so that

$$C\varepsilon_0^{\frac{3}{q}} + C\varepsilon_0^{\frac{3}{2q}} \le \frac{C_1}{8}$$

Step 2: Proof of (3.6).

By (1.4)1, (3.3), (3.4), the interpolation inequality, Lemma 2.1 and Lemma 2.5, one has for any ^t > 0, ∥∇u(t)∥*L*∞(^R 3

$$\leq Ct^{-\frac{5}{4}} |u(\frac{t}{2})|{L^{2}(\mathbb{R}^{3}{+})} + C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} |(u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d))(s)|{L^{r}(\mathbb{R}^{3}_{+})} ds$$

$$\leq C{2}t^{-2} + C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} (|u(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla u(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2}d(s)|{L^{2r}(\mathbb{R}^{3}{+})}) ds$$

$$\leq C{2}t^{-2} + C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} |\nabla u(s)|{L^{2r}(\mathbb{R}^{3}{+})} |u(s)|{L^{3}(\mathbb{R}^{3}{+})}^{\frac{3r}{2r}} |u(s)|{L^{\infty}(\mathbb{R}^{3}{+})}^{\frac{2r-3}{2r}} ds$$

$$+ C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} |\nabla^{2}d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d(s)|{L^{3}(\mathbb{R}^{3}{+})}^{\frac{3r}{2r}} |\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}_{+})}^{\frac{2r-3}{2r}} ds$$

$$\leq C{2}t^{-2} + C\varepsilon{0}^{\frac{3r}{2r}} \int_{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} s^{-\frac{7}{2} + \frac{3}{r}} ds$$

$$\leq C{2}t^{-2} + C\varepsilon{0}^{\frac{3r}{2r}} t^{-2},$$

here, we choose ^{r ∈} (3, ^∞) so that ⁻ − 2r ^&gt; ⁻1. Thus (3.6) follows by choosing ^ε0 sufficiently small so that *^C*ε 2r 0 ≤ *C*2 2

Now we can utilize the continuity argument to obtain the subsequent corollary.

Corollary 3.1. *Assume u*0, *^d*0−e^{3 ∈ L} 1 (R 3 + ), if (u, ^d) is the global strong solution obtained by Proposition 1.1, then

$$\begin{split} |\nabla^2 d(t)|{L^p(\mathbb{R}^3+)} & \leq C t^{-1-\frac{3}{2}(1-\frac{1}{p})} \ |\nabla u(t)|{L^r(\mathbb{R}^3+)} & \leq C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r})} \end{split}$$

hold for any t > ⁰, where p ^∈ [1, ^∞], r ^∈ (1, ^∞].

In order to obtain the L r -decay rates of the second-order derivatives of u, we need the following lemma to overcome the difficulty of integration singularity.

Lemma 3.3. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ), if (u, ^d) is the global strong solution obtained by Proposition 1.1, then it holds for any t > ⁰ that

$$|\mathbb{A}^{\alpha}u(t+h) - \mathbb{A}^{\alpha}u(t)|{L^{r}(\mathbb{R}^{3}{+})} \le C\left(h^{\delta}t^{-\alpha-\delta-\frac{3}{2}+\frac{3}{2r}} + h^{1-\alpha}t^{-\frac{7}{2}+\frac{3}{2r}} + h^{\delta}t^{-\alpha-\delta-\frac{5}{2}+\frac{3}{2r}}\right);\tag{3.9}$$

$$|(-\Delta)^{\alpha}d(t+h) - (-\Delta)^{\alpha}d(t)|{L^{r}(\mathbb{R}^{3}{+})} \le C\left(h^{\delta}t^{-\alpha-\delta-\frac{3}{2}+\frac{3}{2r}} + h^{1-\alpha}t^{-\frac{7}{2}+\frac{3}{2r}} + h^{\delta}t^{-\alpha-\delta-\frac{5}{2}+\frac{3}{2r}}\right); \tag{3.10}$$

$$|(-\Delta)^{\alpha} \nabla d(t+h) - (-\Delta)^{\alpha} \nabla d(t)|{L^{r}(\mathbb{R}^{3}{+})} \le C \Big( h^{\delta} t^{-\alpha-\delta-2+\frac{3}{2r}} + h^{1-\alpha} t^{-4+\frac{3}{2r}} + h^{\delta} t^{-\alpha-\delta-3+\frac{3}{2r}} \Big), \tag{3.11}$$

where ¹ < ^r < ^∞, h > ⁰, ⁰ < α < ¹ and ⁰ < δ < ^{1 −} α.

Proof. Step 1: Proof of (3.9).

It is easy to verify that the ^u equation of system (1.1) can be transformed into the integral form as for any 0 ^≤ τ < ^t,

$$u(t) = e^{-(t-\tau)\mathbb{A}}u(\tau) - \int_{\tau}^{t} e^{-(t-s)\mathbb{A}} \mathbb{P}\left(u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d(s) \odot \nabla d(s))\right) ds.$$

Then, for any ^t > 0, ^h > 0,

$$u(t+h) = e^{-(t+h-\frac{t}{2})\mathbb{A}}u(\frac{t}{2}) - \int_{\frac{t}{2}}^{t+h} e^{-(t+h-s)\mathbb{A}} \mathbb{P}\Big(u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d(s) \odot \nabla d(s))\Big) ds$$

□

$$=e^{-h\mathbb{A}}e^{-\frac{t}{2}\mathbb{A}}u(\frac{t}{2})-\Big(\int{t}^{t+h}+\int{\frac{t}{2}}^{t}\Big)e^{-(t-s)\mathbb{A}}e^{-h\mathbb{A}}\mathbb{P}\Big(u(s)\cdot\nabla u(s)+\nabla\cdot\big(\nabla d(s)\odot\nabla d(s)\big)\Big)ds. \quad (3.12)$$

For any φ ^{∈ D}(^A ) and 1 < ^q < ^∞, we have

$$\left|(e^{-h\mathbb{A}}-I)\varphi\right|{L^q(\mathbb{R}^3+)}=h\left|\int0^1\mathbb{A}^{1-\delta}e^{-sh\mathbb{A}}\mathbb{A}^\delta\varphi\,ds\right|{L^q(\mathbb{R}^3+)}\leq \frac{C}{\delta}h^\delta|\mathbb{A}^\delta\varphi|{L^q(\mathbb{R}^3_+)},\tag{3.13}$$

where we use the fact that A e ⁻t^A = ´ ∞ 0 λ e ⁻*t*λ*dE*λ is a bounded operator with operator norm ^∥A e ⁻*t*A∥ ≤ t −α (see [47, p.206]). It is worth mentioning that (3.13) still holds if we replace the operator ^A by ⁻∆.

Whence, from (3.12), (3.13), Lemma 2.4, Lemma 2.5 and Corollary 3.1, we deduce for 1 < ^r < ^∞ and any ^t > 0, ^h > 0,

$$\begin{split} & \left| \mathbb{A}^{\alpha} u(t+h) - \mathbb{A}^{\alpha} u(t) \right|{L^{r}(\mathbb{R}^{3}{+})} \ \leq & \left| \mathbb{A}^{\alpha} (e^{-h\mathbb{A}} - I) e^{-\frac{t}{2}\mathbb{A}} u(\frac{t}{2}) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \int{t}^{t+h} \left| \mathbb{A}^{\alpha} e^{-(t+h-s)\mathbb{A}} \mathbb{P} (u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d))(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ & + \int{\frac{t}{2}}^{t} \left| (e^{-h\mathbb{A}} - I) \mathbb{A}^{\alpha} e^{-(t-s)\mathbb{A}} \mathbb{P} (u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d))(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ \leq & Ch^{\delta} \left| \mathbb{A}^{\alpha+\delta} e^{-\frac{t}{2}\mathbb{A}} u(\frac{t}{2}) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + C \int{t}^{t+h} (t+h-s)^{-\alpha} \left( | u(s) |{L^{r{1}}(\mathbb{R}^{3}{+})} | \nabla u(s) |{L^{r{2}}(\mathbb{R}^{3}{+})} + | \nabla d(s) |{L^{r{1}}(\mathbb{R}^{3}{+})} | \nabla^{2} d(s) |{L^{r{2}}(\mathbb{R}^{3}{+})} \right) ds \ & + Ch^{\delta} \int{\frac{t}{2}}^{t} (t-s)^{-\alpha-\delta} \left( | u(s) |{L^{r{1}}(\mathbb{R}^{3}{+})} | \nabla u(s) |{L^{r{2}}(\mathbb{R}^{3}{+})} + | \nabla d(s) |{L^{r{1}}(\mathbb{R}^{3}{+})} | \nabla^{2} d(s) |{L^{r{2}}(\mathbb{R}^{3}{+})} \right) ds \ \leq & Ch^{\delta} t^{-\alpha-\delta} | u(\frac{t}{2}) |{L^{r}(\mathbb{R}^{3}{+})} \ & + C \int{t}^{t+h} (t+h-s)^{-\alpha} \left( s^{-\frac{3}{2}(1-\frac{1}{r{1}})-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r{2}})} + s^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r{1}})-1-\frac{3}{2}(1-\frac{1}{r{2}})} \right) ds \ & + Ch^{\delta} \int{\frac{t}{2}}^{t} (t-s)^{-\alpha-\delta} \left( s^{-\frac{3}{2}(1-\frac{1}{r{1}})-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r{2}})} + s^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r{1}})-1-\frac{3}{2}(1-\frac{1}{r{2}})} \right) ds \ \leq & C \left( h^{\delta} t^{-\alpha-\delta-\frac{3}{2}+\frac{3}{2}r} + h^{1-\alpha} t^{-\frac{7}{2}+\frac{3}{2}r} + h^{\delta} t^{-\alpha-\delta-\frac{5}{2}+\frac{3}{2r}} \right), \end{split}$$

where ¹ r = *r*1 + *r*2 , *^r*1,r^{2 ∈} (1, ^∞).

Step 2: Proof of (3.10).

Applying a similar argument as in the proof of (3.9), we are able to obtain, after a straightforward computation, for any ^t > 0, ^h > 0 that

$$(d - e_3)(t + h) = e^{h\Delta} e^{\frac{t}{2}\Delta} (d - e_3)(\frac{t}{2}) - \left(\intt^{t+h} + \int{\frac{t}{2}}^t \right) e^{(t-s)\Delta} e^{h\Delta} \left(u(s) \cdot \nabla d(s) - |\nabla d(s)|^2 d(s)\right) ds.$$ (3.14)

Therefore, from (3.13), (3.14), Lemma 2.4, Lemma 2.5 and Corollary 3.1, we conclude that for all ¹ < ^r < ^∞ and any ^t > 0, ^h > 0,

$$\begin{split} & \left| (-\Delta)^{\alpha} d(t+h) - (-\Delta)^{\alpha} d(t) \right|{L^{r}(\mathbb{R}^{3}{+})} \ \leq & \left| (-\Delta)^{\alpha} (e^{h\Delta} - I) e^{\frac{t}{2}\Delta} (d - e{3}) (\frac{t}{2}) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \int{t}^{t+h} \left| (-\Delta)^{\alpha} e^{(t+h-s)\Delta} (u \cdot \nabla d + |\nabla d|^{2} d)(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ & + \int{\frac{t}{2}}^{t} \left| (e^{h\Delta} - I) (-\Delta)^{\alpha} e^{(t-s)\Delta} (u \cdot \nabla d + |\nabla d|^{2} d)(s) \right|{L^{r}(\mathbb{R}^{3}_{+})} ds \end{split}$$

$$\begin{split} & \leq Ch^{\delta} \Big| (-\Delta)^{\alpha+\delta} e^{\frac{t}{2}\Delta} (d-e3) (\frac{t}{2}) \Big|{L^r(\mathbb{R}^3_+)} \ & + C \intt^{t+h} (t+h-s)^{-\alpha} \Big( |u(s)|{L^{r1}(\mathbb{R}^3+)} |\nabla d(s)|_{L^{r2}(\mathbb{R}^3+)} + |\nabla d(s)|{L^{2r}(\mathbb{R}^3+)}^2 \Big) \, ds \ & + Ch^{\delta} \int{\frac{t}{2}}^t (t-s)^{-\alpha-\delta} \Big( |u(s)|{L^{r1}(\mathbb{R}^3+)} |\nabla d(s)|_{L^{r2}(\mathbb{R}^3+)} + |\nabla d(s)|{L^{2r}(\mathbb{R}^3+)}^2 \Big) \, ds \ & \leq Ch^{\delta} t^{-\alpha-\delta} |(d-e3) (\frac{t}{2})|{L^r(\mathbb{R}^3_+)} \ & + C \intt^{t+h} (t+h-s)^{-\alpha} s^{-\frac{1}{2}-3+\frac{3}{2r}} \, ds + Ch^{\delta} \int{\frac{t}{2}}^t (t-s)^{-\alpha-\delta} s^{-\frac{1}{2}-3+\frac{3}{2r}} \, ds \ & \leq C \Big( h^{\delta} t^{-\alpha-\delta-\frac{3}{2}+\frac{3}{2r}} + h^{1-\alpha} t^{-\frac{7}{2}+\frac{3}{2r}} + h^{\delta} t^{-\alpha-\delta-\frac{5}{2}+\frac{3}{2r}} \Big), \end{split}$$

where ¹ r = 1 *r*1 + *r*2 , *^r*1,r^{2 ∈} (1, ^∞).

