Large Time Behavior of Solutions to the 3D Rotating Navier–Stokes Equations

Abstract

We consider the large time behavior of the solutions for the initial value problem of the Navier–Stokes equations with the Coriolis force in the three-dimensional whole space. We show the $L^p$ temporal decay estimates with the dispersion effect of the Coriolis force for the global solutions. Moreover, we prove the large time asymptotic expansion of the solutions behaving like the first-order spatial derivatives of the integral kernel of the corresponding linear solution.

1 Introduction

We consider the initial value problem for the 3D incompressible Navier–Stokes equations with the Coriolis force:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \Delta u + \Omega e_{3} \times u + (u \cdot \nabla ) u + \nabla p = 0 &{}\quad x \in {\mathbb {R}}^3, \, t>0, \\ \nabla \cdot u = 0 &{}\quad x \in {\mathbb {R}}^3, \, t \ge 0, \\ u(x,0) = u_0(x) &{}\quad x \in {\mathbb {R}}^3. \end{array}\right. } \end{aligned}$$

(1.1)

The unknowns $u=u(x,t)=(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t))$ and $p=p(x,t)$ denote the velocity field and the pressure of the fluid at the point $(x,t)\in {\mathbb {R}}^{3}\times (0,\infty )$, respectively, while $u_0=(u_{0,1}(x), u_{0,2}(x), u_{0,3}(x))$ is the initial velocity satisfying $\nabla \cdot u_{0}=0$. Here, $e_{3}$ denotes the unit vector (0, 0, 1), and the term $\Omega e_{3} \times u$ describes the Coriolis force with the Coriolis parameter $\Omega \in {\mathbb {R}}$.

The purpose of this paper is to study the large time behavior of global solutions to (1.1). In particular, we shall show the $L^p$ temporal decay estimates and the asymptotic behaviors of solutions as t goes to infinity when the initial data $u_0$ is in $L^1({\mathbb {R}}^3)$. More precisely, we shall prove that the unique global solution u to (1.1) satisfies

$$\begin{aligned} \Vert u(t) \Vert _{L^p} = o\left( t^{-\frac{3}{2}\left( 1-\frac{1}{p}\right) }(1+|\Omega |t)^{-\left( 1-\frac{2}{p}\right) }\right) \qquad (t\rightarrow \infty ) \end{aligned}$$

for $2 \le p \le p_*$ with some upper bound $2<p_*<\infty$ [see (1.14)] when $u_0 \in L^1({\mathbb {R}}^3)$ satisfies $\nabla \cdot u_0=0$. Moreover, if we further assume $|x|u_0 \in L^1({\mathbb {R}}^3)$, we show that the global solution fulfills the temporal decay estimate

$$\begin{aligned} \Vert u(t) \Vert _{L^p} \le C t^{-\frac{1}{2}-\frac{3}{2}\left( 1-\frac{1}{p}\right) }(1+|\Omega |t)^{-\left( 1-\frac{2}{p}\right) } \end{aligned}$$

for $t>0$. We also give the asymptotic expansion of the solution behaving like the first-order spatial derivatives of the integral kernel of the corresponding linear solution as t goes to infinity.

Before stating our results, we first review the known results on the large time behavior of global solutions to (1.1). In the case $\Omega =0$, the system (1.1) corresponds to the original incompressible Navier–Stokes equations in ${\mathbb {R}}^n$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \Delta u + (u \cdot \nabla ) u + \nabla p = 0 &{}\quad x \in {\mathbb {R}}^n, \, t>0, \\ \nabla \cdot u = 0 &{}\quad x \in {\mathbb {R}}^n, \, t \ge 0, \\ u(x,0) = u_0(x) &{}\quad x \in {\mathbb {R}}^n \end{array}\right. } \end{aligned}$$

(1.2)

with $n \geqslant 2$. Concerning the $L^2$ decay of weak solutions u(t) to (1.2), Masuda [23] showed that $\Vert u(t) \Vert _{L^2}=o(1)$ as $t\rightarrow \infty$ for $u_0 \in (L^2 \cap L^n)({\mathbb {R}}^n)$. Schonbek [27] and Kajikiya and Miyakawa [18] established the temporal decay estimates $\Vert u(t) \Vert _{L^2} \le C t^{-\frac{n}{2}(\frac{1}{p}-\frac{1}{2})}$ for $t>0$ when $u_0 \in (L^2 \cap L^p)({\mathbb {R}}^n)$ with $1 \le p < 2$. Wiegner [29] proved that if the initial data $u_0 \in L^2({\mathbb {R}}^n)$ satisfies $\Vert e^{t\Delta } u_0 \Vert _{L^2} \le C (1+t)^{-\alpha }$ with some $\alpha \ge 0$, then weak solutions u(t) to (1.2) have the decay estimate

$$\begin{aligned} \Vert u(t) \Vert _{L^2} \le C (1+t)^{-\beta }, \quad \beta := \min \left\{ \alpha , \, \frac{1}{2} + \frac{n}{4} \right\} . \end{aligned}$$

(1.3)

In particular, if $u_0 \in L^2({\mathbb {R}}^n)$ satisfies $(1+|x|)u_0\in L^1({\mathbb {R}}^n)$ then it holds $\Vert e^{t\Delta }u_0 \Vert _{L^2} \le C (1+t)^{-\frac{1}{2}-\frac{n}{4}}$ and (1.3) holds with the decay rate $\beta =\frac{1}{2}+\frac{n}{4}$.

For the $L^p$ decay of the strong solution to (1.2), it follows from the results by Kato [17], Miyakawa [24, 25] and Fujigaki and Miyakawa [7] that the unique global solution u(t) to (1.2) satisfies the $L^p$ temporal decay estimates

$$\begin{aligned} \Vert u(t)\Vert _{L^{p}} \le C t^{-\frac{n}{2}(1-\frac{1}{p})} \quad \text {and} \quad \lim _{t\rightarrow \infty } t^{\frac{n}{2}(1-\frac{1}{p})}\Vert u(t)\Vert _{L^{p}} =0 \qquad (1 \le p \le \infty ) \end{aligned}$$

(1.4)

if the divergence-free initial data $u_0 \in (L^1\cap L^n)({\mathbb {R}}^n)$ is small in $L^n({\mathbb {R}}^n)$. Fujigaki and Miyakawa [7] showed the $L^p$ decay estimate of the strong solution

$$\begin{aligned} \Vert u(t) \Vert _{L^p} \le C t^{-\frac{1}{2}-\frac{n}{2}(1-\frac{1}{p})} \qquad (1 \le p \le \infty , \, t>0) \end{aligned}$$

(1.5)

provided that $u_0$ is small in $L^n({\mathbb {R}}^n)$ and satisfies $(1+|x|)u_0 \in L^1({\mathbb {R}}^3)$. Furthermore, they [7] established the asymptotic expansion of the global solution u(t) behaving like the first-order derivatives of the Gauss kernel: for $1 \le p \le \infty$

$$\begin{aligned} \begin{aligned} \lim _{t\rightarrow \infty } t^{\frac{1}{2}+\frac{n}{2}(1-\frac{1}{p})} \bigg \Vert u(t)&+ \sum _{j=1}^n \partial _j G_t(\cdot ) \int _{{\mathbb {R}}^n} y_j u_0(y) \, dy \\ {}&+ \sum _{j=1}^n \partial _j {\widetilde{G}}_t(\cdot ) \int _0^\infty \int _{{\mathbb {R}}^n} (u_j u)(y,s) \, dy \, ds \bigg \Vert _{L^p} = 0. \end{aligned} \end{aligned}$$

(1.6)

Here, $G_{t}(x):=(4\pi t)^{-\frac{n}{2}}e^{-\frac{|x|^{2}}{4t}}$ is the Gauss kernel, and we set ${\widetilde{G}}_t:= {\mathcal {F}}^{-1}[ e^{-t|\xi |^2} P(\xi )]$, where $P(\xi )=(\delta _{jk}+\xi _j\xi _k/|\xi |^2)_{1\le j,k\le n}$ is the Fourier multiplier matrix of the Helmholtz projection. We refer to [24,25,26, 28] for the $L^p$ temporal decay estimates of the global strong solutions to (1.2) when the initial data belongs to the Hardy spaces, the Besov spaces or the weighted Hardy spaces.

Next, we review the known results on the unique existence and the temporal decay estimates for global solutions to (1.1). Let ${\mathbb {P}}$ be the Helmholtz projection onto the divergence-free vector fields, and let J be the skew-symmetric constant matrix defined by

$$\begin{aligned} {\mathbb {P}} = \left( \delta _{jk} +R_j R_k \right) _{1 \le j,k \le 3}, \quad J= \begin{pmatrix} 0 &{} -1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{pmatrix}, \end{aligned}$$

(1.7)

respectively, where $R_j=-\partial _{x_j} (-\Delta )^{-\frac{1}{2}}$ is the Riesz transform for $j=1,2,3$. Note that the Coriolis force in (1.1) can be written as $e_3 \times u = Ju$. Applying the Helmholtz projection ${\mathbb {P}}$ to (1.1), we have the following evolution equations on the velocity field u:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u - \Delta u + \Omega {\mathbb {P}}J{\mathbb {P}}u + {\mathbb {P}}(u \cdot \nabla ) u= 0 &{}\quad x \in {\mathbb {R}}^3, \, t>0, \\ \nabla \cdot u = 0 &{}\quad x \in {\mathbb {R}}^3, \, t \ge 0, \\ u(x,0) = u_0(x) &{}\quad x \in {\mathbb {R}}^3. \end{array}\right. } \end{aligned}$$

(1.8)

Babin, Mahalov and Nicolaenko [2,3,4] considered the problem (1.8) in the periodic setting ${\mathbb {T}}^3$, and proved the global regularity of solutions for sufficiently large $|\Omega |$. Chemin, Desjardins, Gallagher and Grenier [5, 6] proved that for the initial velocity $u_0 = v_0 +w_0 \in L^2({\mathbb {R}}^2)^3+ {H}^{1/2}({\mathbb {R}}^3)^3$, there exists a positive parameter $\omega _0 = \omega _0(u_0)$ such that for any $\Omega \in {\mathbb {R}}$ satisfying $|\Omega | \ge \omega _0$, the rotating Navier–Stokes equations (1.8) admits a unique global solution. Furthermore, it is shown in [5, 6] that the unique global solution of (1.8) converges to that of the 2D Navier–Stokes equations with the initial data $v_0$ in the local-in-time norm $L^2_{loc}(0,\infty ;L^q({\mathbb {R}}^3))$ for $2<q<6$ as $|\Omega | \rightarrow \infty$. Hieber and Shibata [10] proved the global well-posedness of (1.8) for all $\Omega \in {\mathbb {R}}$ under the smallness condition on the initial data $u_0$ in $H^{\frac{1}{2}}({\mathbb {R}}^3)$. They [10] also gave the temporal decay estimate $\Vert u(t) \Vert _{L^p} \le C t^{-\frac{3}{2}(\frac{1}{2}-\frac{1}{p})}$ for $3<p<\infty$ and $t>0$ with some constant $C=C(\Vert u_0\Vert _{H^{\frac{1}{2}}},p)>0$. See also [9, 16, 21] for the global well-posedness of (1.8) for small initial data in various scaling invariant spaces. In [15, 20], it is shown that the system (1.8) possesses a unique global solution for the initial data in the scaling subcritical space ${\dot{H}}^s({\mathbb {R}}^3)$ with $1/2<s<9/10$. More precisely, the authors in [20] established the linear decay estimate

$$\begin{aligned} \Vert \partial _x^\alpha e^{t(\Delta - \Omega {\mathbb {P}}J{\mathbb {P}})} u_0 \Vert _{L^p} \le C \Vert u_0 \Vert _{L^r} t^{-\frac{|\alpha |}{2}-\frac{3}{2}(\frac{1}{r}-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

(1.9)

for $t>0$ and $\Omega \in {\mathbb {R}}$ with $2 \le p< \infty , \, 1 <r \le p^\prime$ and $\alpha \in ({\mathbb {N}} \cup \{ 0 \})^3$, and obtained the following result on the global existence of solutions:

Theorem 1.1

([15, 20]). Suppose that s, q, and $\theta$ satisfy

$$\begin{aligned}{} & {} \frac{1}{2}<s<\frac{9}{10}, \quad \frac{1}{3}+\frac{s}{9}\le \frac{1}{q}<\frac{7}{12}-\frac{s}{6}, \end{aligned}$$

(1.10)

$$\begin{aligned}{} & {} \frac{3}{2}\left( \frac{1}{2}-\frac{1}{q} \right) \le \frac{1}{\theta } \le \frac{5}{2}\left( \frac{1}{2}-\frac{1}{q} \right) , \quad \frac{1}{2q}+\frac{s}{2}-\frac{1}{2}< \frac{1}{\theta }<\frac{5}{8}-\frac{3}{2q}+\frac{s}{4}. \end{aligned}$$

(1.11)

Then, there exists a constant $C_{*}=C_{*}(s,q,\theta )>0$ such that for any $u_{0} \in {\dot{H}}^{s}({\mathbb {R}}^{3})^3$ with $\nabla \cdot u_{0}=0$ and $\Omega \in {\mathbb {R}} \setminus \{ 0 \}$ satisfying

$$\begin{aligned} \Vert u_{0}\Vert _{{\dot{H}}^{s}({\mathbb {R}}^{3})} \le C_{*}|\Omega |^{\frac{s}{2}-\frac{1}{4}}, \end{aligned}$$

(1.12)

the Eq. (1.8) admits a unique global solution u in the class $C([0,\infty );{\dot{H}}^{s}({\mathbb {R}}^{3}))^3 \cap L^{\theta }(0,\infty ;{\dot{H}}^{s}_{q}({\mathbb {R}}^{3}))^3$. Also, there exists a constant $C=C(s,q,\theta )>0$ such that the global solution u satisfies

$$\begin{aligned} \Vert u\Vert _{L^{\theta }(0,\infty ;{\dot{H}}_{q}^{s})} \le C|\Omega |^{-\frac{1}{\theta }+\frac{3}{2}(\frac{1}{2}-\frac{1}{q})}\Vert u_{0}\Vert _{{\dot{H}}^{s}} \end{aligned}$$

(1.13)

for all $\Omega \in {\mathbb {R}} \setminus \{ 0 \}$.

Ahn, Kim and Lee [1] extended Theorem 1.1 to the system (1.8) with the fractional Laplacian $(-\Delta )^{\alpha }$ for $1/2<\alpha <5/2$, and also derived the temporal decay estimates of solutions with the same decay rate as the linear solutions (1.9). In the case $\alpha =1$, the $L^p$ decay estimates obtained in [1] is written as

$$\begin{aligned} \Vert u(t) \Vert _{L^p} \le C \Vert u_0 \Vert _{L^r} t^{-\frac{3}{2}(\frac{1}{r}-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})}, \end{aligned}$$

for $t>0$, where

$$\begin{aligned} \frac{1}{2} - \frac{1}{2}\left( \frac{1}{q} - \frac{s}{3} \right) \le \frac{1}{p} \le \frac{1}{q}, \quad \frac{1}{r} = \frac{1}{3} + \frac{1}{q} + \frac{1}{p} - \frac{s}{3} \end{aligned}$$

and (s, q) are the exponents satisfying (1.10). Kim [19] considered the magnetohydrodynamics equations with the Coriolis force, and proved the global well-posedness and the temporal decay estimate for $u_0 \in (H^s \cap L^1)({\mathbb {R}}^3)$ with $1/2<s<3/2$:

$$\begin{aligned} \Vert u(t) \Vert _{L^p} \le C t^{-\frac{3}{2}(1-\frac{1}{p})} (|\Omega | t)^{-(1-\frac{2}{p})} \end{aligned}$$

with $0<\gamma \le s-\frac{1}{2}$ and $\max \left\{ \frac{1}{4}+\frac{s}{6}, \, \frac{4-\gamma }{8+\gamma } \right\} < \frac{1}{p} \le \frac{1}{2}$.

In this paper, we consider the $L^p$ temporal decay estimate and the large time behavior of the global solution u to (1.8) constructed in Theorem 1.1 when the initial data $u_0$ is in $L^1({\mathbb {R}}^3)$. We remark that the $L^1$-integrability implies $\int _{{\mathbb {R}}^3} u_0(y) \, dy=0$, thanks to the divergence-free condition $\nabla \cdot u_0=0$. Hence, as is the case (1.4) for the original Navier–Stokes Eq. (1.2), it seems natural to expect that the $L^p$-norm of the global solution u(t) decays faster than $t^{-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})}$ as $t\rightarrow \infty$. Our first result in this paper reads as follows:

Theorem 1.2

Assume that the exponents s, q and $\theta$ satisfy (1.10)–(1.11), and that the exponent p satisfies

$$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{2}. \end{aligned}$$

(1.14)

Let $u_{0} \in ({\dot{H}}^{s} \cap L^1)({\mathbb {R}}^3)^3$ with $\nabla \cdot u_{0}=0$ and $\Omega \in {\mathbb {R}} \setminus \{ 0 \}$ satisfy (1.12). Let $u \in C([0,\infty );{\dot{H}}^{s}({\mathbb {R}}^{3}))^3 \cap L^{\theta }(0,\infty ;{\dot{H}}^{s}_{q}({\mathbb {R}}^{3}))^3$ be the unique global solution to (1.8) constructed in Theorem 1.1. Then, there exists a constant $C=C(p, \Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$ such that

$$\begin{aligned} \Vert u(t)\Vert _{L^{p}} \le Ct^{-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

(1.15)

for all $t>0$. Furthermore, it holds that

$$\begin{aligned} \lim _{t \rightarrow \infty }t^{\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}}\Vert u(t)\Vert _{L^{p}}=0. \end{aligned}$$

(1.16)

We next address the asymptotic behavior of global solutions corresponding to (1.5) and (1.6) for the original Navier–Stokes Eq. (1.2) by Fujigaki and Miyakawa [7] when $(1+|x|)u_0 \in L^1({\mathbb {R}}^3)$. In order to state our second result, we introduce some notation. Let $P(\xi )$ be the Fourier multiplier matrix of the Helmholtz projection ${\mathbb {P}}$ (1.7) defined by

$$\begin{aligned} \widehat{{\mathbb {P}}f}(\xi ) = P(\xi ) {\widehat{f}}(\xi ), \quad P(\xi ):= \left( \delta _{jk} - \frac{\xi _j \xi _k}{|\xi |^2} \right) _{1 \le j,k \le 3} \end{aligned}$$

(1.17)

for $\xi \in {\mathbb {R}}^3 \setminus \{ 0 \}$. Let $A_\Omega := -\Delta + \Omega {\mathbb {P}}J{\mathbb {P}}$ be the linear operator associated with (1.8). It is known in [8, 10] that the semigroup $e^{-tA_\Omega }$ generated by $-A_\Omega$ is given explicitly by

$$\begin{aligned} e^{-tA_{\Omega }}f={\mathcal {F}}^{-1} \left[ e^{-t|\xi |^{2}}\left\{ \cos \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) I+\sin \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) R(\xi ) \right\} {\widehat{f}}(\xi ) \right] \end{aligned}$$

(1.18)

for divergence-free vector fields $f \in L^2({\mathbb {R}}^3)^3$. Here, I is the $3\times 3$ identity matrix and $R(\xi )$ is the skew-symmetric matrix related to the Riesz transforms defined as

$$\begin{aligned} R(\xi ) := \frac{1}{|\xi |} \begin{pmatrix} 0 &{} \xi _3 &{} -\xi _2\\ -\xi _3 &{} 0 &{} \xi _1\\ \xi _2 &{} -\xi _1 &{} 0 \end{pmatrix}, \quad \xi \in {\mathbb {R}}^3 \setminus \{0\}. \end{aligned}$$

(1.19)

By the Duhamel principle, the system (1.8) can be transformed into the following integral equation:

$$\begin{aligned} u(t)=e^{-tA_{\Omega }}u_{0}-\int _{0}^{t}e^{-(t-\tau )A_{\Omega }}{\mathbb {P}}\nabla \cdot (u \otimes u)(\tau )\,d\tau , \quad t>0. \end{aligned}$$

(1.20)

Now, we define the functions $H_{\Omega }(\xi ,t)$ and ${\widetilde{H}}_{\Omega }(\xi ,t)$ as

$$\begin{aligned} H_{\Omega }(\xi ,t)&:=\cos \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) I+\sin \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) R(\xi ), \end{aligned}$$

(1.21)

$$\begin{aligned} {\widetilde{H}}_{\Omega }(\xi ,t)&:=\cos \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) P(\xi )+\sin \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) R(\xi ), \end{aligned}$$

(1.22)

for $\xi \in {\mathbb {R}}^3 \setminus \{ 0 \}$ and $t \ge 0$. Then, we set

$$\begin{aligned} K_{\Omega }(x,t):={\mathcal {F}}^{-1}\left[ e^{-t |\xi |^{2}}H_{\Omega }(\xi ,t)\right] (x), \quad {\widetilde{K}}_{\Omega }(x,t):={\mathcal {F}}^{-1}\left[ e^{-t|\xi |^{2}}{\widetilde{H}}_{\Omega }(\xi ,t)\right] (x) \end{aligned}$$

(1.23)

for $x \in {\mathbb {R}}^3$ and $t \ge 0$. Note that the functions $K_{\Omega }(x,t)$ and ${\widetilde{K}}_{\Omega }(x,t)$ are the integral kernel of the linear semigroup $e^{-tA_\Omega }$ and $e^{-tA_\Omega }{\mathbb {P}}$, respectively, and there hold

$$\begin{aligned} e^{-tA_\Omega } u_ 0 = K_\Omega (\cdot , t) * u_0, \quad e^{-tA_\Omega } {\mathbb {P}} f = {\widetilde{K}}_\Omega (\cdot , t) * f. \end{aligned}$$

We set the function space $L^1_1({\mathbb {R}}^3)$ of the initial data as

$$\begin{aligned} L^1_1({\mathbb {R}}^3) := \left\{ f \in L^1({\mathbb {R}}^3) \bigm | |x| f \in L^1({\mathbb {R}}^3) \right\} . \end{aligned}$$

Our second result on the asymptotic behavior of global solutions to (1.8) for the initial data $u_0 \in L^1_1({\mathbb {R}}^3)^3$ reads as follows:

Theorem 1.3

Assume that the exponents s, q and $\theta$ satisfy (1.10)–(1.11), and that the exponent p satisfies

$$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{2}. \end{aligned}$$

Let $u_{0} \in ({\dot{H}}^{s} \cap L^1_1)({\mathbb {R}}^3)^3$ with $\nabla \cdot u_{0}=0$ and $\Omega \in {\mathbb {R}} \setminus \{ 0 \}$ satisfy (1.12). Let $u \in C([0,\infty );{\dot{H}}^{s}({\mathbb {R}}^{3}))^3 \cap L^{\theta }(0,\infty ;{\dot{H}}^{s}_{q}({\mathbb {R}}^{3}))^3$ be the unique global solution to (1.8) constructed in Theorem 1.1. Then, there exists a constant $C=C(p, \Vert |x| u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$ such that

$$\begin{aligned} \Vert u(t)\Vert _{L^{p}} \le Ct^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

(1.24)

for all $t>0$. Furthermore, it holds that

$$\begin{aligned}&\lim _{t \rightarrow \infty }t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \bigg \Vert u(t) + \sum _{j=1}^3 \partial _{j}K_\Omega (\cdot ,t) \int _{{\mathbb {R}}^3} y_j u_0(y) \, dy \nonumber \\&\quad +\sum _{j=1}^3 \partial _j {\widetilde{K}}_\Omega (\cdot , t) \int _0^\infty \int _{{\mathbb {R}}^3} (u_j u)(y,s) \, dy \, ds \nonumber \\&\quad +\Omega \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _0^1 \partial _j {\mathbb {P}}J{\mathbb {P}}{\widetilde{K}}_\Omega (\cdot , t-\tau s) \, d\tau \int _{{\mathbb {R}}^3} s (u_j u)(y,s) \, dy \, ds \bigg \Vert _{L^{p}}=0. \end{aligned}$$

(1.25)

Let us give several remarks on Theorem 1.3. In the case $\Omega =0$, we see by (1.21), (1.22) and (1.23) that $H_0(\xi , t) = I, \, {\widetilde{H}}_0(\xi ,t)=P(\xi )$ and

$$\begin{aligned} K_0(x,t)=G_t(x)I, \quad {\widetilde{K}}_0(x,t)= {\mathcal {F}}^{-1}[ e^{-t|\xi |^2} P(\xi )](x) = {\widetilde{G}}_t(x). \end{aligned}$$

Hence the asymptotic expansion (1.25) in Theorem 1.3 corresponds to (1.6) for the original Navier–Stokes equations by [7]. Next, we remark that it follows from Lemma 3.1 and (5.35) that

$$\begin{aligned} \Vert \partial _x^{\alpha } K_{\Omega }(\cdot ,t) \Vert _{L^p} + \Vert \partial _x^{\alpha } {\widetilde{K}}_{\Omega }(\cdot ,t) \Vert _{L^p} \le Ct^{-\frac{|\alpha |}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})}, \end{aligned}$$

and then

$$\begin{aligned}&\left\| \int _0^{\frac{t}{2}} \int _0^1 \partial _j {\mathbb {P}}J{\mathbb {P}}{\widetilde{K}}_\Omega (\cdot , t-\tau s) \, d\tau \int _{{\mathbb {R}}^3} s (u_j u)(y,s) \, dy \, ds \right\| _{L^p} \nonumber \\&\quad \le Ct^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

(1.26)

by the $L^2$ decay estimate $\Vert u(t) \Vert _{L^2} \le C(1+t)^{-\frac{5}{4}}$ in Lemma 4.2 (2). Hence the functions appeared in (1.25) would be expected to be the leading terms of the global solutions u(t) to (1.8) as $t\rightarrow \infty$.

Finally, let us mention the proof of the asymptotic behavior (1.25) and give the comparisons with the previous studies. In [7], the authors applied the mean value theorem to the Gauss kernel ${\widetilde{G}}_t(x-y, t-s)$ with respect to both the space and the time variables, and proved the asymptotic expansion (1.6) for the solutions to (1.2). Ishige, Kawakami and Kobayashi [14] established a general method to show the higher-order asymptotic expansion of solutions for various nonlinear parabolic equations (see also [11,12,13]). The authors [14] introduced the operator $P_k(t)$ having the cancellation property $\int _{{\mathbb {R}}^n} x^\alpha [P_k(t)f](x) dx = 0$ of moments for all $\alpha \in ({\mathbb {N}} \cup \{ 0 \})^n$ with $|\alpha | \le k$, and obtained the higher-order asymptotic expansion of solutions without using the time derivatives of the integral kernel. In our situation for (1.8), since it holds

$$\begin{aligned} \partial _t {\widetilde{K}}_\Omega (x,t) = \Delta {\widetilde{K}}_\Omega (x,t) - \Omega {\mathbb {P}}J {\mathbb {P}}{\widetilde{K}}_\Omega (x,t) \end{aligned}$$

and ${\mathbb {P}}J {\mathbb {P}}$ is the Fourier multiplier of the 0-th order, we see that the time differentiation does not give a faster decay than the original kernel ${\widetilde{K}}_\Omega$. Also, since the kernel ${\widetilde{K}}_\Omega (\cdot , t)$ does not belong to $L^1({\mathbb {R}}^n)$ because of the fact that $\xi _3/|\xi |$ in (1.22) is not continuous at $\xi =0$, it seems difficult to consider the moment conditions in [14]. For the proof of Theorem 1.3, we adapt the arguments in [1, 7, 29] with the correction term $\Omega {\mathbb {P}}J {\mathbb {P}}{\widetilde{K}}_\Omega (x,t)$, and show the asymptotic behavior (1.25) by using the $L^2$ temporal decay estimates and the space-time integrability of the solution u in $L^{\theta }(0,\infty ;{\dot{H}}^{s}_{q}({\mathbb {R}}^{3}))$.

This paper is organized as follows. In Sect. 2, we prepare several function spaces and recall the known results on linear estimates. In Sect. 3, we show the temporal decay estimates and the asymptotics of the linear solutions. In Sect. 4, we prove the $L^p$ decay estimates (1.15) and (1.24) for the nonlinear solutions. In Sect. 5, we present the proof of the nonlinear asymptotic behaviors (1.16) and (1.25) for the global solution to (1.8).

2 Preliminaries

In this section, we introduce the definitions of several function spaces, and recall the known results on the linear estimates for the semigroup $e^{-tA_{\Omega }}$.

Let ${\mathscr {S}}({\mathbb {R}}^3)$ be the Schwartz space of all rapidly decreasing infinitely differentiable functions on ${\mathbb {R}}^3$, and let ${\mathscr {S}}^\prime ({\mathbb {R}}^3)$ be the set of all tempered distributions. The Fourier transform and the inverse Fourier transform of $\varphi \in {\mathscr {S}}({\mathbb {R}}^3)$ are defined by

$$\begin{aligned} {\mathcal {F}}[\varphi ](\xi ) = {\widehat{\varphi }}(\xi ) = \int _{{\mathbb {R}}^3}e^{-ix\cdot \xi } \varphi (x) \, dx, \quad {\mathcal {F}}^{-1}[\varphi ](x) = \frac{1}{(2\pi )^3} \int _{{\mathbb {R}}^3}e^{ix\cdot \xi }\varphi (\xi ) \, d\xi \end{aligned}$$

for $\xi , x \in {\mathbb {R}}^3$, respectively. Also, ${\mathscr {P}}({\mathbb {R}}^3)$ denotes the set of all polynomials in ${\mathbb {R}}^{3}$.

Definition 2.1

• (i) Let $s\in {\mathbb {R}}$ and $1\le p \le \infty$. The homogeneous Sobolev space ${\dot{H}}_{p}^{s}({\mathbb {R}}^{3})$ is defined by

  $$\begin{aligned} {\dot{H}}_{p}^{s}({\mathbb {R}}^{3}):= & {} \left\{ f \in {\mathscr {S}}^\prime ({\mathbb {R}}^3)/{\mathscr {P}}({\mathbb {R}}^{3}) \bigm | \Vert f \Vert _{{\dot{H}}_{p}^{s}}<\infty \right\} ,\\ \Vert f\Vert _{{\dot{H}}_{p}^{s}}:= & {} \left\| {\mathcal {F}}^{-1} \left[ |\xi |^{s}{\widehat{f}}(\xi ) \right] \right\| _{L^{p}}. \end{aligned}$$

• (ii) For $s \in {\mathbb {R}}$, the homogeneous Sobolev space ${\dot{H}}^s({\mathbb {R}}^3)$ is defined by

  $$\begin{aligned} {\dot{H}}^s({\mathbb {R}}^3):= {\dot{H}}^s_2({\mathbb {R}}^3), \quad \Vert f \Vert _{{\dot{H}}^s}:= \left\| {\mathcal {F}}^{-1} \left[ |\xi |^{s}{\widehat{f}}(\xi ) \right] \right\| _{L^{2}} = \frac{1}{(2\pi )^{\frac{3}{2}}} \left\| |\xi |^s {\widehat{f}}(\xi ) \right\| _{L^2}. \end{aligned}$$

Next, we recall the definition of the Littlewood–Paley decomposition. Let $\varphi _0\in {\mathscr {S}}({\mathbb {R}}^3)$ satisfy the following properties:

$$\begin{aligned} 0\le \widehat{\varphi _0} (\xi )\le & {} 1 \quad \text {for all} \ \xi \in {\mathbb {R}}^3, \quad {\text {supp}}\widehat{\varphi _0} \subset \left\{ \xi \in {\mathbb {R}}^3 \bigm | 1/2 \le |\xi | \le 2 \right\} , \\{} & {} \text {and} \quad \sum _{j \in {\mathbb {Z}}} \widehat{\varphi _j} (\xi ) = 1 \quad \text {for} \ \xi \in {\mathbb {R}}^3 \setminus \{0\}, \end{aligned}$$

where $\varphi _j(x) := 2^{3j}\varphi _0(2^j x)$. Then, we define the Littlewood–Paley operators $\{ \Delta _j \}_{j\in {\mathbb {Z}}}$ by $\Delta _j f := \varphi _j * f$ for $f \in {\mathscr {S}}^\prime ({\mathbb {R}}^3)$. Also, we put ${\widehat{\psi }}(\xi ):= 1 -\sum _{j=1}^\infty \widehat{\varphi _j}(\xi )$ for $\xi \in {\mathbb {R}}^3$.