Step 3: Proof of (3.11).

where ¹ r

*r*1

*r*2

In the same way, it follows from (3.7) that

$$\nabla d(t+h) = e^{h\Delta} e^{\frac{t}{2}\Delta} \nabla d(\frac{t}{2}) + \left( \int{t}^{t+h} + \int{\frac{t}{2}}^{t} \right) e^{(t-s)\Delta} e^{h\Delta} \left( -\nabla (u \cdot \nabla d) + \nabla (|\nabla d|^2 d) \right) (s) \, ds. \tag{3.15}$$

Therefore, from (3.13), (3.15), Lemma 2.4, Lemma 2.5 and Corollary 3.1, we conclude that for all ¹ < ^r < ^∞ and any ^t > 0, ^h > 0,

$$\begin{split} & \left| (-\Delta)^{\alpha} \nabla d(t+h) - (-\Delta)^{\alpha} \nabla d(t) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq \left| (-\Delta)^{\alpha} (e^{h\Delta} - I) e^{\frac{t}{2}\Delta} \nabla d(\frac{t}{2}) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \int{t}^{t+h} \left| (-\Delta)^{\alpha} e^{(t+h-s)\Delta} (\nabla (u \cdot \nabla d) + \nabla (|\nabla d|^{2}d))(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ & + \int{t}^{t} \left| (e^{h\Delta} - I) (-\Delta)^{\alpha} e^{(t-s)\Delta} (\nabla (u \cdot \nabla d) + \nabla (|\nabla d|^{2}d))(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ & \leq Ch^{\delta} \left| (-\Delta)^{\alpha+\delta} e^{\frac{t}{2}\Delta} \nabla d(\frac{t}{2}) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + C \int{t}^{t+h} (t+h-s)^{-\alpha} (\left| \nabla u(s) \right|{L^{r{1}}(\mathbb{R}^{3}{+})} \left| \nabla d(s) \right|{L^{r{2}}(\mathbb{R}^{3}{+})} + \left| u(s) \right|{L^{r{1}}(\mathbb{R}^{3}{+})} \left| \nabla^{2} d(s) \right|{L^{r{2}}(\mathbb{R}^{3}{+})} \ & + \left| \nabla d(s) \right|{L^{r{1}}(\mathbb{R}^{3}{+})} \left| \nabla^{2} d(s) \right|{L^{r{2}}(\mathbb{R}^{3}{+})} + \left| \nabla d(s) \right|{L^{3}(\mathbb{R}^{3}{+})} ds \ & + Ch^{\delta} \int{\frac{t}{2}}^{t} (t-s)^{-\alpha-\delta} (\left| \nabla u(s) \right|{L^{r{1}}(\mathbb{R}^{3}{+})} \left| \nabla d(s) \right|{L^{r{2}}(\mathbb{R}^{3}{+})} + \left| u(s) \right|{L^{r{1}}(\mathbb{R}^{3}{+})} \left| \nabla^{2} d(s) \right|{L^{r{2}}(\mathbb{R}^{3}{+})} \ & + \left| \nabla d(s) \right|{L^{r{1}}(\mathbb{R}^{3}{+})} \left| \nabla^{2} d(s) \right|{L^{r{2}}(\mathbb{R}^{3}{+})} + \left| \nabla d(s) \right|{L^{3}r(\mathbb{R}^{3}{+})} ds \ & \leq Ch^{\delta} t^{-\alpha-\delta} \left| \nabla d(\frac{t}{2}) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + C \int{t}^{t+h} (t+h-s)^{-\alpha} s^{-\frac{1}{2}-\frac{1}{2}-3+\frac{3}{2r}} ds + Ch^{\delta} \int{\frac{t}{2}}^{t} (t-s)^{-\alpha-\delta} s^{-\frac{1}{2}-\frac{1}{2}-3+\frac{3}{2r}} ds \ & \leq C \left( h^{\delta} t^{-2-\alpha-\delta+\frac{3}{2r}} + h^{1-\alpha} t^{-4+\frac{3}{2r}} + h^{\delta} t^{-\alpha-\delta-3+\frac{3}{2r}} \right), \ & = \frac{1}{r{1}} + \frac{1}{r{2}}, r{1}, r{2} \in (1,\infty). \end{aligned}$$

Base on Lemma 3.3, we can deduce the following decay rates.

Lemma 3.4. *Assume u*0, *^d*0−e^{3 ∈ L} 1 (R 3 + ), if (u, ^d, ^p) is the global strong solution obtained by Proposition 1.1, then it holds that for any t > ⁰ and ¹ < ^r < ^∞,

$$|\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} + |\partialt u(t)|{L^r(\mathbb{R}^3+)} + |\nabla p(t)|{L^r(\mathbb{R}^3_+)} + |\partialt d(t)|{L^r(\mathbb{R}^3_+)} \le Ct^{-1-\frac{3}{2}(1-\frac{1}{r})}$$

Proof. By a direct calculation, we obtain for 1 < ^r < ^∞ and ^t > 0 that

$$\mathbb{A}u(t) = \mathbb{A}e^{-\frac{3t}{4}\mathbb{A}}u(\frac{t}{4}) - (I - e^{-\frac{t}{2}\mathbb{A}})\mathbb{P}\left(u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d)\right)(t)$$

$$-\int_{\frac{t}{4}}^{\frac{t}{2}}\mathbb{A}e^{-(t-s)\mathbb{A}}\mathbb{P}\left(u \cdot \nabla u + \nabla \cdot (\nabla d \odot \nabla d)\right)(s) ds$$

$$-\int_{\frac{t}{2}}^{t}\mathbb{A}e^{-(t-s)\mathbb{A}}\left(\mathbb{P}(u \cdot \nabla u)(s) - \mathbb{P}(u \cdot \nabla u)(t)\right) ds$$

$$-\int_{\frac{t}{2}}^{t}\mathbb{A}e^{-(t-s)\mathbb{A}}\left(\mathbb{P}\nabla \cdot (\nabla d \odot \nabla d)(s) - \mathbb{P}\nabla \cdot (\nabla d \odot \nabla d)(t)\right) ds$$

$$=J{1}(t) + J{2}(t) + J{3}(t) + J{4}(t) + J_{5}(t). \tag{3.16}$$

According to Lemma 2.5 and Corollary 3.1, we have for any $$1 < r < \infty$$ and $t > 0$ , $$||J{1}(t)||{L^{r}(\mathbb{R}^{3}{+})} \leq Ct^{-1}||u(\frac{t}{4})||{L^{r}(\mathbb{R}^{3}_{+})} \leq Ct^{-1-\frac{3}{2}(1-\frac{1}{r})};$$

$$||J{2}(t)||{L^{r}(\mathbb{R}^{3}{+})} \leq 2||\mathbb{P}(u \cdot \nabla u)(t) + \mathbb{P}\nabla \cdot (\nabla d \odot \nabla d)(t)||{L^{r}(\mathbb{R}^{3}_{+})}$$

$$\leq C||u(t)||{L^{2r}(\mathbb{R}^{3}{+})}||\nabla u(t)||{L^{2r}(\mathbb{R}^{3}{+})} + C||\nabla d(t)||{L^{2r}(\mathbb{R}^{3}{+})}||\nabla^{2}d(t)||{L^{2r}(\mathbb{R}^{3}{+})}$$

$$\leq Ct^{-2-\frac{3}{2}(1-\frac{1}{r})};$$

$$(3.18)$$

$$||J{3}(t)||{L^{r}(\mathbb{R}^{3}{+})} \leq C\int{\frac{t}{4}}^{\frac{t}{2}} (t-s)^{-1} \Big( ||\mathbb{P}(u \cdot \nabla u)(s)||{L^{r}(\mathbb{R}^{3}{+})} + ||\mathbb{P}(\nabla \cdot (\nabla d \odot \nabla d))(s)||{L^{r}(\mathbb{R}^{3}{+})} \Big) ds$$

$$\leq C\int{\frac{t}{4}}^{\frac{t}{2}} (t-s)^{-1} \Big( ||u(s)||{L^{2r}(\mathbb{R}^{3}{+})}||\nabla u(s)||{L^{2r}(\mathbb{R}^{3}{+})} + ||\nabla d(s)||{L^{2r}(\mathbb{R}^{3}{+})}||\nabla^{2}d(s)||{L^{2r}(\mathbb{R}^{3}_{+})} \Big) ds$$

$$\leq C\int_{\frac{t}{4}}^{\frac{t}{4}} (t-s)^{-1} \Big( s^{-\frac{1}{2}-3+\frac{3}{2r}} + s^{-\frac{3}{2}-3+\frac{3}{2r}} \Big) ds$$

$$\leq Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}.$$

$$(3.19)$$

By Lemma 3.3 and calculations similar to [11, p.3950], we have

$$||J4(t)||{L^r(\mathbb{R}^3_+)} \le Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}.$$ (3.20)

Note that ∥∇d(t)∥^L (R + ) ≈ ∥(−∆) 1 ² d(t)∥^L (R + ) , using Lemma 2.4, Lemma 2.5, Lemma 3.3 and Corollary ³.1, we obtain the upper bound of the time decay rate for *^J*5(t) as

$$||J{5}(t)||{L^{r}(\mathbb{R}^{3}{+})} \leq C \int{\frac{t}{2}}^{t} (t-s)^{-1} ||\mathbb{P}\nabla \cdot (\nabla d \odot \nabla d)(s) - \mathbb{P}\nabla \cdot (\nabla d \odot \nabla d)(t)||{L^{r}(\mathbb{R}^{3}{+})} ds$$

$$\leq C \int{\frac{t}{2}}^{t} (t-s)^{-1} ||\mathbb{P}\nabla \cdot \left(\nabla d(s) \odot (\nabla d(s) - \nabla d(t))\right)||{L^{r}(\mathbb{R}^{3}_{+})} ds$$

$$+ C \int{\frac{t}{2}}^{t} (t-s)^{-1} ||\mathbb{P}\nabla \cdot \left((\nabla d(s) - \nabla d(t)) \odot \nabla d(t)\right)||{L^{r}(\mathbb{R}^{3}_{+})} ds$$

$$\leq C \int{\frac{t}{2}}^{t} (t-s)^{-1} ||\nabla^{2} d(s)||{L^{2r}(\mathbb{R}^{3}{+})} ||\nabla d(s) - \nabla d(t)||{L^{2r}(\mathbb{R}^{3}_{+})} ds$$

$$+ C \int{\frac{t}{2}}^{t} (t-s)^{-1} ||\nabla d(s)||{L^{2r}(\mathbb{R}^{3}{+})} ||\nabla (\nabla d(s) - \nabla d(t))||{L^{2r}(\mathbb{R}^{3}_{+})} ds$$

$$\leq C \int_{\frac{t}{2}}^{t} \left((t-s)^{\delta-1} s^{-\frac{9}{2}-\delta+\frac{3}{2t}} + (t-s)^{-\frac{1}{2}} s^{-6+\frac{3}{2t}} + (t-s)^{\delta-1} s^{-\delta-\frac{11}{2}+\frac{3}{2t}}\right) ds$$

$$\leq C t^{-\frac{9}{2}+\frac{3}{2t}}. \tag{3.21}$$

A combination of (3.16)−(3.21) shows that for any 1 < ^r < ^∞ and ^t > 0,

$$|\mathbb{A}u(t)|{L^r(\mathbb{R}^3+)} \le Ct^{-1-\frac{3}{2}(1-\frac{1}{r})},$$

which implies that (see [2])

$$|\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} \leq C |\mathbb{A} u(t)|{L^r(\mathbb{R}^3+)} \leq C t^{-1-\frac{3}{2}(1-\frac{1}{r})}.$$

Note that ∂tu(t) ^{= −}Au(t) ^{− P}(^u · ∇u)(t) ^{− P}∇ · (∇^d ⊙ ∇d)(t), we have

$$\begin{split} |\partialt u(t)|{L^r(\mathbb{R}^3+)} &\leq |\mathbb{A}u(t)|{L^r(\mathbb{R}^3+)} + |u(t)|{L^{r1}(\mathbb{R}^3+)} |\nabla u(t)|_{L^{r2}(\mathbb{R}^3+)} + |\nabla d(t)|_{L^{r1}(\mathbb{R}^3+)} |\nabla^2 d(t)|_{L^{r2}(\mathbb{R}^3+)} \ &\leq C \Big(t^{-1-\frac{3}{2}(1-\frac{1}{r})} + t^{-\frac{7}{2}+\frac{3}{2r}} + t^{-\frac{9}{2}+\frac{3}{2r}}\Big) \leq C t^{-1-\frac{3}{2}(1-\frac{1}{r})}, \end{split}$$

and then

$$\begin{split} |\nabla p(t)|{L^r(\mathbb{R}^3+)} \leq & |\partialt u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} + |u(t)|{L^{r1}(\mathbb{R}^3+)} |\nabla u(t)|_{L^{r2}(\mathbb{R}^3+)} \ & + |\nabla d(t)|_{L^{r1}(\mathbb{R}^3+)} |\nabla^2 d(t)|_{L^{r2}(\mathbb{R}^3+)} \ \leq & Ct^{-1-\frac{3}{2}(1-\frac{1}{r})}, \end{split}$$

where 1 < ^r < ^∞, 1 r = 1 *r*1 + 1 *r*2 , *^r*1,r^{2 ∈} (1, ^∞).

On the other hand, by (1.1)2, it follows from Lemma 2.5 and Corollary 3.1 that

$$\begin{split} |\partialt d(t)|{L^r(\mathbb{R}^3+)} &\leq C \Big( |u(t)|{L^{2r}(\mathbb{R}^3+)} |\nabla d(t)|{L^{2r}(\mathbb{R}^3+)} + |\nabla^2 d(t)|{L^r(\mathbb{R}^3+)} + |\nabla d(t)|{L^{2r}(\mathbb{R}^3_+)}^2 \Big) \ &\leq C t^{-1-\frac{3}{2}(1-\frac{1}{r})}. \end{split}$$

Thus, we complete the proof of Lemma 3.4. □

Next we will prove the Theorem 1.2 in the case ^{k = 1}, 2 by using the similar arguments for the incompressible Navier-Stokes equations in the half-space (see [15]).

Lemma 3.5. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ). Then there exists t^{1 &}gt; ⁰ such that for ^{1 &}lt; ^{r &}lt; ^∞ and t ^≥ *^t1, the strong solution* (u, ^d) of (1.1)-(1.2) obtained in Proposition 1.¹ satisfies

$$|\nabla^3 d(t)|{L^r(\mathbb{R}^3+)} \leq C t^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}$$

Proof. Set *^d*1(t) ⁼ ∂td(t). It is not difficult to verify that *^d*1(t) satisfies

$$\begin{cases} \partial_t d_1 + u_t \cdot \nabla d + u \cdot \nabla d_1 = \Delta d_1 + |\nabla d|^2 d_1 + 2(\nabla d : \nabla d1) d & \text{in } \mathbb{R}^3+ \times (1, \infty), \ \frac{\partial d_1}{\partial x3} = 0 & \text{on } \partial \mathbb{R}^3+ \times (1, \infty), \ d_1(x, 1) = \partialt d(1) & \text{in } \mathbb{R}^3+. \end{cases}$$ (3.22)

Using the Duhamel's principle and the solution operators to the heat equation in R 3 + (see [50]), we can know that

$$d{1}(x,t) = \int{\mathbb{R}^{3}{+}} W(x,y,\frac{t}{2}) d{1}(y,\frac{t}{2}) dy$$ $$- \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} W(x,y,t-s) (u{t} \cdot \nabla d + u \cdot \nabla d{1} - |\nabla d|^{2} d{1} - 2(\nabla d : \nabla d_{1}) d)(y,s) dy ds, \qquad (3.23)$$

where ^W(x, ^y, ^t) ^{= G}t(^{x − y}) ⁻Gt(^{x − y} ∗ ), y ^{∗ =} (*y*1, *^y*2, ⁻*y*3). Moreover, ^Gt(^{x − y} ∗ ) satisfies the following estimate (see [48, p.346]):

$$|\partial_t^{\widetilde{s}} \partial_x^{\widetilde{\ell}} \partial_y^{\widetilde{m}} G_t(x - y^)| \le C t^{-s - \frac{\widetilde{m}_3}{2}} (t + x_3^2)^{-\frac{\widetilde{\ell}_3}{2}} (|x - y^|^2 + t)^{-\frac{3 + |\widetilde{\ell}'| + |\widetilde{m}'|}{2}} e^{-\frac{c\gamma_3^2}{t}}, \tag{3.24}$$

where ^{me =} (*m*e1, *^m*e2, *^m*e3) ⁼ (m^e ′ , *^m*e3), ^eℓ ⁼ (eℓ1,eℓ2,eℓ3) ⁼ (ℓe′ ,eℓ3).

Let 1 < ^r < ^∞. Applying Corollary 3.1, Lemma 2.4, Lemma 2.5, Lemma 2.11 and Lemma 3.4, together with (3.24), we obtain that for any integer ^{m ≥} 1 and ^t > 2,

$$\begin{split} &|\nablax^m \int{\mathbb{R}^3_+} W(\cdot, y, \frac{t}{2}) d1(y, \frac{t}{2}) \, dy|{L^r(\mathbb{R}^3_+)} \ \leq & C |\nabla^m Gt|{L^1(\mathbb{R}^3_+)} |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} \times C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d1(\frac{t}{2})|{L^r(\mathbb{R}^3_+)} + C |d_1(\frac{$$

$$\sum_{m=m3+|m'|} \int{\mathbb{R}^3_+} (x_3 + \sqrt{t})^{-m_3} (|x'| + x_3 + \sqrt{t})^{-3-|m'|} dx' dx_3$$

$$\leq Ct^{-\frac{m+2}{2} - \frac{3}{2}(1 - \frac{1}{r})}; \tag{3.25}$$

and

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} W(x, y, t - s) (u{t} \cdot \nabla d + u \cdot \nabla d{1} - |\nabla d|^{2} d{1} - 2(\nabla d : \nabla d{1}) d)(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} |\nabla G{t-s}(\cdot)|{L^{1}(\mathbb{R}^{3}{+})} |(u{t} \cdot \nabla d + u \cdot \nabla d{1} + |\nabla d|^{2} d{1} + (\nabla d : \nabla d{1}) d)(s)|{L^{r}(\mathbb{R}^{3}{+})} \, ds \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (|\partial{t} u(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |u(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} \ & + |\nabla d(s)|{L^{4r}(\mathbb{R}^{3}{+})}^{2} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} \, ds \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (s^{-1 - \frac{3}{2}(1 - \frac{1}{2r}) - \frac{1}{2} - \frac{3}{2}(1 - \frac{1}{2r})} + s^{-1 - 3(1 - \frac{1}{4r}) - 1 - \frac{3}{2}(1 - \frac{1}{2r})} \, ds \ & + C f{1}(t) \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (s^{-\frac{3}{2} - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} + s^{-2 - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}) \, ds \ & \leq C t^{-\frac{5}{2} - \frac{3}{2}(1 - \frac{1}{r})} + C{1} t^{-\frac{5}{2} - \frac{3}{2}(1 - \frac{1}{r})} f{1}(t), \end{split} \tag{3.26}$$

where *f*1(t) = sup [s + (1− ∥∇*d*1(s)∥^L (R + ].

⁰<*s≤^t* By (3.23), (3.25) and (3.26), we have

$$t^{\frac{3}{2} + \frac{3}{2}(1 - \frac{1}{r})} |\nabla d1(t)|{L^r(\mathbb{R}^3_+)} \le C + C_1 t^{-1} f_1(t),$$

from which,

$$f_1(t) \le C + C_1 t^{-1} f_1(t).$$

This implies that *f*1(t) is bounded for some t¹ ≤ t, namely,

$$|\nabla \partialt d(t)|{L^r(\mathbb{R}^3_+)} = |\nabla d1(t)|{L^r(\mathbb{R}^3_+)} \le Ct^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}.$$ (3.27)

In view of Lemma 2.5 and Corollary 3.1, we find that

$$|\nabla(u \cdot \nabla d)(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq C(|\nabla u(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla d(t)|{L^{2r}(\mathbb{R}^{3}{+})} + |u(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla^{2} d(t)|{L^{2r}(\mathbb{R}^{3}{+})})$$

$$\leq Ct^{-\frac{5}{2} - \frac{3}{2}(1 - \frac{1}{r})}, \tag{3.28}$$

and

$$\begin{split} |\nabla(|\nabla d|^2 d)(t)|{L^r(\mathbb{R}^3+)} &\leq C(|\nabla d(t)|{L^{2r}(\mathbb{R}^3+)} |\nabla^2 d(t)|{L^{2r}(\mathbb{R}^3+)} + |\nabla d(t)|{L^{3r}(\mathbb{R}^3+)}^3) \ &\leq Ct^{-3-\frac{3}{2}(1-\frac{1}{t})} \end{split} \tag{3.29}$$

hold for any 1 < ^r < ^∞ and ^{t ≥} *^t*1.

Applying the standard elliptic estimates (see [40]) to system (1.1)-(1.2), we can know that for each integer ℓ ^≥ 0,

$$|\nabla^{\ell+2}d(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq C(|\nabla^{\ell}(u \cdot \nabla d)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{\ell}\partial{t}d(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{\ell}(|\nabla d|^{2}d)(t)|{L^{r}(\mathbb{R}^{3}_{+})}). \tag{3.30}$$

Putting (3.27)-(3.30) together, we obtain for 1 < ^r < ^∞ and ^{t ≥} *^t*1,

$$\begin{split} |\nabla^3 d(t)|{L^r(\mathbb{R}^3+)} &\leq C (|\nabla (u \cdot \nabla d)(t)|{L^r(\mathbb{R}^3+)} + |\nabla \partialt d(t)|{L^r(\mathbb{R}^3+)} + |\nabla (|\nabla d|^2 d)(t)|{L^r(\mathbb{R}^3_+)}) \ &\leq C t^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}. \end{split}$$

□

Lemma 3.6. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ). Then there exists t^{2 &}gt; ⁰ such that for ^{1 &}lt; ^{r &}lt; ^∞ and t ^≥ *^t2, the strong solution* (u, ^d, ^p) of (1.1)-(1.2) obtained in Proposition 1.¹ satisfies

$$\begin{split} |\nabla^3 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 p(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}; \ |\nabla^4 d(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-2 - \frac{3}{2}(1 - \frac{1}{r})}. \end{split}$$

Proof. We will divide the proof into three steps.

Step 1: The estimate of ^∇∂tu(t).

Set *^u*1(t) ^{= ∂}tu(t). Taking ^∂t over problem (1.1)1, we find that *^u*1(t) satisfies

$$\begin{cases} \partial_t u_1 - \Delta u_1 + (u_1 \cdot \nabla u + u \cdot \nabla u_1) + \nabla \partial_t p + \nabla \cdot (\nabla d_1 \odot \nabla d) + \nabla \cdot (\nabla d \odot \nabla d1) = 0 & \text{in} \quad \mathbb{R}+^3 \times (1, \infty), \ \nabla \cdot u1 = 0 & \text{in} \quad \mathbb{R}+^3 \times (1, \infty), \ u1(x, t) = 0 & \text{on} \quad \partial \mathbb{R}+^3 \times (1, \infty), \ u_1(x, t) = \partialt u(1) & \text{in} \quad \mathbb{R}+^3. \end{cases}$$

Applying the Duhamel's principle and the nonstationary Stokes solution formula (see [48, 49]), we can know that

$$u{1}(x,t) = \int{\mathbb{R}^{3}{+}} \mathcal{M}(x,y,\frac{t}{2})u{1}(y,\frac{t}{2}) dy$$ $$-\int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \mathcal{M}(x,y,t-s)\mathbb{P}(u{1} \cdot \nabla u + u \cdot \nabla u{1} + \nabla \cdot (\nabla d{1} \odot \nabla d) + \nabla \cdot (\nabla d \odot \nabla d_{1}))(y,s) dyds,$$

$$(3.32)$$

where ^{M =} (Mi j)i, ^j=1,2,³ is defined as follows for 1 ^{≤ i}, ^{j ≤ 3}

$$M{ij}(x, y, t) = \delta{ij}(G_t(x - y) - Gt(x - y^*)) + M{ij}^(x, y, t) = \delta_{ij}Gt(x - y) + N{ij}^(x, y, t).$$

Here

$$M{ij}^*(x, y, t) = 4(1 - \delta{j3}) \frac{\partial}{\partial x_j} \int_0^{x3} \int{\mathbb{R}^2} \frac{\partial E(x - z)}{\partial x_i} G_t(z - y^*) dz,$$

and

$$N{ij}^{*}(x, y, t) = -\delta{ij}G{t}(x - y^{*}) + M{ij}^{*}(x, y, t),$$

E(z) = − 4π|z| is the fundamental solution of the Laplace equation. In addition, M^∗ = (M^∗ i j)3׳ and N^∗ = (N ∗ i j)3׳ still satisfy (3.24) (see [48]), namely,

$$\begin{aligned} &|\partial{t}^{\widetilde{s}} \overline{\partial{x}^{\widetilde{t}}} \partial{y}^{\widetilde{m}} M^{*}(x, y, t)| + |\partial{t}^{\widetilde{s}} \overline{\partial{x}^{\widetilde{t}}} \partial{y}^{\widetilde{m}} \mathcal{N}^{}(x, y, t)| \ \leq &C t^{-s - \frac{\widetilde{m}{3}}{2}} (t + x{3}^{2})^{-\frac{\widetilde{\zeta}_{3}}{2}} (|x - y^{}|^{2} + t)^{-\frac{3 + |\widetilde{t}'| + |\widetilde{m}'|}{2}} e^{-\frac{cy_{3}^{2}}{t}}. \end{aligned}$$ (3.33)

Let 1 < ^r < ^∞. Similarly, using Corollary 3.1, Lemmas 2.4-2.5, Lemmas 3.4-3.5 and Lemma 2.11, with the help of (3.33), we can establish the estimates for any integer ^{m ≥} 1 and ^{t ≥ 2}*t*1,

$$\begin{split} |\nabla{x}^{m} \int{\mathbb{R}{+}^{3}} \mathcal{M}(\cdot, y, \frac{t}{2}) u{1}(y, \frac{t}{2}) dy |{L^{r}(\mathbb{R}{+}^{3})} \ \leq C |\nabla^{m} G{t}|{L^{1}(\mathbb{R}{+}^{3})} |u{1}(\frac{t}{2})|{L^{r}(\mathbb{R}{+}^{3})} + C |u{1}(\frac{t}{2})|{L^{r}(\mathbb{R}{+}^{3})} \times \ \sum{m=m{3}+|m'|} \int{\mathbb{R}{+}^{3}} (x{3} + \sqrt{t})^{-m{3}} (|x'| + x{3} + \sqrt{t})^{-3-|m'|} dx' dx_{3} \ \leq C t^{-\frac{m+2}{2} - \frac{3}{2}(1 - \frac{1}{r})}; \end{split}$$

$$(3.34)$$

and

$$\left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} \mathcal{M}(x, y, t - s) \mathbb{P}(u{1} \cdot \nabla u + u \cdot \nabla u{1} + \nabla \cdot (\nabla d{1} \odot \nabla d) + \nabla \cdot (\nabla d \odot \nabla d{1}))(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})}$$

$$\leq C \int{\frac{t}{2}}^{t} \left| \nabla G{t-s} \right|{L^{1}(\mathbb{R}^{3}{+})} \left| \mathbb{P}(u{1} \cdot \nabla u + u \cdot \nabla u{1} + \nabla \cdot (\nabla d{1} \odot \nabla d) + \nabla \cdot (\nabla d \odot \nabla d{1}))(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds$$

$$+ C \int{\frac{t}{2}}^{t} \left( \sup{x \in \mathbb{R}^{3}{+}} |\nabla{x} \mathcal{N}^{}(x, \cdot, t - s)|{L^{1}(\mathbb{R}^{3}{+})} + \sup{y \in \mathbb{R}^{3}{+}} |\nabla_{x} \mathcal{N}^{}(\cdot, y, t - s)|{L^{1}(\mathbb{R}^{3}{+})} \right)$$

$$\times |\mathbb{P}(u{1} \cdot \nabla u + u \cdot \nabla u{1} + \nabla \cdot (\nabla d{1} \odot \nabla d) + \nabla \cdot (\nabla d \odot \nabla d{1}))(s)|{L^{r}(\mathbb{R}^{3}{+})} ds$$

$$\leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (|u{1}|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla u(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |u(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |\nabla u{1}(s)|{L^{r}(\mathbb{R}^{3}_{+})}$$

$$+ |\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |\nabla^{2} d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2} d(s)|{L^{2r}(\mathbb{R}^{3}{+})} \right) ds$$

$$\leq C \int_{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{3}{2} - \frac{3}{2} - \frac{3}{2} (1 - \frac{1}{r})} ds$$

$$+ C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{3}{2} - \frac{3}{2} (1 - \frac{1}{r})} f{2}(t),$$

$$\leq Ct^{-\frac{5}{2} - \frac{3}{2} (1 - \frac{1}{r})} + Ct^{-\frac{5}{2} - \frac{3}{2} (1 - \frac{1}{r})} f_{2}(t),$$

$$(3.35)$$

where *f*2(t) = sup [s + (1− 1 (∥∇*u*1(s)∥^L (R + ) + ∥∇2*d*1(s)∥^L (R + )].