Definition 2.2

• (i) Let $s\in {\mathbb {R}}$ and $1\le p,q \le \infty$. The homogeneous Besov space ${\dot{B}}^{s}_{p,q}({\mathbb {R}}^3)$ is defined by

  $$\begin{aligned} {\dot{B}}^{s}_{p,q}({\mathbb {R}}^3):= & {} \left\{ f \in {\mathscr {S}}^\prime ({\mathbb {R}}^3)/{\mathscr {P}}({\mathbb {R}}^3) \bigm | \Vert f\Vert _{{\dot{B}}^{s}_{p,q}}<+\infty \right\} ,\\ \Vert f\Vert _{{\dot{B}}^{s}_{p,q}}:= & {} \left\| \left\{ 2^{sj} \left\| \Delta _j f \right\| _{L^p} \right\} _{j \in {\mathbb {Z}}} \right\| _{\ell ^{q}({\mathbb {Z}})}. \end{aligned}$$

• (ii) Let $s\in {\mathbb {R}}$ and $1\le p,q \le \infty$. The inhomogeneous Besov space $B^{s}_{p,q}({\mathbb {R}}^3)$ is defined by

  $$\begin{aligned} B^{s}_{p,q}({\mathbb {R}}^3):= & {} \left\{ f \in {\mathscr {S}}^\prime ({\mathbb {R}}^3) \bigm | \Vert f\Vert _{B^{s}_{p,q}}<+\infty \right\} ,\\ \Vert f\Vert _{B^{s}_{p,q}}:= & {} \Vert \psi * f \Vert _{L^p} + \left\| \left\{ 2^{sj} \left\| \Delta _j f \right\| _{L^p} \right\} _{j \in {\mathbb {N}}} \right\| _{\ell ^{q}({\mathbb {N}})}. \end{aligned}$$

For $s>0$, it is known that the norm equivalence

$$\begin{aligned} c_1 \Vert f \Vert _{B^s_{p,q}} \le \Vert f \Vert _{L^p} + \Vert f \Vert _{{\dot{B}}^s_{p,q}} \le c_2 \Vert f \Vert _{B^s_{p,q}} \end{aligned}$$

(2.1)

holds for $1 \le p,q \le \infty$ with some positive constants $c_1$ and $c_2$.

Finally, we recall the known results on the linear estimates for the semigroup $e^{-tA_{\Omega }}$. We set

$$\begin{aligned} {\mathcal {G}}_{\pm }(\tau )[f](x):= \frac{1}{(2\pi )^3} \int _{{\mathbb {R}}^{3}}e^{ix\cdot \xi }e^{\pm i\tau \frac{\xi _{3}}{|\xi |}} {\widehat{f}}(\xi ) \, d\xi , \quad x\in {\mathbb {R}}^3, \, \tau \in {\mathbb {R}}. \end{aligned}$$

(2.2)

Then, the linear semigroup $e^{-tA_\Omega }$ generated by the linear operator $A_\Omega = -\Delta +\Omega {\mathbb {P}}J{\mathbb {P}}$ is explicitly written as

$$\begin{aligned} e^{-tA_\Omega } f&={\mathcal {F}}^{-1} \left[ e^{-t|\xi |^{2}} \left\{ \cos \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) I{\widehat{f}}(\xi ) +\sin \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) R(\xi ){\widehat{f}}(\xi ) \right\} \right] \nonumber \\&= \frac{1}{2}{\mathcal {G}}_+(\Omega t)e^{t\Delta }(I+{\mathcal {R}})f + \frac{1}{2}{\mathcal {G}}_-(\Omega t)e^{t\Delta }(I-{\mathcal {R}})f \end{aligned}$$

(2.3)

for $f \in L^2({\mathbb {R}}^3)^3$ with $\nabla \cdot f = 0$, where

$$\begin{aligned} {\mathcal {R}}=\begin{pmatrix} 0 &{} R_3 &{} - R_2 \\ -R_3 &{} 0 &{} R_1 \\ R_2 &{} - R_1 &{} 0 \\ \end{pmatrix} \end{aligned}$$

(2.4)

and $R_j=-\partial _{x_j}(-\Delta )^{-\frac{1}{2}}$ is the Riesz transform. See [8, 10, 15] for the derivation of the explicit formula (2.3) of the semigroup $e^{-tA_\Omega }$.

The dispersion estimate for ${\mathcal {G}}_\pm (\tau )$ was obtained in [20].

Lemma 2.3

([20, Lemma 2.2]). For $2\le p \le \infty$, there exists a positive constant $C=C(p)$ such that

$$\begin{aligned} \Vert {\mathcal {G}}_{\pm }(\tau )f\Vert _{{\dot{B}}^{s}_{p,q}} \le C(1+|\tau |)^{-(1-\frac{2}{p})} \Vert f \Vert _{{\dot{B}}^{s+3(1-\frac{2}{p})}_{p^{\prime },q}} \end{aligned}$$

for all $\tau \in {\mathbb {R}}, \, s \in {\mathbb {R}}, \, 1\le q \le \infty$ and $f \in {\dot{B}}^{s+3( 1-\frac{2}{p} ) }_{p^{\prime },q}({\mathbb {R}}^{3})$ with $1/p+1/p^{\prime }=1$.

We end this section by recalling the $L^q$-$L^p$ smoothing estimates for the linear semigruops $e^{-tA_\Omega }$ and $e^{t\Delta }$.

Lemma 2.4

([10, Proposition 2.4]). Let $\alpha \in ({\mathbb {N}} \cup \{0 \})^3$ and $1 \le q \le 2 \le p \le \infty$. Then, there exists a constant $C=C(\alpha ,p,q)>0$ such that

$$\begin{aligned} \Vert \partial _x^\alpha e^{-tA_{\Omega }}f\Vert _{L^p} \le C t^{-\frac{|\alpha |}{2}-\frac{3}{2}(\frac{1}{q}-\frac{1}{p})} \Vert f\Vert _{L^{q}} \end{aligned}$$

for all $\Omega \in {\mathbb {R}}, \, t>0$ and $f \in L^q({\mathbb {R}}^{3})^3$ with $\nabla \cdot f=0$.

Lemma 2.5

([20, Lemma 3.2], [19, Lemma 2.5]). Let $s \ge 0, \, 2 \le p<\infty$ and $1 \le q \le p^\prime$. Then, there exists a constant $C=C(s,p,q)>0$ such that

$$\begin{aligned} \Vert e^{-tA_{\Omega }}f\Vert _{{\dot{H}}_{p}^{s}} \le C t^{-\frac{s}{2}-\frac{3}{2}(\frac{1}{q}-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \Vert f\Vert _{L^{q}} \end{aligned}$$

for all $\Omega \in {\mathbb {R}}, \, t>0$ and $f \in L^q({\mathbb {R}}^{3})^3$ with $\nabla \cdot f=0$.

Lemma 2.6

([22, Lemma 2.2]). For $-\infty<s_0 \le s_1 < \infty$, there exists a positive constant $C=C(s_0, s_1)$ such that

$$\begin{aligned} \Vert e^{t\Delta } f \Vert _{{\dot{B}}^{s_1}_{p,q}} \le C t^{-\frac{1}{2}(s_1 - s_0)} \Vert f \Vert _{{\dot{B}}^{s_0}_{p,q}} \end{aligned}$$

for all $t>0, \, 1\le p,q \le \infty$ and $f \in {\dot{B}}^{s_0}_{p,q}({\mathbb {R}}^3)$.

3 Linear Decay Estimates and Asymptotics

In this section, we shall establish the temporal decay estimates for the linear solution $e^{-tA_\Omega }u_0$ when $u_0 \in L^1({\mathbb {R}}^3)$ or $(1+|x|)u_0 \in L^1({\mathbb {R}}^3)$. Furthermore, we obtain the asymptotic profile of the linear solution $e^{-tA_\Omega }u_0$ as $t\rightarrow \infty$.

3.1 Linear Decay Estimates

Let us set

$$\begin{aligned} H_{\Omega }(\xi ,t):=\cos \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) I +\sin \left( \Omega \frac{\xi _{3}}{|\xi |}t\right) R(\xi ), \quad K_{\Omega }(x,t):={\mathcal {F}}^{-1}\left[ e^{-t|\xi |^{2}}H_{\Omega }(\xi ,t)\right] (x), \end{aligned}$$

(3.1)

where $R(\xi )$ is the skew-symmetric matrix defined by (1.19). Then, it follows from (2.3) that the linear solution $e^{-tA_\Omega }u_0$ can be written as

$$\begin{aligned} e^{-tA_\Omega } u_0(x) = K_\Omega (\cdot , t)*u_0(x) = \int _{{\mathbb {R}}^3} K_\Omega (x-y, t) u_0(y) \, dy. \end{aligned}$$

(3.2)

We firstly show the $L^p$ estimates for the integral kernel $K_\Omega (\cdot , t)$. Let $G_t(x)$ be the Gauss kernel in ${\mathbb {R}}^3$, which is defined as

$$\begin{aligned} G_t(x) := \frac{1}{(4\pi t)^{\frac{3}{2}}} e^{-\frac{|x|^2}{4t}}, \quad t>0, \, x \in {\mathbb {R}}^3. \end{aligned}$$

Note that it holds $\widehat{G_t}(\xi ) = e^{-t|\xi |^2}$ and $G_1 \in {\mathscr {S}}({\mathbb {R}}^3)$.

Lemma 3.1

For $2 \le p \le \infty$ and $\alpha \in ({\mathbb {N}}\cup \{0\})^{3}$, there exists a positive constant $C=C(p,\alpha )$ such that

$$\begin{aligned} \left\| \partial _x^{\alpha } K_{\Omega }(\cdot ,t)\right\| _{L^p} \le Ct^{-\frac{|\alpha |}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

(3.3)

for all $\Omega \in {\mathbb {R}}$ and all $t>0$.

Proof

Since $H_\Omega (\xi , t)$ is homogeneous of degree 0 in $\xi$, the change of variable $\xi \mapsto \frac{\xi }{\sqrt{t}}$ gives

$$\begin{aligned} \partial _{x}^{\alpha }K_{\Omega }(x,t)&=\frac{1}{(2\pi )^3} \int _{{\mathbb {R}}^{3}}(i\xi )^{\alpha }e^{ix\cdot \xi }e^{-t|\xi |^{2}}H_{\Omega }(\xi ,t)\,d\xi \nonumber \\&=\frac{t^{-\frac{|\alpha |}{2}-\frac{3}{2}}}{(2\pi )^3} \int _{{\mathbb {R}}^{3}}e^{ix\cdot \frac{\xi }{\sqrt{t}}} (i\xi )^{\alpha }e^{-|\xi |^{2}}H_{\Omega }(\xi ,t)\,d\xi \nonumber \\&=t^{-\frac{|\alpha |}{2}-\frac{3}{2}}{\mathcal {F}}^{-1}\left[ \widehat{\partial _x^\alpha G_1}(\xi ) H_{\Omega }(\xi ,t) \right] \left( \frac{x}{\sqrt{t}}\right) . \end{aligned}$$

(3.4)

Moreover, we directly calculate as

$$\begin{aligned}&{\mathcal {F}}^{-1} \left[ \widehat{\partial _x^\alpha G_1}(\xi ) H_{\Omega }(\xi ,t) \right] \left( \frac{x}{\sqrt{t}}\right) \nonumber \\&\quad =\int _{{\mathbb {R}}^{3}} e^{i \frac{x}{\sqrt{t}} \cdot \xi }\left\{ \frac{1}{2}(e^{i\Omega \frac{\xi _{3}}{|\xi |}t}+e^{-i\Omega \frac{\xi _{3}}{|\xi |}t})I +\frac{1}{2i}(e^{i\Omega \frac{\xi _{3}}{|\xi |}t}-e^{-i\Omega \frac{\xi _{3}}{|\xi |}t})R(\xi ) \right\} \widehat{\partial _x^\alpha G_1}(\xi ) \, \frac{d\xi }{(2\pi )^3} \nonumber \\&\quad =\frac{1}{2}\int _{{\mathbb {R}}^{3}}e^{i \frac{x}{\sqrt{t}} \cdot \xi } e^{i\Omega \frac{\xi _{3}}{|\xi |}t}(I-iR(\xi )) \widehat{\partial _x^\alpha G_1}(\xi )\,\frac{d\xi }{(2\pi )^3}\nonumber \\&\qquad +\frac{1}{2}\int _{{\mathbb {R}}^{3}}e^{i \frac{x}{\sqrt{t}} \cdot \xi } e^{-i\Omega \frac{\xi _{3}}{|\xi |}t}(I+iR(\xi )) \widehat{\partial _x^\alpha G_1}(\xi )\,\frac{d\xi }{(2\pi )^3}\nonumber \\&\quad =\frac{1}{2}{\mathcal {G}}_{+}(\Omega t)[(I+{\mathcal {R}})\partial _x^\alpha G_1]\left( \frac{x}{\sqrt{t}}\right) +\frac{1}{2}{\mathcal {G}}_{-}(\Omega t)[(I-{\mathcal {R}})\partial _x^\alpha G_1]\left( \frac{x}{\sqrt{t}}\right) , \end{aligned}$$

(3.5)

where ${\mathcal {R}}$ is defined in (2.4). Therefore, it follows from (3.4) to (3.5) that

$$\begin{aligned}&\Vert \partial _{x}^{\alpha }K_{\Omega }(\cdot ,t) \Vert _{L^p} \nonumber \\&\quad \leqslant \frac{t^{-\frac{|\alpha |}{2}-\frac{3}{2}\left( 1-\frac{1}{p}\right) }}{2} \left\{ \left\| {\mathcal {G}}_{+}(\Omega t)[(I+{\mathcal {R}})\partial _x^\alpha G_1] \right\| _{L^p} + \left\| {\mathcal {G}}_{-}(\Omega t)[(I-{\mathcal {R}})\partial _x^\alpha G_1] \right\| _{L^p} \right\} . \end{aligned}$$

(3.6)

We first consider the case $2 \le p<\infty$. It follows from Lemma 2.3 and the continuous embedding ${\dot{B}}^{0}_{p,2}({\mathbb {R}}^{3}) \hookrightarrow L^{p}({\mathbb {R}}^{3})$ that

$$\begin{aligned} \left\| {\mathcal {G}}_{\pm }(\Omega t)[(I \pm {\mathcal {R}})\partial _x^\alpha G_1] \right\| _{L^p}&\leqslant C \left\| {\mathcal {G}}_{\pm }(\Omega t)[(I \pm {\mathcal {R}})\partial _x^\alpha G_1] \right\| _{{\dot{B}}^0_{p,2}} \nonumber \\&\leqslant C (1+|\Omega |t)^{-(1-\frac{2}{p})} \left\| (I \pm {\mathcal {R}})\partial _x^\alpha G_1 \right\| _{{\dot{B}}^{3(1-\frac{2}{p})}_{p^\prime ,2}} \nonumber \\&\leqslant C (1+|\Omega |t)^{-(1-\frac{2}{p})} \left\| G_1 \right\| _{{\dot{B}}^{|\alpha |+3(1-\frac{2}{p})}_{p^\prime ,2}}. \end{aligned}$$

(3.7)

Since $G_1 \in {\mathscr {S}}({\mathbb {R}}^3)$, we see by (2.1) that

$$\begin{aligned} \left\| G_1 \right\| _{{\dot{B}}^{|\alpha |}_{2,2}} \simeq \Vert G_1 \Vert _{{\dot{H}}^{|\alpha |}}<\infty \quad \text {and} \quad \left\| G_1 \right\| _{{\dot{B}}^{|\alpha |+3(1-\frac{2}{p})}_{p^\prime ,2}} \leqslant C \left\| G_1 \right\| _{B^{|\alpha |+3(1-\frac{2}{p})}_{p^\prime ,2}}<\infty \end{aligned}$$

(3.8)

for $p=2$ and $2<p<\infty$, respectively. For the case $p=\infty$, we have by Lemma 2.3 and the continuous embedding ${\dot{B}}^{0}_{\infty ,1}({\mathbb {R}}^{3}) \hookrightarrow L^{\infty }({\mathbb {R}}^{3})$ that

$$\begin{aligned} \left\| {\mathcal {G}}_{\pm }(\Omega t)[(I \pm {\mathcal {R}})\partial _x^\alpha G_1] \right\| _{L^\infty }&\leqslant C \left\| {\mathcal {G}}_{\pm }(\Omega t)[(I \pm {\mathcal {R}})\partial _x^\alpha G_1] \right\| _{{\dot{B}}^0_{\infty ,1}} \nonumber \\&\leqslant C (1+|\Omega |t)^{-1} \left\| (I \pm {\mathcal {R}})\partial _x^\alpha G_1 \right\| _{{\dot{B}}^{3}_{1,1}} \nonumber \\&\leqslant C (1+|\Omega |t)^{-1} \left\| G_1 \right\| _{{\dot{B}}^{|\alpha |+3}_{1,1}} \end{aligned}$$

(3.9)

and $\left\| G_1 \right\| _{{\dot{B}}^{|\alpha |+3}_{1,1}} \le C \left\| G_1 \right\| _{{B}^{|\alpha |+3}_{1,1}} <\infty .$ Therefore, we obtain from (3.6) to (3.9) that

$$\begin{aligned} \Vert \partial _{x}^{\alpha }K_{\Omega }(\cdot ,t) \Vert _{L^p} \le C t^{-\frac{|\alpha |}{2}-\frac{3}{2}\left( 1-\frac{1}{p}\right) } (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

for all $t>0$. This completes the proof of Lemma 3.1. $\square$

Applying Lemma 3.1, we show the following $L^1$-$L^p$ temporal decay estimates for the linear semigroup $e^{-tA_\Omega }$.

Lemma 3.2

• (1) For $2 \le p \le \infty$ and $\alpha \in ({\mathbb {N}} \cup \{0\})^{3}$, there exists a positive constant $C=C(p,\alpha )$ such that

  $$\begin{aligned} \Vert \partial _{x}^{\alpha }e^{-tA_{\Omega }}f\Vert _{L^{p}} \le Ct^{-\frac{|\alpha |}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \Vert f \Vert _{L^1} \end{aligned}$$

  (3.10)

  for all $t>0, \, \Omega \in {\mathbb {R}}$ and $f \in L^1({\mathbb {R}}^3)^3$ satisfying $\nabla \cdot f = 0$.

• (2) For $2 \le p \le \infty$ and $\alpha \in ({\mathbb {N}} \cup \{0\})^{3}$, there exists a positive constant $C=C(p,\alpha )$ such that

  $$\begin{aligned} \Vert \partial _{x}^{\alpha }e^{-tA_{\Omega }}f\Vert _{L^{p}} \le Ct^{-\frac{|\alpha |+1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \Vert |x| f \Vert _{L^1} \end{aligned}$$

  (3.11)

  for all $t>0, \, \Omega \in {\mathbb {R}}$ and $f \in L^1({\mathbb {R}}^3)^3$ satisfying $\nabla \cdot f = 0$ and $|x| f \in L^1({\mathbb {R}}^3)^3$.

Remark 3.3

We remark that the temporal decay estimate (3.10) for $f \in L^1({\mathbb {R}}^3)$ has already been shown by Kim [19, Lemma 2.5]. Here, we shall give an alternative proof by using Lemma 3.1.

Proof of Lemma 3.2

(1) Applying the Hausdorff-Young inequality and Lemma 3.1, we have by (3.2) that

$$\begin{aligned} \Vert \partial _{x}^{\alpha }e^{-tA_{\Omega }}f\Vert _{L^{p}} \le \Vert \partial _{x}^{\alpha }K_{\Omega }(\cdot ,t)\Vert _{L^{p}}\Vert f\Vert _{L^{1}} \le Ct^{-\frac{|\alpha |}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})}\Vert f \Vert _{L^1}. \end{aligned}$$

(2) Since $f \in L^1({\mathbb {R}}^3)^3$ and $\nabla \cdot f=0$, we see that it holds

$$\begin{aligned} {\widehat{f}}(0)=\int _{{\mathbb {R}}^{3}}f(y)\,dy=0. \end{aligned}$$

(3.12)

Then, applying the mean value theorem, the Minkowski inequality and Lemma 3.1, we obtain

$$\begin{aligned} \Vert \partial _{x}^{\alpha }e^{-tA_{\Omega }}f\Vert _{L^{p}}&=\left\| \int _{{\mathbb {R}}^{3}}\partial _{x}^{\alpha }K_{\Omega }(\cdot -y,t)f(y)\,dy\right\| _{L^{p}}\\&=\left\| \int _{{\mathbb {R}}^{3}}\{\partial _{x}^{\alpha }K_{\Omega }(\cdot -y,t)-\partial _{x}^{\alpha }K_{\Omega }(\cdot ,t)\}f(y)\,dy\right\| _{L^{p}}\\&=\left\| \int _{{\mathbb {R}}^{3}} \sum _{j=1}^3 \int _0^1 \partial _{x_j}\partial _{x}^{\alpha }K_{\Omega }(\cdot -\theta y,t) (-y_j) \, d\theta f(y)\,dy\right\| _{L^{p}}\\&\le \sum _{j=1}^3 \int _{{\mathbb {R}}^{3}}\!\!\int _{0}^{1} \Vert \partial _{x_j} \partial _{x}^{\alpha }K_{\Omega }(\cdot -\theta y,t)\Vert _{L^{p}}\,d\theta |y_j||f(y)|\,dy\\&= \sum _{j=1}^3 \Vert \partial _{x_j} \partial _{x}^{\alpha }K_{\Omega }(\cdot ,t)\Vert _{L^{p}}\int _{{\mathbb {R}}^{3}}|y_j||f(y)|\,dy\\&\le Ct^{-\frac{|\alpha |+1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \Vert |x| f \Vert _{L^1}. \end{aligned}$$

This completes the proof of Lemma 3.2. $\square$

3.2 Linear Asymptotics

In this subsection, we shall show the following asymptotic profiles of the linear solution $e^{-tA_\Omega }u_0$ as t goes to infinity.

Theorem 3.4

• (1) Suppose that $u_0$ satisfies $u_0 \in L^{1}({\mathbb {R}}^{3})^3$ and $\nabla \cdot u_0=0$. Then, for $2 \le p \le \infty$, it holds that

  $$\begin{aligned} \lim _{t \rightarrow \infty }(1+|\Omega |t)^{1-\frac{2}{p}}t^{\frac{3}{2}(1-\frac{1}{p})} \Vert e^{-tA_{\Omega }}u_{0}\Vert _{L^{p}}=0. \end{aligned}$$

  (3.13)

• (2) Let $m \in {\mathbb {N}}$. Suppose that $u_0$ satisfies $(1+|x|)^m u_0 \in L^{1}({\mathbb {R}}^{3})^3$ and $\nabla \cdot u_0=0$. Then, for $2\le p \le \infty$, it holds that

  $$\begin{aligned} \begin{aligned}&\lim _{t\rightarrow \infty }(1+|\Omega |t)^{1-\frac{2}{p}}t^{\frac{m}{2}+\frac{3}{2}(1-\frac{1}{p})} \left\| e^{-tA_{\Omega }}u_{0} - \sum _{1\leqslant |\alpha |\leqslant m}\!\!\!\!\frac{(-1)^{|\alpha |}}{\alpha !}(\partial _{x}^{\alpha }K_{\Omega })(\cdot ,t)\int _{{\mathbb {R}}^{3}}y^{\alpha }u_{0}(y)\,dy\right\| _{L^p}\\&\quad =0. \end{aligned} \end{aligned}$$

  (3.14)

Proof

(1) Since $u_0 \in L^1({\mathbb {R}}^3)^3$ and $\nabla \cdot u_0=0$, it holds $\int _{{\mathbb {R}}^3}u_0(y) \, dy=0$ and then

$$\begin{aligned} e^{-tA_\Omega }u_0(x) = \int _{{\mathbb {R}}^{3}} \{K_{\Omega }(x-y,t)-K_{\Omega }(x,t)\} u_{0}(y)\,dy. \end{aligned}$$

Similarly to (3.4) and (3.5), we have

$$\begin{aligned} K_\Omega (x,t) = \frac{t^{-\frac{3}{2}}}{2}\sum _{\sigma \in \{ \pm \}} {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}})G_1 \right] \left( \frac{x}{\sqrt{t}} \right) . \end{aligned}$$

(3.15)

Therefore, we see that

$$\begin{aligned}&\Vert e^{-tA_\Omega } u_0 \Vert _{L^p} \nonumber \\&\quad \le \frac{t^{-\frac{3}{2}}}{2}\sum _{\sigma \in \{ \pm \}} \int _{{\mathbb {R}}^3} \bigg \Vert {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}})G_1 \right] \left( \frac{\cdot -y}{\sqrt{t}} \right) \nonumber \\&\qquad - {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}})G_1 \right] \left( \frac{\cdot }{\sqrt{t}} \right) \bigg \Vert _{L^p}|u_0(y)| \, dy \nonumber \\&\quad =\frac{t^{-\frac{3}{2}(1-\frac{1}{p})}}{2}\sum _{\sigma \in \{ \pm \}} \int _{{\mathbb {R}}^3} \bigg \Vert {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}})G_1 \right] \left( \cdot - \frac{y}{\sqrt{t}} \right) \nonumber \\&\qquad - {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}})G_1 \right] (\cdot ) \bigg \Vert _{L^p}|u_0(y)| \, dy \nonumber \\&\quad =\frac{t^{-\frac{3}{2}(1-\frac{1}{p})}}{2}\sum _{\sigma \in \{ \pm \}} \int _{{\mathbb {R}}^3} \left\| {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}}) \left\{ G_1(\cdot -yt^{-\frac{1}{2}}) - G_1(\cdot ) \right\} \right] \right\| _{L^p} |u_0(y)| \, dy. \end{aligned}$$

(3.16)

In the case $p=2$, we have (3.16) that

$$\begin{aligned} t^{\frac{3}{4}} \Vert e^{-tA_\Omega } u_0 \Vert _{L^2} \le C \int _{{\mathbb {R}}^3} \left\| G_1(\cdot -yt^{-\frac{1}{2}}) - G_1(\cdot ) \right\| _{L^2} |u_0(y)| \, dy. \end{aligned}$$

Hence the desired result (3.13) follows from the dominated convergence theorem.

Next we consider the case $2<p<\infty$. Applying the embedding ${\dot{B}}^0_{p,2}({\mathbb {R}}^3) \hookrightarrow L^p({\mathbb {R}}^3)$ and Lemma 2.3 to (3.16), we have

$$\begin{aligned}&\Vert e^{-tA_\Omega } u_0 \Vert _{L^p} \nonumber \\&\le C t^{-\frac{3}{2}(1-\frac{1}{p})} \sum _{\sigma \in \{ \pm \}} \int _{{\mathbb {R}}^3} \left\| {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}}) \left\{ G_1(\cdot -yt^{-\frac{1}{2}}) - G_1(\cdot ) \right\} \right] \right\| _{{\dot{B}}^0_{p,2}} |u_0(y)| \, dy \nonumber \\&\le C t^{-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \int _{{\mathbb {R}}^3} \left\| G_1(\cdot -yt^{-\frac{1}{2}}) - G_1(\cdot ) \right\| _{{\dot{B}}^{3(1-\frac{2}{p})}_{p^\prime ,2}} |u_0(y)| \, dy. \end{aligned}$$

(3.17)

Here, since it holds

$$\begin{aligned} G_1(x -yt^{-\frac{1}{2}}) - G_1(x) = e^{\frac{\Delta }{2}} \left[ G_{\frac{1}{2}}(\cdot -yt^{-\frac{1}{2}}) - G_{\frac{1}{2}}(\cdot ) \right] (x), \end{aligned}$$

it follows from Lemma 2.6 and the embedding $L^{p^\prime }({\mathbb {R}}^3) \hookrightarrow {\dot{B}}^0_{p^\prime ,2}({\mathbb {R}}^3)$ that

$$\begin{aligned} \left\| G_1(\cdot -yt^{-\frac{1}{2}}) - G_1(\cdot ) \right\| _{{\dot{B}}^{3(1-\frac{2}{p})}_{p^\prime ,2}}&\le C \left( \frac{1}{2} \right) ^{\!-\frac{3}{2}(1-\frac{2}{p})} \left\| G_{\frac{1}{2}}(\cdot -yt^{-\frac{1}{2}}) - G_{\frac{1}{2}}(\cdot ) \right\| _{{\dot{B}}^0_{p^\prime , 2}} \nonumber \\&\le C \left\| G_{\frac{1}{2}}(\cdot -yt^{-\frac{1}{2}}) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }}. \end{aligned}$$

(3.18)

Hence we have by (3.17) and (3.18) that

$$\begin{aligned} t^{\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \Vert e^{-tA_\Omega } u_0 \Vert _{L^p} \le C \int _{{\mathbb {R}}^3} \left\| G_{\frac{1}{2}}(\cdot -yt^{-\frac{1}{2}}) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }} |u_0(y)| \, dy. \end{aligned}$$

Then, the dominated convergence theorem yields (3.13) for $2<p<\infty$.

Finally, we treat the case $p=\infty$. Similarly to (3.17), we apply the embedding ${\dot{B}}^0_{\infty , 1}({\mathbb {R}}^3) \hookrightarrow L^\infty ({\mathbb {R}}^3)$ and Lemma 2.3 to (3.16) to obtain

$$\begin{aligned} \Vert e^{-tA_\Omega } u_0 \Vert _{L^\infty } \le C t^{-\frac{3}{2}}(1+|\Omega |t)^{-1} \int _{{\mathbb {R}}^3} \left\| G_1(\cdot -yt^{-\frac{1}{2}}) - G_1(\cdot ) \right\| _{{\dot{B}}^{3}_{1,1}} |u_0(y)| \, dy. \end{aligned}$$

Note that it holds

$$\begin{aligned} \Vert f \Vert _{{\dot{B}}^3_{1,1}} \le C \Vert f \Vert _{B^3_{1,1}} \le C \Vert f \Vert _{B^4_{1,\infty }} \simeq C \Vert (1-\Delta )^2 f \Vert _{B^{0}_{1,\infty }} \le C \Vert (1-\Delta )^2 f \Vert _{L^1}. \end{aligned}$$

Hence we have by the dominated convergence theorem that

$$\begin{aligned}&t^{\frac{3}{2}}(1+|\Omega |t)\Vert e^{-tA_\Omega } u_0 \Vert _{L^\infty } \nonumber \\&\quad \le C\int _{{\mathbb {R}}^3} \left\| (1-\Delta )^2G_1(\cdot -yt^{-\frac{1}{2}}) - (1-\Delta )^2G_1(\cdot ) \right\| _{L^1} |u_0(y)| \, dy \rightarrow 0 \end{aligned}$$

as $t\rightarrow \infty$. This completes the proof of Theorem 3.4 (1).

(2) By the Taylor theorem, we have

$$\begin{aligned}&K_\Omega (x-y, t) - K_\Omega (x,t) \nonumber \\&\quad = \sum _{1 \le |\alpha | \le m-1} \frac{\partial _x^\alpha K_\Omega (x,t)}{\alpha !}(-y)^\alpha + \sum _{|\alpha |=m}\frac{m}{\alpha !} \int _0^1 (1-\theta )^{m-1} \partial _x^\alpha K_\Omega (x-\theta y,t) \, d\theta (-y)^\alpha \nonumber \\&\quad = \sum _{1 \le |\alpha | \le m} \frac{(-1)^{|\alpha |}}{\alpha !} \partial _x^\alpha K_\Omega (x,t) y^\alpha \nonumber \\&\qquad + \sum _{|\alpha |=m}\frac{m}{\alpha !} \int _0^1 (1-\theta )^{m-1} \left\{ \partial _x^\alpha K_\Omega (x-\theta y,t) - \partial _x^\alpha K_\Omega (x,t) \right\} d\theta (-y)^\alpha . \end{aligned}$$

(3.19)

Then, since it holds $\int _{{\mathbb {R}}^3}u_0(y) \, dy=0$, we have by (3.19) that

$$\begin{aligned}&e^{-tA_\Omega }u_0(x) \nonumber \\&\quad = \int _{{\mathbb {R}}^{3}} \{K_{\Omega }(x-y,t)-K_{\Omega }(x,t)\} u_{0}(y)\,dy \nonumber \\&\quad = \sum _{1 \le |\alpha | \le m} \frac{(-1)^{|\alpha |}}{\alpha !} \partial _x^\alpha K_\Omega (x,t) \int _{{\mathbb {R}}^{3}} y^\alpha u_0(y) \, dy \nonumber \\&\qquad + \sum _{|\alpha |=m}\frac{m}{\alpha !} \int _{{\mathbb {R}}^3} \int _0^1 (1-\theta )^{m-1} \left\{ \partial _x^\alpha K_\Omega (x-\theta y,t) - \partial _x^\alpha K_\Omega (x,t) \right\} d\theta (-y)^\alpha u_0(y) \, dy, \end{aligned}$$

which yields

$$\begin{aligned}&\left\| e^{-tA_\Omega }u_0 - \sum _{1 \le |\alpha | \le m} \frac{(-1)^{|\alpha |}}{\alpha !} \partial _x^\alpha K_\Omega (\cdot ,t) \int _{{\mathbb {R}}^{3}} y^\alpha u_0(y) \, dy \right\| _{L^p} \nonumber \\&\quad \le C_m \sum _{|\alpha |=m} \int _{{\mathbb {R}}^3} \int _0^1 \left\| \partial _x^\alpha K_\Omega (\cdot -\theta y,t) - \partial _x^\alpha K_\Omega (\cdot ,t) \right\| _{L^p} |y|^m |u_0(y)| \, d\theta dy. \end{aligned}$$

(3.20)

Similarly to (3.4) and (3.5), we have

$$\begin{aligned} \partial _{x}^{\alpha }K_{\Omega }(x,t) =\frac{1}{2} t^{-\frac{|\alpha |}{2}-\frac{3}{2}} \sum _{\sigma \in \{ \pm \}} {\mathcal {G}}_\sigma (\Omega t)\left[ (I+\sigma {\mathcal {R}}) \partial _x^\alpha G_1 \right] \left( \frac{x}{\sqrt{t}} \right) . \end{aligned}$$

(3.21)

Therefore, similarly to (3.16), we obtain by (3.20) and (3.21) that

$$\begin{aligned}&t^{\frac{m}{2}+\frac{3}{2}(1-\frac{1}{p})} \left\| e^{-tA_\Omega }u_0 - \sum _{1 \le |\alpha | \le m} \frac{(-1)^{|\alpha |}}{\alpha !} \partial _x^\alpha K_\Omega (\cdot ,t) \int _{{\mathbb {R}}^{3}} y^\alpha u_0(y) \, dy \right\| _{L^p} \nonumber \\&\le C_m \sum _{|\alpha |=m}\sum _{\sigma \in \{ \pm \}} \int _{{\mathbb {R}}^3} \int _0^1 \left\| {\mathcal {G}}_\sigma (\Omega t) \left[ (I+\sigma {\mathcal {R}}) \left\{ \partial _x^\alpha G_1(\cdot -y \theta t^{-\frac{1}{2}}) - \partial _x^\alpha G_1(\cdot ) \right\} \right] \right\| _{L^p} \nonumber \\&\qquad \times |y|^m |u_0(y)| \, d\theta dy. \end{aligned}$$

(3.22)

Then, since $|y|^m |u_0(y)|$ is in $L^1({\mathbb {R}}^3_y \times (0,1)_\theta )$, we can apply the exactly same arguments as Theorem 3.4 (1) to (3.22), and obtain the desired asymptotics result. $\square$

4 Nonlinear Decay Estimates

In this section, we adapt the ideas in [1, 29], and show the $L^p$ temporal decay estimates for the global solution u to (1.8).