⁰<*s≤^t* Step 2: The estimate of ∇ 2∂td(t).

Recall that the following results hold (see [15, p.16]): For 1 ^{≤ k ≤} 3 and 0 < ^s < ^t, we have

$$\int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} \mathcal{W}(x, y, t - s) g(y, s) dy$$

$$= \int{\mathbb{R}^{3}{+}} \partial{x{k}} \mathcal{W}(x, y, t - s) \partial{y{3}} g(y, s) dy - 2 \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} G_{t-s}(x - y^{*}) g(y, s) dy \tag{3.36}$$

and

$$\int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} G_{t-s}(x-y) g(y,s) dy$$

$$= \int{\mathbb{R}^{3}{+}} \partial{x{k}} (G{t-s}(x-y) - G{t-s}(x-y^{})) \partial{y{3}} g(y,s) dy - \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} G_{t-s}(x-y^{}) g(y,s) dy. \tag{3.37}$$

Let 1 ^&lt; ^{r &}lt; ^∞ and set ^D(x, ^t) :^{= u1} · ∇^{d + u} · ∇d¹ − |∇d^{| 2}d¹ − 2(∇d : ∇*d*1)d. Using one-dimensional Hardy inequality yields for 0 < θ < 1,

$$\begin{aligned} &|x{3}^{-\theta}D(x,t)|{L^{r}(\mathbb{R}^{3}{+})}^{r} \ &\leq \int{\mathbb{R}^{2}} \int{0}^{1} x{3}^{r-r\theta} \frac{|D(x',x{3})|^{r}}{x{3}^{r}} dx{3} dx' + |D(x,t)|{L^{r}(\mathbb{R}^{2} \times [1,+\infty))}^{r} \ &\leq C(|D(x,t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla{x}D(x,t)|{L^{r}(\mathbb{R}^{3}_{+})})^{r}. \end{aligned} \tag{3.38}$$

Using (3.24), (3.27), (3.36)-(3.38), Lemmas 2.4-2.5, Lemmas 3.4-3.5, Lemma 2.11 and Corollary 3.1, we find for 1 ≤ k ≤ 3, 1 ≤ j ≤ 2 and t ≥ 2t¹ that

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{j}} W(x, y, t - s) (u{1} \cdot \nabla d + u \cdot \nabla d{1} - |\nabla d|^{2} d{1} - 2(\nabla d : \nabla d{1}) d)(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} W(x, y, t - s) (u{1} \cdot \nabla d + u \cdot \nabla d{1} - |\nabla d|^{2} d{1} - 2(\nabla d : \nabla d{1}) d)(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} \left| \partial{x{k}} G{t - s}(\cdot) \right|{L^{1}(\mathbb{R}^{3}{+})} (\left| \partial{3} D(s) \right|{L^{r}(\mathbb{R}^{3}{+})} + \left| \partial{j} D(s) \right|{L^{r}(\mathbb{R}^{3}{+})}) \, ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2} \partial{x{k}} \partial{x{3}} G{t - s}(\cdot) \right|{L^{1}(\mathbb{R}^{3}{+})} \left| y{3}^{-2\varepsilon} D(s) \right|{L^{r}(\mathbb{R}^{3}{+})} \, ds \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (\left| u{1}(s) \right|{L^{2r}(\mathbb{R}^{3}{+})} \left| \nabla^{2} d(s) \right|{L^{2r}(\mathbb{R}^{3}{+})} + \left| \nabla u(s) \right|{L^{2r}(\mathbb{R}^{3}{+})} \left| \nabla d{1}(s) \right|{L^{2r}(\mathbb{R}^{3}{+})} \ & + \left| \nabla d(s) \right|{L^{4r}(\mathbb{R}^{3}{+})}^{2} \left| \nabla d{1}(s) \right|{L^{2r}(\mathbb{R}^{3}{+})} + \left| \nabla d(s) \right|{L^{3r}(\mathbb{R}^{3}{+})} \left| \nabla^{2} d(s) \right|{L^{3r}(\mathbb{R}^{3}{+})} \left| d{1}(s) \right|{L^{3r}(\mathbb{R}^{3}_{+})} \end{split}$$

$$\begin{split} &+|\nabla^{2}d(s)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})})\,ds\ &+C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}}\big(|\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})}|\nabla d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}+(|u(s)|{L^{\infty}(\mathbb{R}^{3}{+})}+|\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})})|\nabla^{2}d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}\big)\,ds\ &+C\int{\frac{t}{2}}^{t}(t-s)^{-1+\varepsilon}\big(|u{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})}+|u(s)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})}\ &+|\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2}|d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})}+|\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})})\,ds\ &\leq C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{2}{3}-\frac{3}{2}(1-\frac{1}{r})}ds+C\int{\frac{t}{2}}^{t}(t-s)^{-1+\varepsilon}s^{-\frac{3}{2}-\frac{3}{2}-\frac{3}{2}(1-\frac{1}{r})}ds\ &+C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}}s^{-\frac{3}{2}}\big(|\nabla u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}+|\nabla^{2}d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}\big)\,ds\ &\leq C{\varepsilon}t^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})}+Ct^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})}f{2}(t), \end{split}$$

where ε ^∈ (0, 2 ), *f*2(t) = sup [s 3 + 3 (1− 1 r ) (∥∇*u*1(s)∥^L (R + ) + ∥∇2*d*1(s)∥^L (R + )].

⁰<*s≤^t* Step 3: The estimates of ∇ ³u(t) and ∇ ⁴d(t).

Using (3.23), (3.25), (3.32), (3.34), (3.35) and (3.39), we conclude that for ^{t ≥ 2}*t*1,

$$t^{\frac{3}{2} + \frac{3}{2}(1 - \frac{1}{r})} (|\nabla u1(t)|{L^r(\mathbb{R}^3_+)} + |\nabla^2 d1(t)|{L^r(\mathbb{R}^3_+)}) \le C + C_2 t^{-1} f_2(t). \tag{3.40}$$

From (3.40), we can know that for some ^{t2 ≤ t},

$$|\nabla \partialt u(t)|{L^r(\mathbb{R}^3_+)} + |\nabla^2 \partialt d(t)|{L^r(\mathbb{R}^3_+)} \le Ct^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}.$$ (3.41)

Substituting (3.41) into (3.39) and running the same argument as in (3.39), combined with (3.25), we derive for t ≥ *t*2,

$$|\nabla^2 \partialt d(t)|{L^r(\mathbb{R}^3_+)} \le Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}.$$ (3.42)

By applying (3.30) with ℓ ⁼ 2, together with (3.42), Lemma 2.5, Lemma 3.4, Lemma 3.5 and Corollary ³.1, we conclude for 1 < ^r < ^∞ and ^{t ≥} *^t*2,

$$\begin{split} |\nabla^4 d(t)|{L^r(\mathbb{R}^3+)} &\leq C (|\nabla^2 (u \cdot \nabla d)(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 \partialt d(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 (|\nabla d|^2 d)(t)|{L^r(\mathbb{R}^3_+)}) \ &\leq C t^{-2-\frac{3}{2}(1-\frac{1}{r})}. \end{split}$$

On the other hand, applying (2.7) to system (1.1)-(1.2), we get for 1 < ^r < ^∞, ℓ ^≥ 0 and ^{t ≥} *^t*2,

$$|\nabla^{\ell+2}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{\ell+1}p(t)|{L^{r}(\mathbb{R}^{3}{+})}$$

$$\leq C(|\nabla^{\ell}(u \cdot \nabla u)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{\ell}\partial{t}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{\ell}(\nabla \cdot (\nabla d \odot \nabla d))(t)|{L^{r}(\mathbb{R}^{3}_{+})}).$$ (3.43)

Using (3.41), (3.43), Lemma 2.5, Lemma 3.4, Lemma 3.5 and Corollary 3.1, we get for 1 < ^r < ^∞ and t ≥ *t*2,

$$\begin{split} &|\nabla^{3}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}p(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &C(|\nabla(u \cdot \nabla u)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla\partial{t}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla(\nabla \cdot (\nabla d \odot \nabla d))(t)|{L^{r}(\mathbb{R}^{3}_{+})}) \ \leq &Ct^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}. \end{split}$$

Thus, we complete the proof of Lemma 3.6. □

Next, we will establish the L 1 -decay rates for the first-order spatial derivatives of velocity u.

Lemma 3.7. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ), if (u, ^d) is the global strong solution obtained by Proposition 1.1, then

$$|\nabla u(t)|{L^1(\mathbb{R}^3+)} \le Ct^{-\frac{1}{2}}$$

holds for any t > ⁰.

Proof. Using Lemma 2.4, Lemma 2.5, Lemma 3.5 and Corollary 3.1, together with (1.3)1, (2.2) and (2.13), we obtain for any ^t > 0 that

$$\begin{split} &|\nabla u(t)|{L^{1}(\mathbb{R}^{3}{+})} \ &\leq Ct^{-\frac{1}{2}}|u{0}|{L^{1}(\mathbb{R}^{3}{+})} + \int{0}^{t}|\nabla e^{-(t-s)\mathbb{A}}\mathbb{P}\left(u\cdot\nabla u + \nabla\cdot(\nabla d\odot\nabla d)\right)(s)|{L^{1}(\mathbb{R}^{3}{+})}\,ds \ &\leq Ct^{-\frac{1}{2}} + C\int{0}^{t}(t-s)^{-\frac{1}{2}}\big(|u\cdot\nabla u + \nabla\cdot(\nabla d\odot\nabla d)|{L^{1}(\mathbb{R}^{3}{+})} \ &+|\sum{i,j=1}^{3}\nabla N\partial{i}\partial{j}(u{i}u{j} + \langle\partial{i}d,\partial{j}d\rangle)|{L^{1}(\mathbb{R}^{3}{+})}\big)\,ds \ &\leq Ct^{-\frac{1}{2}} + C\int{0}^{t}(t-s)^{-\frac{1}{2}}\big(|u|{H^{1}(\mathbb{R}^{3}{+})}^{2} + |\nabla d|{H^{1}(\mathbb{R}^{3}{+})}^{2} \ &+||\nabla d|\nabla^{2}d|{L^{1}(\mathbb{R}^{3}{+})} + |\nabla d|\nabla^{3}d|{L^{1}(\mathbb{R}^{3}{+})}\big)(s)\,ds \ &\leq Ct^{-\frac{1}{2}} + C\int{0}^{t}(t-s)^{-\frac{1}{2}}\big(|u(s)|{H^{1}(\mathbb{R}^{3}{+})}^{2} + |\nabla d(s)|{H^{1}(\mathbb{R}^{3}{+})}^{2} \ &+|\nabla d(s)|{L^{2}(\mathbb{R}^{3}{+})}|\nabla^{2}d(s)|{L^{2}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2}(\mathbb{R}^{3}{+})}|\nabla^{3}d(s)|{L^{2}(\mathbb{R}^{3}{+})}\big)\,ds \ &\leq Ct^{-\frac{1}{2}} + C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\big((1+s)^{-\frac{3}{2}} + (1+s)^{-\frac{5}{2}} + (1+s)^{-3} + (1+s)^{-\frac{7}{2}}\big)\,ds \ &\leq Ct^{-\frac{1}{2}}. \end{split}$$

Finally, we derive the decay rates of the second-order derivatives of u in L r (R 3 + ) with ^{r = 1}, ^∞.

Lemma 3.8. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ), if (u, ^d) is the global strong solution obtained by Proposition 1.1, then

$$|\nabla^2 u(t)|{L^1(\mathbb{R}^3+)} \le Ct^{-1}$$

holds for any t > ⁰.

Proof. It follows from Lemmas 2.3-2.5, Lemma 3.5, Corollary 3.1 and (2.13) that for any ^t > 0,

$$\begin{split} & \Big| \int0^{\frac{t}{2}} \nabla^2 e^{-(t-s)\mathbb{A}} \mathbb{P} \Big( u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) \Big) ds \Big|{L^1(\mathbb{R}^3_+)} \ \leq & C \int0^{\frac{t}{2}} (t-s)^{-1} \big( | (u \cdot \nabla u)(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) |{L^1(\mathbb{R}^3+)} \big) ds \ & + \Big| \sum{i,j=1}^3 \nabla \mathcal{N} \partial_i \partial_j (u_i u_j + \langle \partial_i d, \partialj d \rangle)(s) \Big|{L^1(\mathbb{R}^3_+)} \big) ds \ \leq & C t^{-1} \int0^{\frac{t}{2}} \big( | u(s) |{H^1(\mathbb{R}^3+)}^2 + | \nabla d(s) |{H^1(\mathbb{R}^3+)}^2 \big) \ & + | | \nabla d(t) | \nabla^3 d(s) |{L^1(\mathbb{R}^3+)} + | | \nabla d(t) | \nabla^2 d(s) |{L^1(\mathbb{R}^3_+)} \big) ds \ \leq & C t^{-1} \int_0^{\frac{t}{2}} (1+s)^{-\frac{3}{2}} ds \ \leq & C t^{-1}. \end{split}$$

In the same way, using Lemmas 2.3-2.5, Lemma 2.9, Lemma 2.12, Lemmas 3.5-3.6 and (2.12), we derive for any 0 < η < 1 and ^t > 0,

$$\Big| \int{\frac{t}{2}}^t \nabla^2 e^{-(t-s)\mathbb{A}} \mathbb{P}\Big(u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) \Big) \, ds \Big|{L^1(\mathbb{R}^3_+)}$$

□

$$\begin{split} &\leq C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}}(\left|\nabla\left(u\cdot\nabla u+\nabla\cdot(\nabla d\odot\nabla d)\right)(s)\right|{L^{1}(\mathbb{R}^{3}{+})}\ &+\left|\nabla\left(\sum{i,j=1}^{3}\nabla\mathcal{N}\partial{i}\partial{j}(u{i}u{j}+\langle\partial{i}d,\partial{j}d\rangle)\right)(s)\right|{L^{1}(\mathbb{R}^{3}{+})})\,ds\ &+C\int{\frac{t}{2}}^{t}(t-s)^{-1+\frac{\eta}{2}}\left(\left|x{3}^{-\eta}\mathbb{P}u(s)\cdot\nabla u(s)\right|{L^{1}(\mathbb{R}^{3}{+})}+\left|x{3}^{-\eta}\mathbb{P}\nabla\cdot(\nabla d\odot\nabla d)(s)\right|{L^{1}(\mathbb{R}^{3}{+})}\right)ds\ &\leq C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}}\left(\left|u\right|{H^{1}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla^{2}u\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla^{2}d\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}\ &+\left|\nabla d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}\left|\nabla^{4}d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}+\left|\nabla^{2}d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}\left|\nabla^{3}d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}\ &+\left|\nabla d\right|{L^{2}(\mathbb{R}^{3}{+})}\left|\nabla^{2}d\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla^{2}u\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla d\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla^{2}d\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}\ &+\left|\nabla d\right|{L^{2}(\mathbb{R}^{3}{+})}\left|\nabla^{2}d\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}\right|\nabla^{3}d(t)\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}+\left|\nabla^{2}d\right|{L^{2}(\mathbb{R}^{3}{+})}^{2}\ &\leq C\int_{\frac{t}{2}}^{t}\left((t-s)^{-\frac{1}{2}}+(t-s)^{-1+\frac{\eta}{2}}\right)s^{-\frac{3}{2}}\,ds\ &\leq Ct^{-1}. \end{split}$$

Using these two estimates, together with (1.3)¹ and Lemma 2.3, we complete the proof of Lemma 3.8. □

Lemma 3.9. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ), if (u, ^d) is the global strong solution obtained by Proposition 1.1, then

$$|\nabla^2 u(t)|{L^\infty(\mathbb{R}^3+)} \leq C t^{-\frac{1}{2}}$$

holds for any t > ⁰.

Proof. Let 0 < θ < 1 and ³ θ < ^r < ^∞. It follows from Lemma 2.2, Lemmas 2.4-2.5, Lemmas 2.9-2.10, Lemma 2.12, Lemmas 3.5-3.6 and Corollary 3.1 that for any ^t > 0,

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \nabla^{2} e^{-(t-s)\mathbb{A}} \mathbb{P} \Big( u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) \Big) ds \right|{L^{\infty}(\mathbb{R}^{3}{+})} \ \leq & C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} \left| \nabla \mathbb{P} \Big( (u \cdot \nabla u)(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) \Big) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} (t-s)^{-1 + \frac{\theta}{2} - \frac{3}{2r}} \left| x{3}^{-\theta} \mathbb{P} \Big( (u \cdot \nabla u)(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) \Big) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ \leq & C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} \Big( \left| \nabla \Big( (u \cdot \nabla u)(s) + \nabla \cdot (\nabla d \odot \nabla d)(s) \Big) \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \left| \nabla \Big( \sum{i,j=1}^{3} \nabla \mathcal{N} \partial{i} \partial{j} (u{i}u{j} + \langle \partial{i}d, \partial{j}d \rangle) \Big)(s) \right|{L^{r}(\mathbb{R}^{3}{+})} \Big) ds \ & + C \int{\frac{t}{2}}^{t} (t-s)^{-1 + \frac{\theta}{2} - \frac{3}{2r}} \Big( \left| x{3}^{-\theta} \mathbb{P} (u \cdot \nabla u)(s) \right|{L^{r}(\mathbb{R}^{3}{+})} + \left| x{3}^{-\theta} \mathbb{P} \nabla \cdot (\nabla d \odot \nabla d)(s) \right|{L^{r}(\mathbb{R}^{3}{+})} \Big) ds \ \leq & C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} \Big( \left| u \right|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + \left| \nabla^{2} d \right|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + \left| \nabla u \right|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + \left| \nabla^{2} u \right|{L^{2r}(\mathbb{R}^{3}{+})}^{2} \Big) ds \ + \left| \nabla d(t) \right|{L^{2r}(\mathbb{R}^{3}{+})} | \nabla^{4} d(t) \right|{L^{2r}(\mathbb{R}^{3}{+})} + \left| \nabla^{2} d(t) \right|{L^{2r}(\mathbb{R}^{3}{+})} | \nabla^{3} d(t) \right|{L^{2r}(\mathbb{R}^{3}{+})} ds \end{split}$$

$$\begin{split} &+C\int{\frac{t}{2}}^{t}(t-s)^{-1+\frac{\theta}{2}-\frac{3}{2r}}(|u(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2}+|\nabla u(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2}+|\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2}+|\nabla^{2}u|{L^{2r}(\mathbb{R}^{3}{+})}^{2}+|\nabla^{2}u|{L^{2r}(\mathbb{R}^{3}{+})}^{2}+|\nabla^{2}u|{L^{2r}(\mathbb{R}^{3}{+})}^{2})\,ds\ &+|\nabla^{2}d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2}+|\nabla^{3}d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2})\,ds\ &\leq C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}-\frac{3}{2r}}s^{-3(1-\frac{1}{2r})}\,ds+C\int_{\frac{t}{2}}^{t}(t-s)^{-1+\frac{\theta}{2}-\frac{3}{2r}}s^{-3(1-\frac{1}{2r})}\,ds\ &\leq Ct^{-\frac{5}{2}}.\end{split}$$

Using the above estimate, combined with (1.4)¹ and (2.1), the proof of Lemma 3.9 is complete. □

Proof of Theorem 1.1. Theorem 1.1 directly follows from Corollary 3.1, Lemma 3.4 and Lemmas 3.7- ³.9. □

4. Proof of ^Theorem 1.²

This section is devoted to establishing the decay rates of ∥(∇ ²+ku, ^{∇ 2}+kd)(t)∥^L (R 3 + , where k ≥ 1, ¹ < ^r ≤ ∞, and (u, ^d) is the strong solution obtained by Proposition 1.1. Firstly, we provide a review of the decay estimates of the strong solution that have been derived in Lemma 2.5, Lemmas 3.5-3.6 and Theorem 1.1, as they are frequently utilized in the proofs of subsequent theorems.

Lemma 4.1. *Assume u*0, *^d*0−e^{3 ∈ L} 1 (R 3 + ), if (u, ^d, ^p) is the global strong solution obtained by Proposition 1.1, then the following estimates

$$\begin{cases} |(\nabla^m u, \nabla^m d)(t)|{L^p(\mathbb{R}^3+)} \leq Ct^{-\frac{m}{2} - \frac{3}{2}(1 - \frac{1}{p})}, \ m = 0, 1, 2, \ |\nabla^l d(t)|{L^q(\mathbb{R}^3+)} \leq Ct^{-\frac{l}{2} - \frac{3}{2}(1 - \frac{1}{q})}, \ l = 3, 4, \ |\nabla^3 u(t)|{L^q(\mathbb{R}^3+)} \leq Ct^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{q})}, \ |\partialt u(t)|{L^q(\mathbb{R}^3+)} + |\nabla p(t)|{L^q(\mathbb{R}^3_+)} + |\partialt d(t)|{L^q(\mathbb{R}^3_+)} \leq Ct^{-1 - \frac{3}{2}(1 - \frac{1}{q})}, \end{cases}$$

hold for any t > ⁰, where p ^∈ (1, ^∞] and q ^∈ (1, ^∞).

Lemma 4.2. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ). Then there exists t^{3 &}gt; ⁰ such that for ^{1 &}lt; ^{r &}lt; ^∞ and t ^≥ *^t3, the strong solution* (u, ^d, ^p) of (1.1)-(1.2) obtained in Proposition 1.¹ satisfies

$$\begin{split} |\nabla^4 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^3 p(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-2-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^5 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^4 p(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^5 d(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^6 d(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-3-\frac{3}{2}(1-\frac{1}{r})}. \end{split}$$

Proof. We will divide the proof into six steps.

Step1:. Using the regular estimate (3.30) with ℓ ⁼ 3, we can obtain that for ^t > 1,

$$|\nabla^5 d(t)|{L^r(\mathbb{R}^3+)} \leq C (|\nabla^3 (u \cdot \nabla d)(t)|{L^r(\mathbb{R}^3+)} + |\nabla^3 \partialt d(t)|{L^r(\mathbb{R}^3+)} + |\nabla^3 (|\nabla d|^2 d)(t)|{L^r(\mathbb{R}^3_+)})$$

So in order to get the decay of ∥∇5d(t)∥^L (R + , we should gain the decay of ∥∇3∂td(t)∥^L (R + at the first time. Applying the elliptic estimates to ^d1 system (3.22), we get for ^{t &}gt; 1,

$$\begin{split} |\nabla^{3}d{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} &\leq C(|\nabla(u{1}\cdot\nabla d+u\cdot\nabla d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} \ &+ |\nabla(|\nabla d|^{2}d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla((\nabla d:\nabla d{1})d)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla\partial{t}d{1}(t)|{L^{r}(\mathbb{R}^{3}_{+})}). \end{split} \tag{4.1}$$

So it is sufficient to establish the decay of ∥∇∂*td*1(t)∥^L (R ) in (4.1).

• Next we try to obtain the decay of ∥∇6d(t)∥^L (R 3 + . Similarly, using the elliptic estimates, we find for ^t > 1,

$$|\nabla^{6}d(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq C(|\nabla^{4}(u \cdot \nabla d)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{4}\partial{t}d(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{4}(|\nabla d|^{2}d)(t)|{L^{r}(\mathbb{R}^{3}{+})}). \tag{4.2}$$ Applying the elliptic estimates to $d{1}$ system again, we get for $t > 1$ ,

$$|\nabla^4 d1(t)|{L^r(\mathbb{R}^3_+)} \leq C (|\nabla^2 (u_1 \cdot \nabla d + u \cdot \nabla d1)(t)|{L^r(\mathbb{R}^3_+)}$$

$$+ |\nabla^{2}(|\nabla d|^{2}d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}((\nabla d:\nabla d{1})d)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}\partial{t}d{1}(t)|{L^{r}(\mathbb{R}^{3}{+})}). \tag{4.3}$$

In order to get the estimate of ∥∇6d(t)∥^L (R + , it is sufficient to establish the decay of ∥∇2∂*td*1(t)∥^L (R + in terms of (4.2) and (4.3).

Step2: The estimate of ∥∇2*u1∥^L* (R 3 .

Using the regularity estimate (3.43) with ℓ ⁼ 2, we get for ^t > 1,

$$\begin{split} &|\nabla^4 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^3 p(t)|{L^r(\mathbb{R}^3+)} \ \leq &C(|\nabla^2 (u\cdot \nabla u)(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 \partialt u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 (\nabla\cdot (\nabla d\odot \nabla d))(t)|{L^r(\mathbb{R}^3_+)}). \end{split}$$

So in order to get the estimate of ∥∇4u(t)∥^L (R , it is key to establish the decay of ∥∇2∂tu(t)∥^L (R 3 .

• + Let 1 ^&lt; ^{r &}lt; ^∞ and set ^U(x, ^t) :^{= u1} · ∇^{u + u} · ∇u^{1 +} ∇ · (∇d¹ ⊙ ∇d) ⁺ ∇ · (∇^d ⊙ ∇*d*1), *^d*2(t) ^{= ∂}ttd(t), *^u*2(t) ⁼ ∂ttu(t). Using (3.27), (3.33), (3.37), (3.41), (3.42), Lemmas 2.9-2.12 and Lemma 4.1, we find for 1 ^{≤ k ≤} 3, 1 ^{≤ j ≤} 2, 0 < ε < ¹ 2 and t ≥ 2*t*2,

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{j}} \mathcal{M}(x, y, t - s) \mathbb{P}U(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} \mathcal{M}(x, y, t - s) \mathbb{P}U(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} \left| \partial{x{k}} G{t - s}(\cdot) \right|{L^{1}(\mathbb{R}^{3}{+})} (\left| \partial{3} \mathbb{P}U(s) \right|{L^{r}(\mathbb{R}^{3}{+})} + \left| \partial{j} \mathbb{P}U(s) \right|{L^{r}(\mathbb{R}^{3}{+})} \right) ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} G{t - s}(\cdot) \right|{L^{1}(\mathbb{R}^{3}{+})} \left| y{3}^{2s} \mathbb{P}U(s) \right|{L^{r}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} G{t - s}(\cdot) \right|{L^{1}(\mathbb{R}^{3}{+})} \left| y{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} \right| ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} \mathcal{N}^{s}(x, y, t - s) \right|{L^{1}(\mathbb{R}^{3}{+})} ds \ & + C \int{\frac{t}{2}}^{t} \left( t - s \right)^{-1+\varepsilon} \left( | u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} \ & + | \nabla d{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla d(s) |{L^{2s}(\mathbb{R}^{3}{+})} + | \nabla^{2} d{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla^{2} d{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} ds \ & + \int{\frac{t}{2}}^{t} \left( t - s \right)^{-1+\varepsilon} \left( | u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} \ & + | \nabla^{2} d{1}(s) |{L^{2s}(\mathbb{R}^{3}{+})} | \nabla^{2} d{1}(s) |{L^{2s}(\mathbb{R}^{3}_{+})} |$$

$$+ (|\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})} + |\nabla^{2}d(s)|{L^{\infty}(\mathbb{R}^{3}{+})})|\nabla^{3}d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})}|\nabla^{4}d{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}) ds$$

$$\leq C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2}} s^{-3-\frac{3}{2}(1-\frac{1}{r})} ds + C \int{\frac{t}{2}}^{t} (t-s)^{-1+\varepsilon} s^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})} ds$$

$$+ C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2}} s^{-\frac{3}{2}} (|\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})}) ds,$$

$$(4.4)$$

where we have used the estimates (4.1) and (4.3) in bounding the final inequality.