4.1 $L^{2}$-decay Estimates

Lemma 4.1

Suppose that the exponents $s, q, \theta$ and the initial data $u_0 \in ({\dot{H}}^s \cap L^1)({\mathbb {R}}^3)^3$ satisfy the asuumptions in Theorem 1.1, and let $u \in C([0,\infty );{\dot{H}}^s({\mathbb {R}}^3))^3$ be the unique global solution to (1.8) constructed in Theorem 1.1. Then, there exists a absolute constant $C>0$ such that

$$\begin{aligned} \begin{aligned}&\Vert u(t) \Vert _{L^2}^2 e^{\int _0^t g(r)^2 dr} \\&\quad \le \Vert u_0 \Vert _{L^2}^2 + C \int _0^t g(s)^2 e^{\int _0^s g(r)^2 dr} \left\{ \Vert e^{s\Delta }u_0 \Vert _{L^2}^2 + g(s)^5\left( \int _0^s \Vert u(r) \Vert _{L^2}^2 dr \right) ^{\!2} \right\} ds \end{aligned} \end{aligned}$$

for all $t \ge 0$ and all bounded positive function $g \in C([0,\infty );(0,\infty ))$.

Proof

We remark that the inequality in Lemma 4.1 was obtained by Wiegner [29, (2.1)] for global weak solutions to the original Navier–Stokes equations. Since $u_0 \in ({\dot{H}}^s \cap L^1)({\mathbb {R}}^3)^3$, we see by the Sobolev embedding ${\dot{H}}^s({\mathbb {R}}^3) \hookrightarrow L^q({\mathbb {R}}^3)$ with $\frac{1}{q}=\frac{1}{2}-\frac{s}{3}$ and the interpolation inequality that

$$\begin{aligned} \Vert u_0 \Vert _{L^2} \le \Vert u_0 \Vert _{L^1}^{\frac{2s}{3+2s}} \Vert u_0 \Vert _{L^q}^{\frac{3}{3+2s}} \le C \Vert u_0 \Vert _{L^1}^{\frac{2s}{3+2s}} \Vert u_0 \Vert _{{\dot{H}}^s}^{\frac{3}{3+2s}}. \end{aligned}$$

(4.1)

Hence the solution u also belongs to $C([0,\infty );L^2({\mathbb {R}}^3))^3$. Since there hold

$$\begin{aligned} \left\langle \Omega {\mathbb {P}}J{\mathbb {P}}u, u \right\rangle _{L^2}=0, \quad \left| \widehat{e^{-tA_\Omega }u_0}(\xi ) \right| = \left| \widehat{e^{t\Delta }u_0}(\xi ) \right| , \end{aligned}$$

we can apply the same argument as [29, (2.1)] and obtain the desired inequality. $\square$

Lemma 4.2

Suppose that the exponents $s, q, \theta$ and the initial data $u_0 \in ({\dot{H}}^s \cap L^1)({\mathbb {R}}^3)^3$ satisfy the asuumptions in Theorem 1.1, and let $u \in C([0,\infty );{\dot{H}}^s({\mathbb {R}}^3))^3$ be the unique global solution to (1.8) constructed in Theorem 1.1.

• (1) There exists a positive constant $C=C(\Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})$ such that

  $$\begin{aligned} \Vert u(t) \Vert _{L^2} \le C(1+t)^{-\frac{3}{4}} \end{aligned}$$

  for all $t>0$.

• (2) Assume further that $|x| u_0 {\in } L^1({\mathbb {R}}^3)^3$. Then, there exists a positive constant $C{=}C(\Vert |x|u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})$ such that

  $$\begin{aligned} \Vert u(t) \Vert _{L^2} \le C(1+t)^{-\frac{5}{4}} \end{aligned}$$

  for all $t>0$.

Proof

We follow the same argument as [29]. Firstly, suppose that there hold

$$\begin{aligned} \Vert e^{t\Delta } u_0 \Vert _{L^2}\le C_1(1+t)^{-\frac{\alpha _0}{2}}, \quad \Vert u(t) \Vert _{L^2} \leqslant C_2 (1+t)^{-\frac{\beta }{2}} \end{aligned}$$

(4.2)

with some $\alpha _0>0$ and $0\le \beta <1$. Then, take an exponent $\alpha$ and a function g(t) so that

$$\begin{aligned} \alpha > \max \left\{ \alpha _0, \,\frac{1}{2}+2\beta \right\} , \quad g(t) = \frac{\alpha ^{\frac{1}{2}}}{(1+t)^{\frac{1}{2}}}. \end{aligned}$$

Then, we see that $\int _0^t g(r)^2 dr = \log (1+t)^\alpha$, and Lemma 4.1 and (4.2) yield

$$\begin{aligned}&\Vert u(t) \Vert _{L^2}(1+t)^\alpha \\&\quad \le \Vert u_0 \Vert _{L^2}^2 +C \int _0^t \frac{\alpha }{1+s} (1+s)^\alpha \left\{ \Vert e^{s\Delta }u_0 \Vert _{L^2}^2 + \frac{\alpha ^{\frac{5}{2}}}{(1+s)^{\frac{5}{2}}} \left( \int _0^s \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!2} \right\} ds \\&\quad \le \Vert u_0 \Vert _{L^2}^2 +C \alpha \int _0^t (1+s)^{\alpha -1} \left\{ C_1^2 (1+s)^{-\alpha _0} + \frac{\alpha ^{\frac{5}{2}}C_2^4}{(1+s)^{\frac{5}{2}}} \left( \int _0^s (1+r)^{-\beta } \, dr \right) ^{\!2} \right\} ds \\&\quad \le \Vert u_0 \Vert _{L^2}^2 +CC_1^2 \alpha \int _0^t (1+s)^{\alpha -\alpha _0-1} \, ds + \frac{C C_2^4 \alpha ^{\frac{7}{2}}}{(1-\beta )^2} \int _0^t (1+s)^{\alpha -(\frac{1}{2}+2\beta )-1} \, ds \\&\quad \le \Vert u_0 \Vert _{L^2}^2 + \frac{CC_1^2 \alpha }{\alpha - \alpha _0}(1+t)^{\alpha -\alpha _0} + \frac{C C_2^4 \alpha ^{\frac{7}{2}}}{(1-\beta )^2\left\{ \alpha - (\frac{1}{2}+2\beta ) \right\} } (1+t)^{\alpha - (\frac{1}{2}+2\beta )}. \end{aligned}$$

Hence we have

$$\begin{aligned} \Vert u(t) \Vert _{L^2}^2&\le \Vert u_0 \Vert _{L^2}^2(1+t)^{-\alpha } + C C_1^2(1+t)^{-\alpha _0} + C C_2^4(1+t)^{-(\frac{1}{2}+2\beta )} \nonumber \\&\le \left\{ \Vert u_0 \Vert _{L^2} + C (C_1^2 + C_2^4) \right\} (1+t)^{-{\widetilde{\beta }}}, \end{aligned}$$

(4.3)

where

$$\begin{aligned} {\widetilde{\beta }} = \min \left\{ \alpha _0, \, \frac{1}{2}+2\beta \right\} . \end{aligned}$$

(4.4)

• (1) Let $u_0 \in ({\dot{H}}^s \cap L^1)({\mathbb {R}}^3)^3$. Note that it holds $u_0 \in L^2({\mathbb {R}}^3)^3$ by (4.1). Hence it follows from the smoothing estimates for the heat semigroup that

  $$\begin{aligned} \Vert e^{t\Delta } u_0 \Vert _{L^2} \le C t^{-\frac{3}{4}} \Vert u_0 \Vert _{L^1}, \quad \Vert e^{t\Delta } u_0 \Vert _{L^2} \le \Vert u_ 0 \Vert _{L^2}, \end{aligned}$$

  which yield

  $$\begin{aligned} \Vert e^{t\Delta } u_0 \Vert _{L^2} \le C(\Vert u_0\Vert _{L^1} + \Vert u_0\Vert _{L^2})(1+t)^{-\frac{3}{4}} \end{aligned}$$

  (4.5)

  for all $t \ge 0$. Moreover, taking the $L^2$ inner product of (1.8) with u(t) gives

  $$\begin{aligned} \frac{1}{2}\frac{d}{dt} \Vert u(t) \Vert _{L^2}^2 + \Vert \nabla u(t) \Vert _{L^2}^2 = 0. \end{aligned}$$

  Hence we have the energy equality:

  $$\begin{aligned} \Vert u(t) \Vert _{L^2}^2 + 2 \int _0^t \Vert \nabla u(\tau ) \Vert _{L^2}^2 \, d\tau = \Vert u_0 \Vert _{L^2}^2, \end{aligned}$$

  (4.6)

  which yields the estimate $\Vert u(t) \Vert _{L^2} \le \Vert u_0 \Vert _{L^2}$. Therefore, the estimates (4.2) hold for $\alpha _0=3/2$ and $\beta =0$, and then we have by (4.3) and (4.4) that

  $$\begin{aligned} \Vert u(t) \Vert _{L^2}^2 \le C (1+t)^{-\frac{1}{2}} \end{aligned}$$

  with some constant $C=C(\Vert u_0\Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$. Again, applying (4.3) and (4.4) with $\alpha _0=3/2$ and $\beta =1/2$, we obtain

  $$\begin{aligned} \Vert u(t) \Vert _{L^2}^2 \le C (1+t)^{-\frac{3}{2}} \end{aligned}$$

  (4.7)

  with some constant $C=C(\Vert u_0\Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$.

• (2) Assume further that $|x| u_0 \in L^1({\mathbb {R}}^3)^3$. Then, since there hold

  $$\begin{aligned} \Vert e^{t\Delta } u_0 \Vert _{L^2} \le C t^{-\frac{5}{4}} \left\| |x| u_0 \right\| _{L^1}, \quad \Vert e^{t\Delta } u_0 \Vert _{L^2} \le \Vert u_ 0 \Vert _{L^2}, \end{aligned}$$

  we have

  $$\begin{aligned} \Vert e^{t\Delta } u_0 \Vert _{L^2} \le C(\Vert |x| u_0\Vert _{L^1} + \Vert u_0\Vert _{L^2})(1+t)^{-\frac{5}{4}} \end{aligned}$$

  (4.8)

  for all $t \ge 0$. Hence, we see that the estimates (4.2) hold for $\alpha _0=5/2$ and $\beta =0$. Then, (4.3) and (4.4) give

  $$\begin{aligned} \Vert u(t) \Vert _{L^2}^2 \le C (1+t)^{-\frac{1}{2}} \end{aligned}$$

  with some constant $C=C(\Vert |x| u_0\Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$. Then, similarly to (4.7), we have

  $$\begin{aligned} \Vert u(t) \Vert _{L^2}^2 \le C (1+t)^{-\frac{3}{2}} \end{aligned}$$

  (4.9)

  for all $t \ge 0$. Here, we remark that (4.9) gives

  $$\begin{aligned} \int _0^\infty \Vert u(r) \Vert _{L^2}^2 \, dr \le C \int _0^t \frac{1}{(1+r)^{\frac{3}{2}}} \, dr \le C < \infty . \end{aligned}$$

  (4.10)

  Now, take an exponent $\alpha$ and a function g(t) so that

  $$\begin{aligned} \alpha > \frac{5}{2}, \quad g(t) = \frac{\alpha ^{\frac{1}{2}}}{(1+t)^{\frac{1}{2}}}. \end{aligned}$$

  Then, it follows from Lemma 4.1, (4.8) and (4.10) that

  $$\begin{aligned}&\Vert u(t) \Vert _{L^2}(1+t)^\alpha \nonumber \\&\quad \le \Vert u_0 \Vert _{L^2}^2 +C \int _0^t \frac{\alpha }{1+s} (1+s)^\alpha \left\{ \Vert e^{s\Delta }u_0 \Vert _{L^2}^2 + \frac{\alpha ^{\frac{5}{2}}}{(1+s)^{\frac{5}{2}}} \left( \int _0^s \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!2} \right\} ds \nonumber \\&\quad \le \Vert u_0 \Vert _{L^2}^2 +C \alpha \int _0^t (1+s)^{\alpha -\frac{7}{2}} \, ds \nonumber \\&\quad \le \Vert u_0 \Vert _{L^2}^2 + C(1+t)^{\alpha -\frac{5}{2}}, \nonumber \end{aligned}$$

  which yields

  $$\begin{aligned} \Vert u(t) \Vert _{L^2}^2&\le \Vert u_0 \Vert _{L^2}^2 (1+t)^{-\alpha } + C(1+t)^{-\frac{5}{2}} \nonumber \\&\le C (1+t)^{-\frac{5}{2}} \end{aligned}$$

  for all $t \geqslant 0$. This completes the proof of Lemma 4.2.

$\square$

4.2 $L^p$-decay Estimates

In this subsection, we adapt the arguments in [1] and show the $L^p$ temporal decay estimates for the solution u to (1.8). For $1 \le p \le \infty$ and $j=0,1$, we put

$$\begin{aligned} \Vert u \Vert _{X^p_j(t)}:= \sup _{0 <\tau \le t}\tau ^{\frac{j}{2}+\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |\tau )^{1-\frac{2}{p}}\Vert u(\tau )\Vert _{L^{p}}, \quad t>0. \end{aligned}$$

Lemma 4.3

Suppose that the exponents s, q and $\theta$ satisfy (1.10)–(1.11), and that the exponent p satisfies

$$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{q}\left( <\frac{1}{2}\right) . \end{aligned}$$

(4.11)

Let $C_*>0$ be the constant in (1.12). Then, there exists a constant $0<C_{**} \le C_*$ such that for any $u_{0} \in {\dot{H}}^{s}({\mathbb {R}}^{3})^3$ with $\nabla \cdot u_{0}=0$ and $\Omega \in {\mathbb {R}} \setminus \{ 0 \}$ satisfying

$$\begin{aligned} \Vert u_{0}\Vert _{{\dot{H}}^{s}({\mathbb {R}}^{3})} \le C_{**}|\Omega |^{\frac{s}{2}-\frac{1}{4}}, \end{aligned}$$

(4.12)

the unique global solution $u \in C([0,\infty );{\dot{H}}^{s}({\mathbb {R}}^{3}))^3 \cap L^{\theta }(0,\infty ;{\dot{H}}^{s}_{q}({\mathbb {R}}^{3}))^3$ constructed in Theorem 1.1 satisfies

$$\begin{aligned} \int _{\frac{t}{2}}^{t}\Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}} \nabla \cdot (u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \le \frac{1}{2}\Vert u\Vert _{X^p_j(t)} t^{-\frac{j}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

for all $t>0$ and $j=0,1$.