Step3: The estimate of ∥∇∂*td1∥^L* (R + .

From (3.22), we find for ^t > 1,

$$\begin{cases} \partial_t d_2 + u_2 \cdot \nabla d + 2u_1 \cdot \nabla d_1 + u \cdot \nabla d_2 - \Delta d_2 - |\nabla d|^2 d_2 \ = 2|\nabla d_1|^2 d + 2(\nabla d: \nabla d_2)d + 4(\nabla d: \nabla d_1)d1 & \text{in } \mathbb{R}^3+ \times (1, \infty), \ \frac{\partial d_2}{\partial x3} = 0 & \text{on } \partial \mathbb{R}^3+ \times (1, \infty), \ d2(x, 1) = \partial{tt} d(1) & \text{in } \mathbb{R}^3_+. \end{cases}$$

Furthermore, it holds for ^t > 2 that

$$d{2}(x,t) = \int{\mathbb{R}^{3}{+}} \mathcal{W}(x,y,\frac{t}{2}) d{2}(y,\frac{t}{2}) dy - \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \mathcal{W}(x,y,t-s) (u{2} \cdot \nabla d + 2u{1} \cdot \nabla d{1} + u \cdot \nabla d{2} - |\nabla d|^{2} d{2} - 2|\nabla d{1}|^{2} d{2} - 2(\nabla d : \nabla d{2})d - 4(\nabla d : \nabla d{1})d_{1})(y,s) dy ds.$$

$$(4.5)$$

From (3.22), (3.27), (3.42) and Lemma 4.1, we find for ^{t ≥ 2}*t*2,

$$\begin{aligned} &|d{2}(t)|{L^{r}(\mathbb{R}^{3}{+})} = |\partial{t}d{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &|(u{1}\cdot\nabla d + u\cdot\nabla d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}d{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} + |(|\nabla d|^{2}d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + 2|(\nabla d:\nabla d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}. \end{aligned} \tag{4.6}$$

Let 1 < ^r < ^∞. Using (3.24) and (4.6) yields for each integer ^{m ≥} 1 and ^{t ≥ 2}*t*2,

$$\begin{split} |\nabla{x}^{m} \int{\mathbb{R}^{3}{+}} W(\cdot, y, \frac{t}{2}) d{2}(y, \frac{t}{2}) dy |{L^{r}(\mathbb{R}^{3}{+})} \ \leq C |\nabla^{m} G{t}|{L^{1}(\mathbb{R}^{3}{+})} |d{2}(\frac{t}{2})|{L^{r}(\mathbb{R}^{3}{+})} + C |d{2}(\frac{t}{2})|{L^{r}(\mathbb{R}^{3}{+})} \ \times \sum{m=m{3}+|m'|} \int{\mathbb{R}^{3}{+}} (x{3} + \sqrt{t})^{-m{3}} (|x'| + x{3} + \sqrt{t})^{-3-|m'|} dx' dx_{3} \ \leq C t^{-2-\frac{m}{2} - \frac{3}{2}(1-\frac{1}{r})}. \end{split} \tag{4.7}$$

Set *^D*1(x, ^t) :^{= u2} · ∇^{d + 2}u¹ · ∇d^{1 + u} · ∇d² − |∇d^{| 2}d² − 2|∇*d*1| ²d − 2(∇d : ∇*d*2)d − 4(∇d : ∇*d*1)*d*1. Using (3.31), we obtain for ^{t ≥ 2}*t*2,

$$\begin{split} &|\partial{tt}u(t)|{L^{r}(\mathbb{R}^{3}{+})} = |\partial{t}u{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &|\mathbb{P}(u{1}\cdot\nabla u + u\cdot\nabla u{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\mathbb{A}u{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\mathbb{P}(\nabla\cdot(\nabla d{1}\odot\nabla d) + \nabla\cdot(\nabla d\odot\nabla d{1}))(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &|\nabla^{2}u{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} + |u{1}(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla u(t)|{L^{2r}(\mathbb{R}^{3}{+})} + |u(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla u(t)|{L^{2r}(\mathbb{R}^{3}{+})} \ &+ |\nabla^{2}d{1}(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla d(t)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d{1}(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla^{2}d(t)|{L^{2r}(\mathbb{R}^{3}{+})}. \end{split} \tag{4.8}$$

And then from (4.8), combined with Lemma 4.1 and Lemma 2.11, we obtain for ^{t ≥ 2}*t*2,

$$\begin{split} & \Big| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} W(x, y, t - s) D{1}(y, s) \, dy ds \Big|{L^{r}(\mathbb{R}^{3}{+})} \ \leq & C \int{\frac{t}{2}}^{t} | \nabla G{t-s}(\cdot) |{L^{1}(\mathbb{R}^{3}{+})} | D{1}(s) |{L^{r}(\mathbb{R}^{3}{+})} \, ds \ \leq & C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (| \nabla d(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | u{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} \ & + | u(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | \nabla d{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} + | \nabla d(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | \nabla d{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} ) \, ds \end{split}$$

$$+ C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2}} (|u{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{3r}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}_{+$$

Step4: The estimate of ∥∇2∂*td1∥^L* (R ) .

• Using Lemma 2.11, (3.24), (3.27), (3.36)-(3.38), (3.41), (3.42) and Lemma 4.1, we find for 1 ^{≤ k ≤} 3, ^{1 ≤ j ≤} 2, 0 < ε < ¹ 2 and t ≥ 2*t*2,

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{t}} \partial{x{j}} W(x, y, t - s) D{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{t}} \partial{x{s}} \partial{x{s}} W(x, y, t - s) D{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} \left| \partial{x{k}} G{t-s} \right|{L^{1}(\mathbb{R}^{3}{+})} (|\partial{3} D{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\partial{j} D{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}) \, ds \ & + C \int{\frac{t}{2}}^{t} \left| x{3}^{2s} \partial{x{k}} \partial{x{3}} G{t-s}(\cdot) \right|{L^{1}(\mathbb{R}^{3}{+})} |z{3}^{-2s} D{1}(\cdot, s)|{L^{r}(\mathbb{R}^{3}{+})} \, ds \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-1+\varepsilon} (|\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |u{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |u{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} \ & + |u(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} \ & + |\nabla d(s)|{L^{\infty}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} \ & + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2} d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} \ & + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2} d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} \ & + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2} d{2}(s)|{L^{2r}(\mathbb{R}^{3}{+})} \ & + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2} d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} \ & + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^{2} d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |d{1}(s)|{L^{3r}(\mathbb{R}^{3}{+})} \ & + |\nabla d{1}(s)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla d{2}(s)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d{1}(s)|{L^{2r}(\mathbb{$$

where we have applied the estimate (4.8) to bound the final inequality.

Step5: The estimate of ∥∇∂*tu1∥^L* (R + .

Now we try to get the estimate of ∥∇5u(t)∥^L (R 3 + ) . Using the regularity estimate (3.43) with ℓ ⁼ 3, we get for ^t > 1,

$$\begin{split} &|\nabla^{5}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{4}p(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &C(|\nabla^{3}(u\cdot\nabla u)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{3}\partial{t}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{3}(\nabla\cdot(\nabla d\odot\nabla d))(t)|{L^{r}(\mathbb{R}^{3}_{+})}). \end{split}$$

Applying the regularity estimate (2.7) to problem (3.31), we obtain for ^t > 1,

$$|\nabla^{3}u{1}(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq C(|\nabla(u{1}\cdot\nabla u + u\cdot\nabla u{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla(\nabla\cdot(\nabla d{1}\odot\nabla d))(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla(\nabla\cdot(\nabla d{1}\odot\nabla d{1})(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla\partial{t}u{1}(t)|{L^{r}(\mathbb{R}^{3}{+})}).$$

$$(4.11)$$

So it is sufficient to establish the decay of ∥∇∂*tu*1(t)∥^L (R + ) in (4.11). From (3.31), we find for ^t > 1,

$$\begin{cases} \partial_t u_2 - \Delta u_2 + u_2 \cdot \nabla u + 2u_1 \cdot \nabla u_1 + u \cdot \nabla u_2 + \nabla \partial_t p \ + \nabla \cdot (\nabla d_2 \odot \nabla d) + \nabla \cdot (\nabla d \odot \nabla d_2) + 2\nabla \cdot (\nabla d_1 \odot \nabla d1) = 0 & \text{in} \quad \mathbb{R}^3+ \times (1, \infty), \ \nabla \cdot u2 = 0 & \text{in} \quad \mathbb{R}^3+ \times (1, \infty), \ u2(x, t) = 0 & \text{on} \quad \partial \mathbb{R}^3+ \times (1, \infty), \ u2(x, t) = \partial{tt} u(1) & \text{in} \quad \mathbb{R}^3_+. \end{cases}$$

Set *^U*1(x, ^t) :^{= u2} · ∇^{u + 2}u¹ · ∇u^{1 + u} · ∇u^{2 +} ∇ · (∇^d ⊙ ∇*d*2) ⁺ ∇ · (∇d² ⊙ ∇d) ^{+ 2}∇ · (∇d¹ ⊙ ∇*d*1), then it holds for ^t > 2,

$$u2(x,t) = \int{\mathbb{R}^3_+} \mathcal{M}(x,y,\frac{t}{2}) u2(y,\frac{t}{2}) dy - \int{\frac{t}{2}}^t \int{\mathbb{R}^3+} \mathcal{M}(x,y,t-s) \mathbb{P} U_1(y,s) \, dy ds. \tag{4.12}$$

Let 1 < ^r < ^∞. Using (3.27), (3.33), (3.37), (3.41), (3.42), Lemma 4.1 and Lemma 2.11, we find for each integer m ≥ 1 and t ≥ 2*t*2,

$$|\nabla{x}^{m} \int{\mathbb{R}{+}^{3}} \mathcal{M}(\cdot, y, \frac{t}{2}) u{2}(y, \frac{t}{2}) dy|{L^{r}(\mathbb{R}{+}^{3})}$$

$$\leq C |\nabla^{m} G{t}|{L^{1}(\mathbb{R}{+}^{3})} |u{2}(\frac{t}{2})|{L^{r}(\mathbb{R}{+}^{3})} + C |u{2}(\frac{t}{2})|{L^{r}(\mathbb{R}_{+}^{3})} \times$$

$$\sum{m=m{3}+|m'|} \int{\mathbb{R}{+}^{3}} (x{3} + \sqrt{t})^{-m{3}} (|x'| + x{3} + \sqrt{t})^{-3-|m'|} dx' dx{3}$$

$$\leq C t^{-\frac{m}{2}} |u{2}(t)|{L^{r}(\mathbb{R}_{+}^{3})}; \tag{4.13}$$

and

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} \mathcal{M}(x, y, t - s) \mathbb{P} U{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ \leq & C \int{\frac{t}{2}}^{t} | \nabla G{t-s} |{L^{1}(\mathbb{R}^{3}{+})} | \mathbb{P} U{1}(s) |{L^{r}(\mathbb{R}^{3}{+})} \, ds \ & + C \int{\frac{t}{2}}^{t} \left( \sup{x \in \mathbb{R}^{3}{+}} | \nabla{x} \mathcal{N}^{}(x, \cdot, t - s) |{L^{1}(\mathbb{R}^{3}{+})} + \sup{y \in \mathbb{R}^{3}{+}} | \nabla_{x} \mathcal{N}^{}(\cdot, y, t - s) |{L^{1}(\mathbb{R}^{3}{+})} \right) \times | \mathbb{P} U{1}(s) |{L^{r}(\mathbb{R}^{3}{+})} \, ds \ \leq & C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (| u{1}(s) |{L^{2r}(\mathbb{R}^{3}{+})} | \nabla u{1}(s) |{L^{2r}(\mathbb{R}^{3}{+})} + | \nabla^{2} d{1}(s) |{L^{2r}(\mathbb{R}^{3}{+})} | \nabla d{1}(s) |{L^{2r}(\mathbb{R}^{3}{+})}) \, ds \ & + \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} (| \nabla u(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | u{2} |{L^{r}(\mathbb{R}^{3}{+})} + | u(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | \nabla u{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} \ & + | \nabla d(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | \nabla^{2} d{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} + | \nabla^{2} d(s) |{L^{\infty}(\mathbb{R}^{3}{+})} | \nabla d{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} \, ds \ \leq & C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-4 - \frac{3}{2}(1 - \frac{1}{r})} \, ds + C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{3}{2}} (| \nabla d{2}(s) |{L^{r}(\mathbb{R}^{3}{+})} \, ds, \end{cases} \tag{4.14} \end{split}$$

where we have used the estimate (4.8) in bounding the final inequality.

Step6: The estimates of ∥∇4u(t)∥^L (R + ) , ∥∇5u(t)∥^L (R + , ∥∇5d(t)∥^L (R + and ∥∇6d(t)∥^L (R + .