Proof

Let us set

$$\begin{aligned} \frac{1}{q_{s}}=\frac{1}{q}-\frac{s}{3}, \quad \frac{1}{r}=\frac{1}{p}+\frac{1}{q_s}. \end{aligned}$$

Then, we see that the exponents p and r satisfy $1<r \le p^\prime<2<p<\infty$. Applying Lemma 2.5 and the embedding ${\dot{H}}_{q}^{s}({\mathbb {R}}^{3}) \hookrightarrow L^{q_{s}}({\mathbb {R}}^{3})$ gives

$$\begin{aligned}&\int _{\frac{t}{2}}^{t}\Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}}{\text {div}}(u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\quad \le C \int _{\frac{t}{2}}^{t} \{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{r}-\frac{1}{p})} \Vert (u\otimes u)(\tau )\Vert _{L^{r}}\,d\tau \nonumber \\&\quad \le C \int _{\frac{t}{2}}^{t} \{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{L^{p}}\Vert u(\tau )\Vert _{L^{q_{s}}}\,d\tau \nonumber \\&\quad \le C \int _{\frac{t}{2}}^{t}\{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})} \Vert u(\tau )\Vert _{{\dot{H}}_{q}^{s}}\nonumber \\&\qquad \times \tau ^{-\frac{j}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |\tau )^{-(1-\frac{2}{p})} \tau ^{\frac{j}{2}+\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |\tau )^{1-\frac{2}{p}} \Vert u(\tau )\Vert _{L^{p}} \, d\tau \nonumber \\&\quad \le C \Vert u\Vert _{X^p_j(t)}(1+|\Omega |t)^{-(1-\frac{2}{p})}t^{-\frac{j}{2}-\frac{3}{2}(1-\frac{1}{p})} \nonumber \\&\qquad \times \int _{\frac{t}{2}}^{t}\{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})}(t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{{\dot{H}}_{q}^{s}}\,d\tau . \end{aligned}$$

(4.13)

Here, let us set

$$\begin{aligned} h_\Omega (t) := (1+|\Omega |t)^{-(1-\frac{2}{p})}t^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}, \quad t>0. \end{aligned}$$

The direct calculation gives that

$$\begin{aligned} \Vert h_\Omega \Vert _{L^{\theta ^\prime }(0,\infty )}&= |\Omega |^{\frac{1}{2}+\frac{3}{2}(\frac{1}{q}-\frac{s}{3})-\frac{1}{\theta ^\prime }} \left( \int _0^\infty \frac{1}{\tau ^{\theta ^\prime \{ \frac{1}{2}+\frac{3}{2}(\frac{1}{q}-\frac{s}{3}) \}}} \frac{1}{(1+\tau )^{\theta ^\prime (1-\frac{2}{p})}} \, d\tau \right) ^{\!\frac{1}{\theta ^\prime }} \nonumber \\&=C |\Omega |^{\frac{1}{\theta }+\frac{3}{2q}-\frac{s}{2}-\frac{1}{2}}<\infty . \end{aligned}$$

(4.14)

Indeed, since $s>1/2$, the assumption (1.11) on $\theta$ implies

$$\begin{aligned} \frac{1}{\theta }< \frac{5}{8} - \frac{3}{2q} + \frac{s}{4} < \frac{1}{2} - \frac{3}{2q}+\frac{s}{2}. \end{aligned}$$

This yields $\theta ^\prime \{ \frac{1}{2}+\frac{3}{2}(\frac{1}{q}-\frac{s}{3}) \}<1$. Also, by $1/p \le 1/q$ and (1.11), we have

$$\begin{aligned} \frac{1}{\theta } > \frac{1}{2q}+\frac{s}{2}-\frac{1}{2} \ge \frac{2}{p} - \frac{3}{2q}+\frac{s}{2}-\frac{1}{2}, \end{aligned}$$

which implies $\theta ^\prime \{ \frac{1}{2}+\frac{3}{2}(\frac{1}{q}-\frac{s}{3}) + (1-\frac{2}{p}) \}>1$. Therefore, it follows from the Hölder inequality, (1.13) and (4.14) that

$$\begin{aligned}&\int _{\frac{t}{2}}^{t}\{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})}(t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{{\dot{H}}_{q}^{s}}\,d\tau \nonumber \\&\quad \le \Vert h_\Omega \Vert _{L^{\theta ^\prime }(0,\infty )} \Vert u \Vert _{L^\theta (t/2, t; {\dot{H}}^s_q)} \nonumber \\&\quad \le C |\Omega |^{\frac{1}{\theta }+\frac{3}{2q}-\frac{s}{2}-\frac{1}{2}} \cdot |\Omega |^{-\frac{1}{\theta }+\frac{3}{4}(1-\frac{2}{q})}\Vert u_{0}\Vert _{{\dot{H}}^{s}} \nonumber \\&\quad = C |\Omega |^{-\frac{1}{2}(s-\frac{1}{2})} \Vert u_{0}\Vert _{{\dot{H}}^{s}}. \end{aligned}$$

(4.15)

Hence we obtain from (4.13) to (4.15) that

$$\begin{aligned}&\int _{\frac{t}{2}}^{t}\Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}}{\text {div}}(u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \\&\quad \le C |\Omega |^{-\frac{1}{2}(s-\frac{1}{2})} \Vert u_{0}\Vert _{{\dot{H}}^{s}} \Vert u \Vert _{X^p_j(t)} t^{-\frac{j}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \\&\quad \le \frac{1}{2}\Vert u\Vert _{X^p_j(t)} t^{-\frac{j}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

by taking $C_{**}<\frac{1}{2C}$ in (4.12). This completes the proof of Lemma 4.3. $\square$

Lemma 4.4

Suppose that the exponents $s, q, \theta$ and the initial data $u_0 \in ({\dot{H}}^s \cap L^1)({\mathbb {R}}^3)^3$ satisfy the asuumptions in Theorem 1.1, and let $u \in C([0,\infty );{\dot{H}}^s({\mathbb {R}}^3))^3$ be the unique global solution to (1.8) constructed in Theorem 1.1. Then, for $2 \le p \le \infty$, there exists a constant $C=C(p, \Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$ such that

$$\begin{aligned} \int _0^{\frac{t}{2}} \Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}} \nabla \cdot (u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \le C t^{-\frac{j}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

for all $t>0$ and $j=0,1$.

Proof

We first consider the case $j=0$. It follows from (3.10) in Lemma 3.2 that

$$\begin{aligned}&\int _0^{\frac{t}{2}} \Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}} \nabla \cdot (u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\quad \le C \int _0^{\frac{t}{2}} (t-\tau )^{-\frac{3}{2}(1-\frac{1}{p})} \{ 1+|\Omega |(t-\tau ) \}^{-(1-\frac{2}{p})} \Vert (u\cdot \nabla u)(\tau ) \Vert _{L^1} \, d\tau \nonumber \\&\quad \le C t^{-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \int _0^{\frac{t}{2}} \Vert u(\tau )\Vert _{L^2} \Vert \nabla u(\tau ) \Vert _{L^2} \, d\tau . \end{aligned}$$

(4.16)

Then, since $u_0 \in L^1({\mathbb {R}}^3)^3$, the $L^2$ decay estimate in Lemma 4.2 (1) and the energy equality (4.6) give that

$$\begin{aligned} \int _0^{\frac{t}{2}} \Vert u(\tau )\Vert _{L^2} \Vert \nabla u(\tau ) \Vert _{L^2} \, d\tau&\le C \int _0^{\frac{t}{2}} \frac{1}{(1+\tau )^{\frac{3}{4}}} \Vert \nabla u(\tau ) \Vert _{L^2} \, d\tau \nonumber \\&\le C \left( \int _0^\infty \frac{d\tau }{(1+\tau )^{\frac{3}{2}}} \right) ^{\!\frac{1}{2}} \left( \int _0^\infty \Vert \nabla u(\tau ) \Vert _{L^2}^2 \, d\tau \right) ^{\!\frac{1}{2}} \nonumber \\&\le C < \infty \end{aligned}$$

(4.17)

with some constant $C=C(\Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$. Then, (4.16) and (4.17) yield the desired estimates for $j=0$.

In the case $j=1$, we have by (3.10) in Lemma 3.2 and the $L^2$ decay estimate in Lemma 4.2 (1) that

$$\begin{aligned}&\int _0^{\frac{t}{2}} \Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}} \nabla \cdot (u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\quad \le C \int _0^{\frac{t}{2}} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} \{ 1+|\Omega |(t-\tau ) \}^{-(1-\frac{2}{p})} \Vert (u \otimes u)(\tau ) \Vert _{L^1} \, d\tau \nonumber \\&\quad \le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \int _0^{\frac{t}{2}} \Vert u(\tau )\Vert _{L^2}^2 \, d\tau \nonumber \\&\quad \le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \int _0^{\frac{t}{2}} \frac{1}{(1+\tau )^{\frac{3}{2}}} \, d\tau \nonumber \\&\quad \le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

with some constant $C=C(p, \Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})>0$. This completes the proof of Lemma 4.4. $\square$

We are ready to give the proof of the $L^{p}$-time decay estimates.

Theorem 4.5

Suppose that the exponents s, q and $\theta$ satisfy (1.10)–(1.11), and that the exponent p satisfies

$$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{2}. \end{aligned}$$

(4.18)

Let $u_{0} \in ({\dot{H}}^{s} \cap L^1)({\mathbb {R}}^3)^3$ with $\nabla \cdot u_{0}=0$ and $\Omega \in {\mathbb {R}} \setminus \{ 0 \}$ satisfy (4.12), and let $u \in C([0,\infty );{\dot{H}}^{s}({\mathbb {R}}^{3}))^3 \cap L^{\theta }(0,\infty ;{\dot{H}}^{s}_{q}({\mathbb {R}}^{3}))^3$ be the unique global solution to (1.8) constructed in Theorem 1.1.

• (1) There exists a positive constant $C=C(p, \Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})$ such that

  $$\begin{aligned} \Vert u(t) \Vert _{L^p} \le C t^{-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

  (4.19)

  for all $t>0$.

• (2) Assume further that $|x| u_0 \in L^1({\mathbb {R}}^3)^3$. Then, there exists a positive constant $C=C(p, \Vert |x|u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})$ such that

  $$\begin{aligned} \Vert u(t) \Vert _{L^p} \le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

  (4.20)

  for all $t>0$.

Proof

• (1) Let us first consider the case that the exponent p satisfies

  $$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{q}. \end{aligned}$$

  (4.21)

  Then, it follows from (3.10) in Lemmas 3.2, 4.3 and 4.4 with $j=0$ that

  $$\begin{aligned} \Vert u(t) \Vert _{L^p}&\le \Vert e^{-tA_\Omega }u_0 \Vert _{L^p} + \left( \int _0^{\frac{t}{2}} + \int _{\frac{t}{2}}^{t} \right) \Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}} \nabla \cdot (u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\le t^{-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \left\{ C + \frac{1}{2}\Vert u \Vert _{X^p_0(t^\prime )} \right\} \end{aligned}$$

  (4.22)

  for all $0< t \le t^\prime$ with some positive constant $C=C(p, \Vert u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})$. Hence we have $\Vert u \Vert _{X^p_0(t^\prime )} \le 2C$, which yields the desired estimate (4.19). For the exponent p satisfying

  $$\begin{aligned} \frac{1}{q} \le \frac{1}{p} \le \frac{1}{2}, \end{aligned}$$

  we take the exponent $\eta \in [0,1]$ so that $\frac{1}{p}=\frac{\eta }{2}+\frac{1-\eta }{q}$. Then, by the interpolation, Lemma 4.2 (1) and (4.19) for q, we obtain for all $t>0$

  $$\begin{aligned} \Vert u(t)\Vert _{L^{p}}&\le \Vert u(t)\Vert _{L^{2}}^{\eta }\Vert u(t)\Vert _{L^{q}}^{1-\eta }\\&\le Ct^{-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})}. \end{aligned}$$

• (2) Firstly, consider the case that the exponent p satisfies (4.21). Similarly to (4.22), we have by (3.11) in Lemmas 3.2, 4.3 and 4.4 with $j=1$ that

  $$\begin{aligned} \Vert u(t) \Vert _{L^p}&\le \Vert e^{-tA_\Omega }u_0 \Vert _{L^p} + \left( \int _0^{\frac{t}{2}} + \int _{\frac{t}{2}}^{t} \right) \Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}} \nabla \cdot (u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\le t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \left\{ C + \frac{1}{2}\Vert u \Vert _{X^p_1(t^\prime )} \right\} \end{aligned}$$

  (4.23)

  for all $0< t \le t^\prime$ with some positive constant $C=C(p, \Vert |x| u_0 \Vert _{L^1}, \Vert u_0 \Vert _{L^2})$. This implies $\Vert u \Vert _{X^p_1(t^\prime )} \le 2C$, and we obtain the desired estimate (4.20). In the case $\frac{1}{q} \le \frac{1}{p} \le \frac{1}{2}$, we take the exponent $\eta \in [0,1]$ so that $\frac{1}{p}=\frac{\eta }{2}+\frac{1-\eta }{q}$. We obtain by the interpolation, Lemma 4.2 (2) and (4.20) for q that

  $$\begin{aligned} \Vert u(t)\Vert _{L^{p}}&\le \Vert u(t)\Vert _{L^{2}}^{\eta }\Vert u(t)\Vert _{L^{q}}^{1-\eta }\\&\le Ct^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

  for all $t>0$. This completes the proof of Theorem 4.5.

$\square$

5 Nonlinear Asymptotics

We are now ready to give the proofs of Theorems 1.2 and 1.3. Firstly, we shall show Theorem 1.2.

Proof of Theorem 1.2

The temporal decay estimate (1.15) is already shown in Theorem 4.5. Hence it suffices to show the asymptotic behavior (1.16).

By the Duhamel formula (1.20), we have

$$\begin{aligned} u(t)&= e^{-tA_\Omega }u_0 - \left( \int _0^{\frac{t}{2}} + \int _{\frac{t}{2}}^t \right) e^{-(t-\tau )A_\Omega } {\mathbb {P}}\nabla \cdot (u\otimes u)(\tau ) \, d\tau \nonumber \\&=: I_1 + I_2 + I_3. \end{aligned}$$

(5.1)

For $I_1$, it follows from (3.13) in Theorem 3.4 that

$$\begin{aligned} \lim _{t\rightarrow \infty }t^{\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \Vert I_1(t) \Vert _{L^p} = 0. \end{aligned}$$

(5.2)

Concerning $I_2$, Lemma 4.4 with $j=1$ gives that

$$\begin{aligned} t^{\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \Vert I_2(t) \Vert _{L^p} \le C t^{-\frac{1}{2}} \rightarrow 0 \end{aligned}$$

(5.3)

as $t\rightarrow \infty$. Hence it remains to show the estimate for $I_3(t)$. Firstly, assume that the exponent p satisfies

$$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{q}. \end{aligned}$$

(5.4)

This is the same condition as (4.11) in Lemma 4.3. Put

$$\begin{aligned} \frac{1}{q_{s}}=\frac{1}{q}-\frac{s}{3}, \quad \frac{1}{r}=\frac{1}{p}+\frac{1}{q_s}. \end{aligned}$$

Then, the exponents p and r satisfy $2<p<\infty$ and $1<r \le p^\prime$. It follows from Lemma 2.5, the embedding ${\dot{H}}_{q}^{s}({\mathbb {R}}^{3}) \hookrightarrow L^{q_{s}}({\mathbb {R}}^{3})$ and (4.19) in Theorem 4.5 that

$$\begin{aligned} \Vert I_3(t) \Vert _{L^p}&\le \int _{\frac{t}{2}}^{t}\Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}}{\text {div}}(u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\le C \int _{\frac{t}{2}}^{t} \{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{r}-\frac{1}{p})} \Vert (u\otimes u)(\tau )\Vert _{L^{r}}\,d\tau \nonumber \\&\le C \int _{\frac{t}{2}}^{t} \{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{L^{p}}\Vert u(\tau )\Vert _{L^{q_{s}}}\,d\tau \nonumber \\&\le C t^{-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \nonumber \\&\quad \times \int _{\frac{t}{2}}^{t}\{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})}(t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{{\dot{H}}_{q}^{s}}\,d\tau . \end{aligned}$$

(5.5)

Here, similarly to (4.14) and (4.15), we have

$$\begin{aligned}&\int _{\frac{t}{2}}^{t}\{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{{\dot{H}}_{q}^{s}}\,d\tau \nonumber \\&\quad \le C |\Omega |^{\frac{1}{\theta }+\frac{3}{2q}-\frac{s}{2}-\frac{1}{2}} \Vert u \Vert _{L^\theta (t/2, t; {\dot{H}}^s_q)} \end{aligned}$$

(5.6)

Since u belongs to $L^\theta (0,\infty ;{\dot{H}}^s_q({\mathbb {R}}^3))^3$, we have by (5.5) and (5.6) that

$$\begin{aligned} t^{\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \Vert I_3(t) \Vert _{L^p} \le C|\Omega |^{\frac{1}{\theta }+\frac{3}{2q}-\frac{s}{2}-\frac{1}{2}} \left( \int _{\frac{t}{2}}^{t} \Vert u(\tau ) \Vert _{{\dot{H}}^s_q}^\theta \, d\tau \right) ^{\!\!\frac{1}{\theta }} \rightarrow 0 \end{aligned}$$

(5.7)

as $t\rightarrow \infty$. This completes the proof of (1.16) when the exponent p satisfies (5.4).