Set *f*3(t) = sup ⁰<*s≤^t* [s 2+ (1− 1 ) (∥∇*d*2(s)∥^L (R + ) + ∥∇*u*2(s)∥^L (R + ) + ∥∇2*u*1(s)∥^L (R + ) + ∥∇2*d*2(s)∥^L (R + )]. By (4.8), (4.13) and Lemma 4.1, we get for ^{t ≥ 2}*t*2,

$$|\nabla{x} \int{\mathbb{R}^{3}{+}} \mathcal{M}(\cdot, y, \frac{t}{2}) u{2}(y, \frac{t}{2}) dy|{L^{r}(\mathbb{R}^{3}{+})}$$

$$\leq C t^{-\frac{1}{2}} |u{2}(t)|{L^{r}(\mathbb{R}^{3}_{+})}$$

$$\leq C t^{-\frac{1}{2}} (|\nabla^{2} u{1}(t)|{L^{r}(\mathbb{R}^{3}_{+})} + t^{-3 - \frac{3}{2}(1 - \frac{1}{r})}). \tag{4.15}$$

Using (4.4), (4.5), (4.7), (4.9), (4.10), (4.12)-(4.15), (3.32), (3.34), we conclude for ^{t ≥ 2}*t*2,

$$(1.16)^{2^{1+\frac{3}{2}(1-\frac{1}{r})}} (|\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla u{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})}) \leq C + C{3}t^{-1}f{3}(t).$$ (4.16)

Note that there exists t³ ≥ 2t² such that *C*3t −1 3 ≤ 2 . Whence (4.16) yields for ^{t ≥ t3} that *^f*3(t) ^{≤ 2}C, which implies for 1 < ^r < ^∞ and ^{t ≥} *^t*3,

$$|\nabla d2(s)|{L^r(\mathbb{R}^3_+)} + |\nabla u2(s)|{L^r(\mathbb{R}^3_+)} + |\nabla^2 u1(s)|{L^r(\mathbb{R}^3_+)} + |\nabla^2 d2(s)|{L^r(\mathbb{R}^3_+)} \le Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}. \tag{4.17}$$

Using (4.17) and running the same argument as in Step1-6, we obtain for 1 < ^r < ^∞ and ^{t ≥} *^t*3,

$$\begin{split} |\nabla^4 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^3 p(t)|{L^r(\mathbb{R}^3+)} &\leq Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^5 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^4 p(t)|{L^r(\mathbb{R}^3+)} &\leq Ct^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^5 d(t)|{L^r(\mathbb{R}^3+)} &\leq Ct^{-\frac{5}{2}-\frac{3}{2}(1-\frac{1}{r})}, \ |\nabla^6 d(t)|{L^r(\mathbb{R}^3+)} &\leq Ct^{-3-\frac{3}{2}(1-\frac{1}{r})}. \end{split}$$

Lemma 4.3. *Assume u*0, ^{d0 − e3 ∈ L} 1 (R 3 + ). Then for every k ^{≥ 1}, there exists some number t(k) > ⁰ such that for t ^{≥ t}(k) and ¹ < ^r ≤ ∞, the strong solution (u, ^d, ^p) of (1.1)-(1.2) obtained in Proposition 1.¹ satisfies

$$|\nabla^{2+k}u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{1+k}p(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{2+k}d(t)|{L^r(\mathbb{R}^3+)} \leq Ct^{-\frac{2+k}{2}-\frac{3}{2}(1-\frac{1}{r})}.$$

Proof. Repeating the proofs of Lemma 3.6 and Lemma 4.2, it can be easily shown that for every integer ^{k ≥} 1, there exists ^t(k) > 0 independent of ^t, such that for 1 < ^r < ^∞ and ^{t ≥ t}(k),

$$|\nabla^{2+k}u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{1+k}p(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{2+k}d(t)|{L^r(\mathbb{R}^3+)} \leq Ct^{-\frac{2+k}{2}-\frac{3}{2}(1-\frac{1}{r})}$$

By virtue of (2.6), we can obtain that for every ^{k ≥} 1 and ^{t ≥ t}(k),

$$\begin{split} &|(\nabla^{2+k}u,\nabla^{2+k}d)(t)|{L^{\infty}(\mathbb{R}^{3}{+})} + |\nabla^{1+k}p(t)|{L^{\infty}(\mathbb{R}^{3}{+})} \ \leq &C|\nabla^{2+k}u(t)|{L^{6}(\mathbb{R}^{3}{+})}^{\frac{1}{2}}|\nabla^{3+k}u(t)|{L^{6}(\mathbb{R}^{3}{+})}^{\frac{1}{2}} \ &+ C|\nabla^{2+k}d(t)|{L^{6}(\mathbb{R}^{3}{+})}^{\frac{1}{2}}|\nabla^{3+k}d(t)|{L^{6}(\mathbb{R}^{3}{+})}^{\frac{1}{2}} \ &+ C|\nabla^{1+k}p(t)|{L^{6}(\mathbb{R}^{3}{+})}^{\frac{1}{2}}|\nabla^{2+k}p(t)|{L^{6}(\mathbb{R}^{3}{+})}^{\frac{1}{2}} \ \leq &Ct^{-\frac{2+k}{2}-\frac{3}{2}}. \end{split}$$

Hence, we finish the proof of Theorem 1.2. □

5. Proof of ^Theorem 1.³

In this section, we shall prove the decay rates of the solution stated in Proposition 1.1 under the additional assumption that the initial datum u⁰ satisfies ∥*x*3*u0∥^L* (R + ) ^&lt; ^∞.

Lemma 5.1. Under the assumptions of Theorem 1.3, if (u, ^d) is the global strong solution obtained by Proposition 1.1, then

$$\begin{split} |\nabla u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} &\leq Ct^{-\frac{5}{2}}, \ |\nabla u(t)|{L^{p}(\mathbb{R}^{3}{+})} &\leq Ct^{-1-\frac{3}{2}(1-\frac{1}{p})} \end{split}$$

hold for any t > ⁰, p ^∈ (1, ^∞).

Proof. Using (1.4)1, Lemma 2.1, Lemma 2.4, Lemma 4.1 and the second part of Lemma 2.5, one has for any ^t > 0,

$$\begin{split} &|\nabla u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} \ \leq &Ct^{-\frac{5}{4}}|u(\frac{t}{2})|{L^{2}(\mathbb{R}^{3}{+})} + C\int{\frac{t}{2}}^{t}(t-s)^{-\frac{1}{2}-\frac{3}{2r}}|(u\cdot\nabla u+\nabla\cdot(\nabla d\odot\nabla d))(s)|{L^{r}(\mathbb{R}^{3}_{+})}\,ds \end{split}$$

□

$$\leq Ct^{-\frac{5}{2}} + C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} \Big( |u(s)|{L^{r1}(\mathbb{R}^3+)} |\nabla u(s)|_{L^{r2}(\mathbb{R}^3+)} + |\nabla d(s)|_{L^{r1}(\mathbb{R}^3+)} |\nabla^2 d(s)|_{L^{r2}(\mathbb{R}^3+)} \Big) ds \ \leq Ct^{-\frac{5}{2}} + C \int_{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2} - \frac{3}{2r}} \Big( s^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{r_1})} s^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{r_2})} + s^{-\frac{1}{2} - \frac{3}{2}(1 - \frac{1}{r_1})} s^{-1 - \frac{3}{2}(1 - \frac{1}{r_2})} \Big) ds \ \leq Ct^{-\frac{5}{2}}, \tag{5.1}$$

where ¹ r = 1 *r*1 + *r*2 , ^{r ∈} (3, ^∞), *^r*1,r^{2 ∈} (1, ^∞).

The conclusion now follows from (5.1), Lemma 2.5 and the interpolation inequality. □

Lemma 5.2. Under the assumptions of Theorem 1.3, if (u, ^d, ^p) is the global strong solution obtained by Proposition 1.1, then it holds for any t > ⁰ and ¹ < ^r < ^∞ that

$$|\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} + |\partialt u(t)|{L^r(\mathbb{R}^3+)} + |\nabla p(t)|{L^r(\mathbb{R}^3_+)} \le C t^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}.$$

Proof. Following the proofs of (3.9)-(3.11), combined with Lemma 2.5 and Lemma 5.1, we obtain for all 1 < ^r < ^∞ and any ^t > 0,

$$|\mathbb{A}^{\alpha}u(t+h) - \mathbb{A}^{\alpha}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |(-\Delta)^{\alpha}\nabla d(t+h) - (-\Delta)^{\alpha}\nabla d(t)|{L^{r}(\mathbb{R}^{3}{+})}$$

$$\leq C\Big(h^{\delta}t^{-\alpha-\delta-2+\frac{3}{2r}} + h^{1-\alpha}t^{-\frac{9}{2}+\frac{3}{2r}} + h^{\delta}t^{-\alpha-\delta-\frac{7}{2}+\frac{3}{2r}}\Big); \tag{5.2}$$

$$|(-\Delta)^{\alpha}d(t+h) - (-\Delta)^{\alpha}d(t)|{L^{r}(\mathbb{R}^{3}{+})} \le C\left(h^{\delta}t^{-\alpha-\delta-\frac{3}{2}+\frac{3}{2r}} + h^{1-\alpha}t^{-4+\frac{3}{2r}} + h^{\delta}t^{-\alpha-\delta-3+\frac{3}{2r}}\right),\tag{5.3}$$

where 0 < α ⁺ δ < 1, and α > 0, δ > 0.

From (5.2), (5.3), Lemma 2.1, Lemma 4.1, Lemma 5.1 and Lemmas 2.4-2.5, we obtain for all 1 < ^r < ^∞ and any ^t > 0,

$$\begin{split} &|\mathbb{A}u(t)|{L^{r}(\mathbb{R}^{3}{+})} \ &\leq Ct^{-1}|u(\frac{t}{4})|{L^{r}(\mathbb{R}^{3}{+})} + C|u(t)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla u(t)|{L^{r{2}}(\mathbb{R}^{3}{+})} + C|\nabla d(t)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla^{2}d(t)|{L^{r{2}}(\mathbb{R}^{3}{+})} \ &+ C\int{\frac{t}{4}}^{\frac{t}{2}}(t-s)^{-1}\Big(|u(s)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla u(s)|{L^{r{2}}(\mathbb{R}^{3}{+})} + |\nabla d(s)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla^{2}d(s)|{L^{r{2}}(\mathbb{R}^{3}{+})}\Big)ds \ &+ C\int{\frac{t}{2}}^{t}(t-s)^{-1}|u(s)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla u(s) - \nabla u(t)|{L^{r{2}}(\mathbb{R}^{3}{+})}ds \ &+ C\int{\frac{t}{2}}^{t}(t-s)^{-1-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}|(u(s)-u(t))|{L^{\frac{3q}{3-q}}(\mathbb{R}^{3}{+})}|\nabla u(t)|{L^{3}(\mathbb{R}^{3}{+})}ds \ &+ C\int{\frac{t}{2}}^{t}(t-s)^{-1}|\nabla^{2}d(s)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla (ds) - \nabla d(t)|{L^{r{2}}(\mathbb{R}^{3}{+})}ds \ &+ C\int{\frac{t}{2}}^{t}(t-s)^{-1}|\nabla d(s)|{L^{r{1}}(\mathbb{R}^{3}{+})}|\nabla (\nabla d(s) - \nabla d(t))|{L^{r{2}}(\mathbb{R}^{3}{+})}ds \ &\leq Ct^{-\frac{3}{2}-\frac{3}{2}(1-\frac{1}{r})} + Ct^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r})}t^{-1-\frac{3}{2}(1-\frac{1}{r{2}})} \ &+ C\int{\frac{t}{4}}^{t}(t-s)^{-1}(1+s)^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{r})}s^{-1-\frac{3}{2}(1-\frac{1}{r{2}})}ds \ &+ Ct^{-2}\int{\frac{t}{2}}^{t}\Big((t-s)^{\delta-1}s^{-\frac{9}{2}-\delta+\frac{3}{2r}} + (t-s)^{-\frac{1}{2}}s^{-\frac{13}{2}+\frac{3}{2r}} + (t-s)^{\delta-1}s^{-6-\delta+\frac{3}{2r}}\Big)ds \ &+ Ct^{-2}\int{\frac{t}{2}}^{t}\Big((t-s)^{\delta-1-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}s^{-\delta-\frac{5}{2}+\frac{3}{2q}} \ &+ (t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}s^{-\frac{9}{2}+\frac{3}{2q}} + (t-s)^{\delta-1-\frac{3}{2}(\frac{1}{q}-\frac{1}{r})}s^{-4-\delta+\frac{3}{2q}}\Big)ds \ &\leq Ct^{-\frac{3}{2}-\frac{3}{2}(1-\frac{1}{r})}. \end{split}$$

Here ¹ r = 1 *r*1 + 1 *r*2 , *^r*1,r^{2 ∈} (1, ^∞), and we take the number

$$q = \begin{cases} r, & \text{if } r \in (1,3), \ 2, & \text{if } r \in [3,\infty). \end{cases}$$

Whence, for all 1 < ^r < ^∞ and any ^t > 0,

$$|\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} \le C |\mathbb{A} u(t)|{L^r(\mathbb{R}^3+)} \le C t^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}.$$

Using the above estimates, combined with Lemma 2.5, Lemma 4.1 and Lemma 5.1, we obtain for ^t > 0,

$$\begin{split} |\partialt u(t)|{L^r(\mathbb{R}^3+)} \leq & |\mathbb{A}u(t)|{L^r(\mathbb{R}^3+)} + |u(t)|{L^{r1}(\mathbb{R}^3+)} |\nabla u(t)|_{L^{r2}(\mathbb{R}^3+)} + |\nabla d(t)|_{L^{r1}(\mathbb{R}^3+)} |\nabla^2 d(t)|_{L^{r2}(\mathbb{R}^3+)} \ \leq & Ct^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}, \end{split}$$

and

$$\begin{split} |\nabla p(t)|{L^r(\mathbb{R}^3+)} \leq & |\partialt u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^2 u(t)|{L^r(\mathbb{R}^3+)} + |u(t)|{L^{r1}(\mathbb{R}^3+)} |\nabla u(t)|_{L^{r2}(\mathbb{R}^3+)} \ & + |\nabla d(t)|_{L^{r1}(\mathbb{R}^3+)} |\nabla^2 d(t)|_{L^{r2}(\mathbb{R}^3+)} \ \leq & Ct^{-\frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})}, \end{split}$$

where ¹ r = 1 *r*1 + 1 *r*2 , *^r*1,r^{2 ∈} (1, ^∞). □

Lemma 5.3. Under the assumptions of Theorem 1.3, if (u, ^d) is the global strong solution obtained by Proposition 1.1, then

$$|\nabla^2 u(t)|{L^\infty(\mathbb{R}^3+)} \leq C t^{-3}$$

holds for any t > ⁰.