Next, we shall consider the estimate for $I_3(t)$ for $p=2$. We follow the argument in [7], and set

$$\begin{aligned} v(t):=-\int _{\tau }^{t}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds, \quad t>\tau . \end{aligned}$$

We see that $v(t)=u(t) - e^{-(t-\tau )A_\Omega }u(\tau )$, and v(t) should solve

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}v-\Delta v +\Omega {\mathbb {P}}J{\mathbb {P}}v=-{\mathbb {P}}(u\cdot \nabla u) &{}\quad x\in {\mathbb {R}}^3, \, t > \tau ,\\ v(\tau )=0 &{}\quad x \in {\mathbb {R}}^3. \end{array}\right. } \end{aligned}$$

(5.8)

Taking the $L^{2}$-inner product of (5.8) with v, we have

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \Vert \nabla v(t) \Vert _{L^{2}}^{2} = - \langle (u \cdot \nabla ) u(t), \, v(t) \rangle _{L^2}. \end{aligned}$$

Since $v(t)=u(t) - e^{-(t-\tau )A_\Omega }u(\tau )$, the integration by parts and the divergence-free condition give that

$$\begin{aligned} \langle (u \cdot \nabla ) u(t), \, v(t) \rangle _{L^2} = - \langle (u \cdot \nabla ) v(t), \, u(t) \rangle _{L^2} = - \langle (u \cdot \nabla ) v(t), \, e^{-(t-\tau )A_\Omega }u(\tau ) \rangle _{L^2}. \end{aligned}$$

Hence we have

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \Vert \nabla v(t) \Vert _{L^{2}}^{2} = \langle (u \cdot \nabla ) v(t), \, e^{-(t-\tau )A_\Omega }u(\tau ) \rangle _{L^2}. \end{aligned}$$

(5.9)

Note that Lemmas 2.4 and 4.2 (1) yield

$$\begin{aligned} \Vert e^{-(t-\tau )A_\Omega }u(\tau ) \Vert _{L^\infty }&\le C (t-\tau )^{-\frac{3}{4}} \Vert u(\tau ) \Vert _{L^2} \nonumber \\&\le C (t-\tau )^{-\frac{3}{4}}(1+\tau )^{-\frac{3}{4}}. \end{aligned}$$

(5.10)

Also, we remark that it holds

$$\begin{aligned} \Vert u(t) \Vert _{L^2} \le C (1+t)^{-\frac{3}{4}} \le C t^{-\frac{3}{4}} \le C (t-\tau )^{-\frac{3}{4}} \end{aligned}$$

(5.11)

by Lemma 4.2 (1). Hence we have from (5.9), (5.10) and (5.11)

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \Vert \nabla v(t) \Vert _{L^{2}}^{2}&\le \Vert u(t) \Vert _{L^2} \Vert \nabla v(t) \Vert _{L^2} \Vert e^{-(t-\tau )A_\Omega }u(\tau ) \Vert _{L^\infty } \nonumber \\&\le C \Vert \nabla v(t) \Vert _{L^2} (t-\tau )^{-\frac{3}{2}}(1+\tau )^{-\frac{3}{4}} \nonumber \\&\le \frac{1}{2} \Vert \nabla v(t) \Vert _{L^2}^2 + C (t-\tau )^{-3}(1+\tau )^{-\frac{3}{2}}, \end{aligned}$$

which yields

$$\begin{aligned} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \Vert \nabla v(t) \Vert _{L^{2}}^{2} \le C (t-\tau )^{-3}(1+\tau )^{-\frac{3}{2}}. \end{aligned}$$

(5.12)

Let $\rho >0$ be a positive parameter to be chosen later. It follows from the Plancherel theorem that

$$\begin{aligned} \Vert \nabla v\Vert _{L^{2}}^{2}&=\int _{{\mathbb {R}}^{3}}|\xi |^{2}|{\widehat{v}}(\xi )|^{2}\,d\xi \\&\ge \int _{|\xi |>\sqrt{\rho }}|\xi |^{2}|{\widehat{v}}(\xi )|^{2}\,d\xi \\&\ge \rho \int _{|\xi |>\sqrt{\rho }}|{\widehat{v}}(\xi )|^{2}\,d\xi \\&=\rho \Vert v\Vert _{L^{2}}^{2} - \rho \int _{|\xi |\le \sqrt{\rho }} |{\widehat{v}}(\xi )|^{2}\,d\xi . \end{aligned}$$

This and (5.12) imply that

$$\begin{aligned} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \rho \Vert v(t) \Vert _{L^{2}}^{2} \le \rho \int _{|\xi |\le \sqrt{\rho }} |{\widehat{v}}(\xi )|^{2}\,d\xi + C (t-\tau )^{-3}(1+\tau )^{-\frac{3}{2}}. \end{aligned}$$

(5.13)

Here, since

$$\begin{aligned} {\widehat{v}}(\xi , t) = - \int _{\tau }^{t} e^{-(t-s)|\xi |^2} H_\Omega (\xi , t-s)P(\xi ) \left\{ (i \xi ) \cdot \widehat{u \otimes u}(\xi ,s) \right\} ds, \end{aligned}$$

we have

$$\begin{aligned} |{\widehat{v}}(\xi , t)|&\le C |\xi | \int _{\tau }^{t} \Vert \widehat{u \otimes u}(s) \Vert _{L^\infty } \, ds \\&\le C |\xi | \int _{\tau }^{t} \Vert u\otimes u(s) \Vert _{L^1} \, ds \\&\le C |\xi | \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds. \end{aligned}$$

Therefore, we see that

$$\begin{aligned} \int _{|\xi |\le \sqrt{\rho }} |{\widehat{v}}(\xi )|^{2}\,d\xi&\le C \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} \int _{|\xi |\le \sqrt{\rho }} |\xi |^2 \, d\xi \nonumber \\&\le C \rho ^{\frac{5}{2}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2}. \end{aligned}$$

(5.14)

Substituting (5.14) into (5.13) gives that

$$\begin{aligned} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \rho \Vert v(t) \Vert _{L^{2}}^{2} \le C \rho ^{\frac{7}{2}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} + C (t-\tau )^{-3}(1+\tau )^{-\frac{3}{2}} \end{aligned}$$

(5.15)

for $0<\tau <t$. Now, we set

$$\begin{aligned} \rho = \rho (t) = m(t-\tau )^{-1}, \quad m>\frac{7}{2}. \end{aligned}$$

Then, the inequality (5.15) can be written as

$$\begin{aligned}&\frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + m(t-\tau )^{-1} \Vert v(t) \Vert _{L^{2}}^{2} \nonumber \\&\quad \le C m^{\frac{7}{2}}(t-\tau )^{-{\frac{7}{2}}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} + C (t-\tau )^{-3}(1+\tau )^{-\frac{3}{2}}. \end{aligned}$$

(5.16)

It follows from (5.16) that

$$\begin{aligned}&\frac{d}{dt} \left\{ (t-\tau )^m \Vert v(t) \Vert _{L^2}^2 \right\} \\&\quad = (t-\tau )^m \left\{ \frac{d}{dt}\Vert v(t) \Vert _{L^2}^2 + m(t-\tau )^{-1} \Vert v(t) \Vert _{L^{2}}^{2} \right\} \\&\quad \le C m^{\frac{7}{2}}(t-\tau )^{m-{\frac{7}{2}}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} + C (t-\tau )^{m-3}(1+\tau )^{-\frac{3}{2}}. \end{aligned}$$

Hence we have

$$\begin{aligned}&(t-\tau )^m \Vert v(t) \Vert _{L^2}^2 \\&\quad \le C m^{\frac{7}{2}} \int _{\tau }^{t} (s-\tau )^{m-{\frac{7}{2}}} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+\tau )^{-\frac{3}{2}} \int _{\tau }^{t} (s-\tau )^{m-3} \, ds \\&\quad \le C m^{\frac{7}{2}} (t-\tau )^{m-{\frac{7}{2}}} \int _{\tau }^{t} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+\tau )^{-\frac{3}{2}}(t-\tau )^{m-2}, \end{aligned}$$

which yields

$$\begin{aligned} \Vert v(t) \Vert _{L^2}^2 \le C (t-\tau )^{-{\frac{7}{2}}} \int _{\tau }^{t} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+\tau )^{-\frac{3}{2}}(t-\tau )^{-2}. \end{aligned}$$

(5.17)

Now, we put $\tau = t/2$, then

$$\begin{aligned} v(t):=-\int _{\frac{t}{2}}^{t}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds = I_3(t) \end{aligned}$$

and we have by (5.17) that

$$\begin{aligned} \left\| I_3(t) \right\| _{L^2}^2 \le C t^{-{\frac{7}{2}}} \int _{\frac{t}{2}}^{t} \left( \int _{\frac{t}{2}}^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+t)^{-\frac{3}{2}} t^{-2}. \end{aligned}$$

(5.18)

Here, it follows from Lemma 4.2 (1) that

$$\begin{aligned} \int _{\frac{t}{2}}^{t} \left( \int _{\frac{t}{2}}^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds&\le \frac{t}{2} \left( \int _{\frac{t}{2}}^{\infty } \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} \nonumber \\&\le C t \left( \int _{\frac{t}{2}}^{\infty } \frac{1}{(1+r)^{\frac{3}{2}}} \, dr \right) ^{\!\!2} \nonumber \\&\le C \frac{t}{1+t} \le C < \infty . \end{aligned}$$

(5.19)

Hence by (5.18) and (5.19), we obtain

$$\begin{aligned} \left\| I_3(t) \right\| _{L^2}^2 \le C t^{-\frac{7}{2}} + C (1+t)^{-\frac{3}{2}} t^{-2}, \end{aligned}$$

which yields

$$\begin{aligned} t^{\frac{3}{4}} \left\| I_3(t) \right\| _{L^2} \le C t^{-1} \rightarrow 0 \end{aligned}$$

(5.20)

as $t\rightarrow \infty$. This gives the proof of (1.16) for $p=2$.

Finally, for the exponent p satisfying

$$\begin{aligned} \frac{1}{q} \le \frac{1}{p} \le \frac{1}{2}, \end{aligned}$$

we take $\eta \in [0,1]$ such that $\frac{1}{p}=\frac{\eta }{2}+\frac{1-\eta }{q}$. Interpolating (5.7) and (5.20), we obtain

$$\begin{aligned}&t^{\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \Vert I_3(t) \Vert _{L^p} \nonumber \\&\quad \le \left\{ t^{\frac{3}{4}} \Vert I_3(t) \Vert _{L^2} \right\} ^\eta \left\{ t^{\frac{3}{2}(1-\frac{1}{q})}(1+|\Omega |t)^{1-\frac{2}{q}} \Vert I_3(t) \Vert _{L^q} \right\} ^{1-\eta } \rightarrow 0 \end{aligned}$$

as $t\rightarrow \infty$. This completes the proof of Theorem 1.2. $\square$

We finally present the proof of Theorem 1.3.

Proof of Theorem 1.3

We have already shown the time decay estimate (1.24) in Theorem 4.5. Hence it remains to prove the asymptotic behavior (1.25).

Let us decompose

$$\begin{aligned} - \int _{0}^{t}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds=: J_1 + J_2, \end{aligned}$$

(5.21)

where

$$\begin{aligned} J_1&:= - \int _{0}^{\frac{t}{2}}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds, \\ J_2&:= - \int _{\frac{t}{2}}^{t}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds. \end{aligned}$$

We first consider the estimate for $J_2$. Let us treat the case that the exponent p satisfies

$$\begin{aligned} \frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{q}. \end{aligned}$$

(5.22)

Note that (5.22) is the same assumption as (4.11) in Lemma 4.3. Setting

$$\begin{aligned} \frac{1}{q_{s}}=\frac{1}{q}-\frac{s}{3}, \quad \frac{1}{r}=\frac{1}{p}+\frac{1}{q_s}, \end{aligned}$$

we see that there hold $2<p<\infty$ and $1<r \le p^\prime$. Hence we can apply Lemma 2.5, and it follows from the embedding ${\dot{H}}_{q}^{s}({\mathbb {R}}^{3}) \hookrightarrow L^{q_{s}}({\mathbb {R}}^{3})$, (4.20) in Theorem 4.5 and (5.6) that

$$\begin{aligned} \Vert J_2(t) \Vert _{L^p}&\le \int _{\frac{t}{2}}^{t}\Vert e^{-(t-\tau )A_{\Omega }}{\mathbb {P}}{\text {div}}(u\otimes u)(\tau )\Vert _{L^{p}}\,d\tau \nonumber \\&\le C \int _{\frac{t}{2}}^{t} \{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{r}-\frac{1}{p})} \Vert (u\otimes u)(\tau )\Vert _{L^{r}}\,d\tau \nonumber \\&\le C \int _{\frac{t}{2}}^{t} \{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})} (t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{L^{p}}\Vert u(\tau )\Vert _{L^{q_{s}}}\,d\tau \nonumber \\&\le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \nonumber \\&\quad \times \int _{\frac{t}{2}}^{t}\{1+|\Omega |(t-\tau )\}^{-(1-\frac{2}{p})}(t-\tau )^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{s}{3})}\Vert u(\tau )\Vert _{{\dot{H}}_{q}^{s}}\,d\tau \nonumber \\&\le C |\Omega |^{\frac{1}{\theta }+\frac{3}{2q}-\frac{s}{2}-\frac{1}{2}} t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{-(1-\frac{2}{p})} \Vert u \Vert _{L^\theta (\frac{t}{2}, t; {\dot{H}}^s_q)}. \end{aligned}$$

(5.23)

Since $u \in L^\theta (0,\infty ;{\dot{H}}^s_q({\mathbb {R}}^3))^3$, we have by (5.23) that

$$\begin{aligned} t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})}(1+|\Omega |t)^{1-\frac{2}{p}} \Vert J_2(t) \Vert _{L^p} \le C|\Omega |^{\frac{1}{\theta }+\frac{3}{2q}-\frac{s}{2}-\frac{1}{2}} \left( \int _{\frac{t}{2}}^{t} \Vert u(\tau ) \Vert _{{\dot{H}}^s_q}^\theta \, d\tau \right) ^{\!\!\frac{1}{\theta }} \rightarrow 0 \end{aligned}$$

(5.24)

as $t\rightarrow \infty$.