Proof. Let 0 < θ < 1 and ³ < ^r < ^∞. Following the proof of Lemma 3.9, we conclude for any ^t > 0,

$$\begin{split} &|\nabla^2 u(t)|{L^{\infty}(\mathbb{R}^{3}{+})} \ \leq &|\nabla^2 e^{-\frac{t}{2}\mathbb{A}} u(\frac{t}{2})|{L^{\infty}(\mathbb{R}^{3}{+})} \ &+ \int{\frac{t}{2}}^{t} \left|\nabla^2 e^{-(t-s)\mathbb{A}} \mathbb{P}\Big(u(s) \cdot \nabla u(s) + \nabla \cdot (\nabla d \odot \nabla d)(s)\Big) \right|{L^{\infty}(\mathbb{R}^{3}{+})} ds \ \leq &Ct^{-1-\frac{3}{4}} |u(t)|{L^{2}(\mathbb{R}^{3}{+})} + C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2}-\frac{3}{2r}} \Big( |u|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla^2 d|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla u|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla^2 u|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla^2 u|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla^2 d(t)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^3 d(t)|{L^{2r}(\mathbb{R}^{3}{+})} + |\nabla d(t)|{L^{2r}(\mathbb{R}^{3}{+})} |\nabla^2 d(t)|{L^{2r}(\mathbb{R}^{3}{+})} \Big) ds \ &+ C \int{\frac{t}{2}}^{t} (t-s)^{-1+\frac{\theta}{2}-\frac{3}{2r}} \Big( |u(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla u(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla^2 u(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} \ &+ |\nabla^2 d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |\nabla^3 d(s)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} \Big) ds \ \leq &Ct^{-3} + C \int{\frac{t}{2}}^{t} (t-s)^{-\frac{1}{2}-\frac{3}{2r}} s^{-1-3(1-\frac{1}{2r})} ds + C \int_{\frac{t}{2}}^{t} (t-s)^{-1+\frac{\theta}{2}-\frac{3}{2r}} s^{-1-3(1-\frac{1}{2r})} ds \ \leq &Ct^{-3}. \end{split}$$

Lemma 5.4. Under the assumptions of Theorem 1.3, for every integer k ^{≥ 1}, there exists t(k) > ⁰ such that for ¹ < ^r ≤ ∞ and t ^{≥ t}(k), the strong solution (u, ^d, ^p) obtained in Proposition 1.¹ satisfies

$$|\nabla^{2+k}u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^{1+k}p(t)|{L^r(\mathbb{R}^3+)} \leq Ct^{-\frac{1}{2}-\frac{2+k}{2}-\frac{3}{2}(1-\frac{1}{r})}$$

□

Proof. Suppose ∥*x*3*u0∥^L* (R 3 + ) ^&lt; ^∞. Using Lemma 2.5, Lemma 4.1, Lemma 4.3 and Lemmas 5.1-5.2, following the proofs of (3.34), (3.35) and (3.39), we conclude for ^{m ≥} 1 and ^t > 2,

$$|\nabla{x}^{m} \int{\mathbb{R}^{3}{+}} \mathcal{M}(\cdot, y, \frac{t}{2}) u{1}(y, \frac{t}{2}) dy|{L^{r}(\mathbb{R}^{3}{+})} \le C t^{-\frac{1}{2} - \frac{m+2}{2} - \frac{3}{2}(1 - \frac{1}{r})}; \tag{5.4}$$

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} \mathcal{M}(x, y, t - s) \mathbb{P} U(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{5}{2} - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds + C \tilde{f}{2}(t) \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-4 - \frac{3}{2}(1 - \frac{1}{r})} \, ds \ & \leq C t^{-\frac{7}{2} - \frac{3}{2}(1 - \frac{1}{r})} + C \tilde{f}{2}(t) t^{-\frac{7}{2} - \frac{3}{2}(1 - \frac{1}{r})}; \end{split}$$

$$(5.5)$$

and

$$\left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{j}} W(x, y, t - s) (u{1} \cdot \nabla d + u \cdot \nabla d{1} - |\nabla d|^{2} d{1} - 2(\nabla d : \nabla d{1}) d)(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}_{+})}$$

$$+ \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} W(x, y, t - s) (u{1} \cdot \nabla d + u \cdot \nabla d{1} - |\nabla d|^{2} d{1} - 2(\nabla d : \nabla d{1}) d)(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}_{+})}$$

$$\leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{5}{2} - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} ds + C \int{\frac{t}{2}}^{t} (t - s)^{-1 + \varepsilon} s^{-2 - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds$$

$$+ C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-2} (|\nabla u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2} d{1}(s)|{L^{r}(\mathbb{R}^{3}_{+})}) \, ds$$

$$\leq C{\varepsilon} t^{-3 - \frac{3}{2}(1 - \frac{1}{r})} + C \tilde{f}{2}(t) t^{-\frac{7}{2} - \frac{3}{2}(1 - \frac{1}{r})},$$

$$(5.6)$$

where ˜*f*2(t) = sup ⁰<*s≤^t* [s 2+ (1− ) (∥∇*u*1(s)∥^L (R 3 + ) + ∥∇2*d*1(s)∥^L (R 3 + ) )].

Putting (5.4)-(5.6) together, following the proof of (3.41), we have for 1 ^&lt; ^{r &}lt; ^∞ and ^{t ≥} *^t*˜¹ with some number *^t*˜^{1 &}gt; 2,

$$|\nabla \partialt u(t)|{L^r(\mathbb{R}^3_+)} = |\nabla u1(t)|{L^r(\mathbb{R}^3_+)} \le Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}.$$ (5.7)

Thanks to Lemma 2.5, Lemma 5.1 and Lemma 5.2, we find for 1 < ^r < ^∞ and ^{t ≥} *^t*˜1,

$$|\nabla(u \cdot \nabla u)(t)|{L^{r}(\mathbb{R}^{3}{+})} \leq C(|\nabla u(t)|{L^{2r}(\mathbb{R}^{3}{+})}^{2} + |u(t)|{L^{2r}(\mathbb{R}^{3}{+})}|\nabla^{2}u(t)|{L^{2r}(\mathbb{R}^{3}{+})})$$

$$\leq Ct^{-\frac{7}{2} - \frac{3}{2}(1 - \frac{1}{r})}.$$ (5.8)

Applying (3.43) with ℓ ⁼ 1, together with (5.7) and (5.8), we obtain for 1 < ^r < ^∞ and ^{t ≥} *^t*˜1,

$$\begin{split} &|\nabla^{3}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2}p(t)|{L^{r}(\mathbb{R}^{3}{+})} \ \leq &C\big(|\nabla(u \cdot \nabla u)(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla\partial{t}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla(\nabla \cdot (\nabla d \odot \nabla d))(t)|{L^{r}(\mathbb{R}^{3}_{+})}\big) \ \leq &Ct^{-2-\frac{3}{2}(1-\frac{1}{r})}. \end{split}$$

Using Lemma 2.5, Lemma 5.1, Lemma 5.2 and (5.7), following the proof of (4.4), we get for 1 ^{≤ k ≤} 3, 1 ^{≤ j ≤} 2, 1 < ^r < ^∞, 0 < ε < ¹ 2 and t ≥ 2*t*˜1,

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{j}} \mathcal{M}(x, y, t - s) \mathbb{P}U(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} \mathcal{M}(x, y, t - s) \mathbb{P}U(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{5}{2} - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} ds + C \int{\frac{t}{2}}^{t} (t - s)^{-1 + \varepsilon} s^{-2 - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds \ & + C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-2} (|\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2} u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2} d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})}) \, ds \ & \leq C{\varepsilon} t^{-3 - \frac{3}{2}(1 - \frac{1}{r})} + C \tilde{f}{3}(t) t^{-4 - \frac{3}{2}(1 - \frac{1}{r})}, \end{split} \tag{5.9}$$

where ˜*f*3(t) = sup [s + (1− (∥∇*d*2(s)∥^L (R 3 + ) + ∥∇*u*2(s)∥^L r (R + ) + ∥∇2*u*1(s)∥^L (R 3 + ) + ∥∇2*d*2(s)∥^L (R 3 + )].

⁰<*s≤^t* Following the proof of (4.9), we obtain that

$$\left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} \mathcal{W}(x, y, t - s) D{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}_{+})}$$

$$\leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-3 - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds + C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-2} (|\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2} u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})}) \, ds$$

$$\leq C t^{-4 - \frac{3}{2}(1 - \frac{1}{r})} + C \tilde{f}_{3}(t) t^{-4 - \frac{3}{2}(1 - \frac{1}{r})}. \tag{5.10}$$

Following the proofs of (4.10) and (4.14), we get for 1 ^{≤ k ≤} 3, 1 ^{≤ j ≤} 2, 1 < ^r < ^∞ and ^{t ≥ 2}*t*˜1,

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{j}} W(x, y, t - s) D{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & + \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \partial{x{k}} \partial{x{3}} W(x, y, t - s) D{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-5 - \frac{3}{2}(1 - \frac{1}{r})} ds + C \int{\frac{t}{2}}^{t} (t - s)^{-1 + \varepsilon} s^{-\frac{9}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds \ & + C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-2} (|\nabla d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla u{2}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2} u{1}(s)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{2} d{2}(s)|{L^{r}(\mathbb{R}^{3}{+})}) \, ds \ & \leq C{\varepsilon} t^{-4 - \frac{3}{2}(1 - \frac{1}{r})} + C \tilde{f}_{3}(t) t^{-4 - \frac{3}{2}(1 - \frac{1}{r})}, \end{split} \tag{5.11}$$

and

$$\begin{split} & \left| \int{\frac{t}{2}}^{t} \int{\mathbb{R}^{3}{+}} \nabla{x} \mathcal{M}(x, y, t - s) \mathbb{P} U{1}(y, s) \, dy ds \right|{L^{r}(\mathbb{R}^{3}{+})} \ & \leq C \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{7}{2} - \frac{3}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds + C \tilde{f}{3}(t) \int{\frac{t}{2}}^{t} (t - s)^{-\frac{1}{2}} s^{-\frac{9}{2} - \frac{3}{2}(1 - \frac{1}{r})} \, ds \ & \leq C t^{-\frac{9}{2} - \frac{3}{2}(1 - \frac{1}{r})} + C \tilde{f}_{3}(t) t^{-4 - \frac{3}{2}(1 - \frac{1}{r})}. \end{split} \tag{5.12}$$

From (4.13), (5.9)-(5.12), following the proof of (4.17), we conclude that for 1 ^&lt; ^{r &}lt; ^∞ and ^{t ≥} *^t*˜² with some number *t*˜² ≥ 2*t*˜1,

$$|\nabla d2(s)|{L^r(\mathbb{R}^3_+)} + |\nabla u2(s)|{L^r(\mathbb{R}^3_+)} + |\nabla^2 u1(s)|{L^r(\mathbb{R}^3_+)} + |\nabla^2 d2(s)|{L^r(\mathbb{R}^3_+)} \le Ct^{-\frac{5}{2} - \frac{3}{2}(1 - \frac{1}{r})}.$$ (5.13)

Using (3.43) and (5.13), running the same argument as in (5.9)-(5.12), we obtain that for 1 < ^r < ^∞ and t ≥ *t*˜2,

$$\begin{split} |\nabla^4 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^3 p(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-\frac{5}{2} - \frac{3}{2}(1 - \frac{1}{r})}, \ |\nabla^5 u(t)|{L^r(\mathbb{R}^3+)} + |\nabla^4 p(t)|{L^r(\mathbb{R}^3+)} &\leq C t^{-3 - \frac{3}{2}(1 - \frac{1}{r})}. \end{split}$$

Repeating the proofs of (5.4)-(5.12), we can get for every integer ^{k ≥} 1,

$$|\nabla^{2+k}u(t)|{L^{r}(\mathbb{R}^{3}{+})} + |\nabla^{1+k}p(t)|{L^{r}(\mathbb{R}^{3}{+})} \le \tilde{C}t^{-\frac{1}{2} - \frac{2+k}{2} - \frac{3}{2}(1 - \frac{1}{r})}$$ (5.14)

holds for 1 < ^r < ^∞ and ^{t ≥} *^t*˜ ^k with some number *t*˜ ^{k &}gt; 0. Using the Gagliardo-Nirenberg inequality (2.6), we conclude that for any ^{k ≥} 1 and ^{t ≥} *^t*˜ ^k, (5.14) is valid for ^{r = ∞}. Thus, we complete the proof of Lemma 5.4. □

Proof of Theorem 1.3. Theorem 1.3 directly follows from Lemmas 5.1-5.4. □

Acknowledgements

This work was partially supported by National Key R&D Program of China (No. 2021YFA1002900), Guangzhou City Basic and Applied Basic Research Fund (No. 2024A04J6336), and the Scientific Research Innovation Project of Graduate School of South China Normal University (No. 2024KYLX084).

Data Availability: Data sharing is not applicable to this article.

Conflict of Interest: The authors declare that they have no conflict of interest.

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School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China. Email address, H. Chen: chenhaokunn@163.com

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China. Email address, Y. Wang: wangyongxmu@163.com