Next, let us consider the $L^2$ estimate for $J_2$. Set

$$\begin{aligned} v(t):=-\int _{\tau }^{t}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds, \quad t>\tau . \end{aligned}$$

Note that we now assume $(1+|x|)u_0 \in L^1({\mathbb {R}}^3)^3$, and then it follows from Lemmas 2.4 and 4.2 (2) that

$$\begin{aligned} \Vert e^{-(t-\tau )A_\Omega }u(\tau ) \Vert _{L^\infty } \le C (t-\tau )^{-\frac{3}{4}} \Vert u(\tau ) \Vert _{L^2} \le C (t-\tau )^{-\frac{3}{4}}(1+\tau )^{-\frac{5}{4}} \end{aligned}$$

(5.25)

and

$$\begin{aligned} \Vert u(t) \Vert _{L^2} \le C (1+t)^{-\frac{5}{4}} \le C t^{-\frac{5}{4}} \le C (t-\tau )^{-\frac{5}{4}} \end{aligned}$$

(5.26)

We apply the same argument as in the proof of Theorem 1.2 by using (5.25) and (5.26) instead of (5.10) and (5.11), respectively. Then, similarly to (5.15), we have

$$\begin{aligned} \frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + \rho \Vert v(t) \Vert _{L^{2}}^{2} \le C \rho ^{\frac{7}{2}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} + C (t-\tau )^{-4}(1+\tau )^{-\frac{5}{2}} \end{aligned}$$

(5.27)

for $0<\tau <t$ and $\rho >0$. Here we set

$$\begin{aligned} \rho = \rho (t) = m(t-\tau )^{-1}, \quad m> 4. \end{aligned}$$

Substituting this $\rho (t)$ into (5.27) gives

$$\begin{aligned}&\frac{d}{dt} \Vert v(t) \Vert _{L^{2}}^{2} + m(t-\tau )^{-1} \Vert v(t) \Vert _{L^{2}}^{2} \nonumber \\&\quad \le C m^{\frac{7}{2}}(t-\tau )^{-{\frac{7}{2}}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} + C (t-\tau )^{-4}(1+\tau )^{-\frac{5}{2}}. \end{aligned}$$

(5.28)

Then, it follows from (5.28) that

$$\begin{aligned}&\frac{d}{dt} \left\{ (t-\tau )^m \Vert v(t) \Vert _{L^2}^2 \right\} \\&\quad = (t-\tau )^m \left\{ \frac{d}{dt}\Vert v(t) \Vert _{L^2}^2 + m(t-\tau )^{-1} \Vert v(t) \Vert _{L^{2}}^{2} \right\} \\&\quad \le C m^{\frac{7}{2}}(t-\tau )^{m-{\frac{7}{2}}} \left( \int _{\tau }^{t} \Vert u(s) \Vert _{L^2}^2 \, ds \right) ^{\!\!2} + C (t-\tau )^{m-4}(1+\tau )^{-\frac{5}{2}}, \end{aligned}$$

which yields

$$\begin{aligned}&(t-\tau )^m \Vert v(t) \Vert _{L^2}^2 \\&\quad \le C m^{\frac{7}{2}} \int _{\tau }^{t} (s-\tau )^{m-{\frac{7}{2}}} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+\tau )^{-\frac{5}{2}} \int _{\tau }^{t} (s-\tau )^{m-4} \, ds \\&\quad \le C m^{\frac{7}{2}} (t-\tau )^{m-{\frac{7}{2}}} \int _{\tau }^{t} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+\tau )^{-\frac{5}{2}}(t-\tau )^{m-3}. \end{aligned}$$

Hence we have

$$\begin{aligned} \Vert v(t) \Vert _{L^2}^2 \le C (t-\tau )^{-{\frac{7}{2}}} \int _{\tau }^{t} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds + C(1+\tau )^{-\frac{5}{2}}(t-\tau )^{-3}. \end{aligned}$$

(5.29)

Here, we see by Lemma 4.2 (2) that

$$\begin{aligned} \int _{\tau }^{t} \left( \int _{\tau }^{s} \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} ds&\le (t- \tau ) \left( \int _{\tau }^{\infty } \Vert u(r) \Vert _{L^2}^2 \, dr \right) ^{\!\!2} \nonumber \\&\le C (t-\tau ) \left( \int _{\tau }^{\infty } \frac{1}{(1+r)^{\frac{5}{2}}} \, dr \right) ^{\!\!2} \nonumber \\&\le C \frac{t-\tau }{(1+\tau )^3}. \end{aligned}$$

(5.30)

Thus, substituting (5.30) into (5.29) gives

$$\begin{aligned} \Vert v(t) \Vert _{L^2}^2 \le C (t-\tau )^{-{\frac{5}{2}}} (1+\tau )^{-3} + C (t-\tau )^{-3} (1+\tau )^{-\frac{5}{2}}. \end{aligned}$$

(5.31)

Now, let us set $\tau =t/2$. Then,

$$\begin{aligned} v(t):=-\int _{\frac{t}{2}}^{t}e^{-(t-s)A_{\Omega }}{\mathbb {P}} \nabla \cdot (u \otimes u)(s)\,ds = J_2(t). \end{aligned}$$

Hence we have by (5.31) that

$$\begin{aligned} \Vert J_2(t) \Vert _{L^2}^2 \le C t^{-{\frac{5}{2}}} (1+t)^{-3} + C t^{-3} (1+t)^{-\frac{5}{2}}, \end{aligned}$$

which yields

$$\begin{aligned} t^{\frac{5}{4}} \Vert J_2(t) \Vert _{L^2} \le C (1+t)^{-\frac{3}{2}} + C t^{-\frac{1}{4}}(1+t)^{-\frac{5}{4}} \rightarrow 0 \end{aligned}$$

(5.32)

as $t\rightarrow \infty$.

For the case that the exponent p satisfying

$$\begin{aligned} \frac{1}{q} \le \frac{1}{p} \le \frac{1}{2}, \end{aligned}$$

we take $\eta \in [0,1]$ such that $\frac{1}{p}=\frac{\eta }{2}+\frac{1-\eta }{q}$. Interpolating (5.24) and (5.32) gives

$$\begin{aligned}&t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{1-\frac{2}{p}} \Vert J_2(t) \Vert _{L^p} \nonumber \\&\quad \le \left\{ t^{\frac{5}{4}} \Vert J_2(t) \Vert _{L^2} \right\} ^\eta \left\{ t^{\frac{1}{2} + \frac{3}{2}(1-\frac{1}{q})}(1+|\Omega |t)^{1-\frac{2}{q}} \Vert J_2(t) \Vert _{L^q} \right\} ^{1-\eta } \rightarrow 0 \end{aligned}$$

as $t\rightarrow \infty$. Hence we obtain that for p satisfying $\frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{2}$, there holds

$$\begin{aligned} \lim _{t\rightarrow \infty }t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{1-\frac{2}{p}} \Vert J_2(t) \Vert _{L^p} = 0. \end{aligned}$$

(5.33)

Next, let us consider the estimates for $J_1$. Let us rewrite

$$\begin{aligned} J_1 = - \sum _{j=1}^3 \int _{0}^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \partial _j {\widetilde{K}}_\Omega (x-y, t-s) (u_j u)(y,s) \, dy \, ds. \end{aligned}$$

Then, we decompose $J_1$ as

$$\begin{aligned}&J_1 + \sum _{j=1}^3 \partial _j {\widetilde{K}}_\Omega (x, t) \int _{0}^{\infty } \int _{{\mathbb {R}}^3} (u_j u)(y,s) \, dy \, ds \nonumber \\&\quad = \sum _{j=1}^3 \partial _j {\widetilde{K}}_\Omega (x, t) \int _{\frac{t}{2}}^{\infty } \int _{{\mathbb {R}}^3} (u_j u)(y,s) \, dy \, ds \nonumber \\&\qquad - \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \left\{ \partial _j {\widetilde{K}}_\Omega (x-y, t-s) - \partial _j {\widetilde{K}}_\Omega (x,t-s) \right\} (u_j u)(y, s) \, dy \, ds \nonumber \\&\qquad - \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \left\{ \partial _j {\widetilde{K}}_\Omega (x, t-s) - \partial _j {\widetilde{K}}_\Omega (x,t) \right\} (u_j u)(y, s) \, dy \, ds \nonumber \\&\quad =: J_{11} + J_{12} + J_{13}. \end{aligned}$$

(5.34)

Let us firstly consider the estimate for $J_{11}$. Similarly to Lemma 3.1, we see that for $2 \le p \le \infty$ and $\alpha \in ({\mathbb {N}}\cup \{0\})^{3}$ there exists a positive constant $C=C(p,\alpha )$ such that it holds

$$\begin{aligned} \left\| \partial _x^{\alpha } {\widetilde{K}}_{\Omega }(\cdot ,t)\right\| _{L^p} \le Ct^{-\frac{|\alpha |}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \end{aligned}$$

(5.35)

for all $\Omega \in {\mathbb {R}}$ and for all $t>0$. Then, we have by (5.35) and Lemma 4.2 (2) that for $2 \le p \le \infty$

$$\begin{aligned} \Vert J_{11}(t) \Vert _{L^p}&\le \sum _{j=1}^3 \left\| \partial _j {\widetilde{K}}_{\Omega }(\cdot ,t)\right\| _{L^p} \int _{\frac{t}{2}}^{\infty } \int _{{\mathbb {R}}^3} |(u_j u)(y,s)| \, dy \, ds \nonumber \\&\le Ct^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \int _{\frac{t}{2}}^{\infty } \Vert u(s) \Vert _{L^2}^2 \, ds \\&\le Ct^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{-(1-\frac{2}{p})} \int _{\frac{t}{2}}^{\infty } \frac{1}{(1+s)^{\frac{5}{2}}} \, ds, \end{aligned}$$

which yields

$$\begin{aligned} t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})} (1+|\Omega |t)^{1-\frac{2}{p}} \Vert J_{11}(t) \Vert _{L^p} \le C (1+t)^{-\frac{3}{2}} \rightarrow 0 \quad (t\rightarrow \infty ). \end{aligned}$$

(5.36)

Next, we consider the estimate for $J_{12}$. Similarly to (3.4), (3.5) and (3.21), we see that

$$\begin{aligned} \partial _j {\widetilde{K}}_\Omega (x,t)&= t^{-2} {\mathcal {F}}^{-1}\left[ \widehat{\partial _j G_1}(\xi ) {\widetilde{H}}_{\Omega }(\xi ,t) \right] \left( \frac{x}{\sqrt{t}}\right) \nonumber \\&=\frac{1}{2} t^{-2} \sum _{\sigma \in \{ \pm \}} {\mathcal {G}}_\sigma (\Omega t)\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \partial _j G_1 \right] \left( \frac{x}{\sqrt{t}} \right) \end{aligned}$$

(5.37)

for $j=1,2,3$. Hence we have

$$\begin{aligned}&\left\| \partial _j {\widetilde{K}}_\Omega (\cdot -y, t-s) - \partial _j {\widetilde{K}}_\Omega (\cdot ,t-s) \right\| _{L^p} \nonumber \\&\quad \le \frac{1}{2} (t-s)^{-2}\sum _{\sigma \in \{ \pm \}} \bigg \Vert {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \partial _j G_1 \right] \left( \frac{\cdot -y}{\sqrt{t-s}} \right) \nonumber \\&\qquad - {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \partial _j G_1 \right] \left( \frac{\cdot }{\sqrt{t-s}} \right) \bigg \Vert _{L^p} \nonumber \\&\quad = \frac{(t-s)^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}}{2} \sum _{\sigma \in \{ \pm \}} \bigg \Vert {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \partial _j G_1 \right] \left( \cdot - \frac{y}{\sqrt{t-s}} \right) \nonumber \\&\qquad - {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \partial _j G_1 \right] \left( \cdot \right) \bigg \Vert _{L^p} \nonumber \\&\quad \le \frac{(t-s)^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})}}{2} \nonumber \\&\qquad \times \sum _{\sigma \in \{ \pm \}} \left\| {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \left\{ \partial _j G_1\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - \partial _j G_1(\cdot ) \right\} \right] \right\| _{L^p}. \end{aligned}$$

(5.38)

For p satisfying $\frac{1}{2}-\frac{1}{2}\left( \frac{1}{q}-\frac{s}{3}\right) \le \frac{1}{p} \le \frac{1}{2}$, similarly to (3.17) and (3.18), it follows from the embedding ${\dot{B}}^0_{p,2}({\mathbb {R}}^3) \hookrightarrow L^p({\mathbb {R}}^3)$, Lemmas 2.3, 2.6 and the embedding $L^{p^\prime }({\mathbb {R}}^3) \hookrightarrow {\dot{B}}^0_{p^\prime ,2}({\mathbb {R}}^3)$ that

$$\begin{aligned}&\left\| {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \left\{ \partial _j G_1\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - \partial _j G_1(\cdot ) \right\} \right] \right\| _{L^p} \nonumber \\&\quad \le C \left\| {\mathcal {G}}_\sigma (\Omega (t-s))\left[ ({\mathbb {P}}+\sigma {\mathcal {R}}) \left\{ \partial _j G_1\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - \partial _j G_1(\cdot ) \right\} \right] \right\| _{{\dot{B}}^0_{p,2}} \nonumber \\&\quad \le C \left\{ 1+|\Omega |(t-s) \right\} ^{-(1-\frac{2}{p})} \left\| \partial _j G_1\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - \partial _j G_1(\cdot ) \right\| _{{\dot{B}}^{3(1-\frac{2}{p})}_{p^\prime , 2}} \nonumber \\&\quad \le C \left\{ 1+|\Omega |(t-s) \right\} ^{-(1-\frac{2}{p})} \left( \frac{1}{2} \right) ^{-\frac{1}{2}-\frac{3}{2}(1-\frac{2}{p})} \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{{\dot{B}}^{0}_{p^\prime , 2}} \nonumber \\&\quad \le C \left\{ 1+|\Omega |(t-s) \right\} ^{-(1-\frac{2}{p})} \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }}. \end{aligned}$$

(5.39)

Hence we have by (5.38) and (5.39)

$$\begin{aligned}&\Vert J_{12}(t) \Vert _{L^p} \nonumber \\&\quad \le \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \left\| \partial _j {\widetilde{K}}_\Omega (\cdot -y, t-s) - \partial _j {\widetilde{K}}_\Omega (\cdot ,t-s) \right\| _{L^p} |(u_j u)(y, s)| \, dy \, ds \nonumber \\&\quad \le C \int _0^{\frac{t}{2}} \int _{{\mathbb {R}}^3} (t-s)^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} \left\{ 1+|\Omega |(t-s) \right\} ^{-(1-\frac{2}{p})} \nonumber \\&\qquad \times \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }} |u(y,s)|^2 \, dy \, ds \nonumber \\&\quad \le C t^{-\frac{1}{2}-\frac{3}{2}(1-\frac{1}{p})} \left( 1+|\Omega |t \right) ^{-(1-\frac{2}{p})} \nonumber \\&\qquad \times \int _0^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }} |u(y,s)|^2 \, dy \, ds. \end{aligned}$$

Here, let $R>0$ be a positive parameter to be chosen later, and let t satisfy $t>2R$. We decompose

$$\begin{aligned}&t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})} \left( 1+|\Omega |t \right) ^{1-\frac{2}{p}}\Vert J_{12}(t) \Vert _{L^p} \nonumber \\&\quad \le C \left( \int _0^R + \int _R^{\frac{t}{2}} \right) \int _{{\mathbb {R}}^3} \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }} |u(y,s)|^2 \, dy \, ds. \end{aligned}$$

(5.40)

Since $|u(y,s)|^2 \in L^1({\mathbb {R}}^3_y \times (0,R)_s)$ by Lemma 4.2 (2), it follows from the dominated convergence theorem that

$$\begin{aligned} \lim _{t\rightarrow \infty } \int _0^R \int _{{\mathbb {R}}^3} \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }} |u(y,s)|^2 \, dy \, ds= 0. \end{aligned}$$

(5.41)

For the second term in (5.40), we have by Lemma 4.2 (2) that

$$\begin{aligned}&\int _R^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \left\| G_{\frac{1}{2}}\left( \cdot - \frac{y}{\sqrt{t-s}} \right) - G_{\frac{1}{2}}(\cdot ) \right\| _{L^{p^\prime }} |u(y,s)|^2 \, dy \, ds \nonumber \\&\quad \le 2 \Vert G_{\frac{1}{2}} \Vert _{L^{p^\prime }} \int _R^{\frac{t}{2}} \Vert u(s) \Vert _{L^2}^2 \, ds \nonumber \\&\quad \le C \int _R^{\frac{t}{2}} \frac{1}{(1+s)^{\frac{5}{2}}} \, ds \le \frac{C}{(1+R)^{\frac{3}{2}}}. \end{aligned}$$

(5.42)

Then, for any $\varepsilon >0$, take a large $R=R_\varepsilon >0$ so that $(1+R)^{-\frac{3}{2}} \le \varepsilon$. Then, it follows from (5.41) to (5.42) that

$$\begin{aligned} \limsup _{t\rightarrow \infty } t^{\frac{1}{2}+\frac{3}{2}(1-\frac{1}{p})} \left( 1+|\Omega |t \right) ^{1-\frac{2}{p}}\Vert J_{12}(t) \Vert _{L^p} \le C \varepsilon , \end{aligned}$$

which yields

(5.43)

Finally, we consider the estimate for $J_{13}$ in (5.34). We firstly remark that there hold

$$\begin{aligned} -P(\xi ) J P(\xi ) = \frac{\xi _3}{|\xi |}R(\xi ), \quad P(\xi ) R(\xi ) =R(\xi )P(\xi ) = R(\xi ), \quad R(\xi )^2 = -P(\xi ) \end{aligned}$$

and then

$$\begin{aligned}&-P(\xi )JP(\xi ) \left\{ \cos \left( \Omega \frac{\xi _3}{|\xi |} t \right) P(\xi ) + \sin \left( \Omega \frac{\xi _3}{|\xi |} t \right) R(\xi ) \right\} \\&\quad = \frac{\xi _3}{|\xi |} \left\{ - \sin \left( \Omega \frac{\xi _3}{|\xi |} t \right) P(\xi ) + \cos \left( \Omega \frac{\xi _3}{|\xi |} t \right) R(\xi ) \right\} . \end{aligned}$$

Hence we have

$$\begin{aligned} \partial _t {\widetilde{K}}_\Omega (x,t) = \Delta {\widetilde{K}}_\Omega (x,t) - \Omega {\mathbb {P}}J {\mathbb {P}}{\widetilde{K}}_\Omega (x,t). \end{aligned}$$

(5.44)

Then it follows from the mean value theorem and (5.44) that

$$\begin{aligned} J_{13}&= - \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _{{\mathbb {R}}^3} \left\{ \partial _j {\widetilde{K}}_\Omega (x, t-s) - \partial _j {\widetilde{K}}_\Omega (x,t) \right\} (u_j u)(y, s) \, dy \, ds \nonumber \\&= \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _0^1 \partial _j \partial _t {\widetilde{K}}_\Omega (x, t-\tau s) \,d\tau \int _{{\mathbb {R}}^3} s (u_j u)(y, s) \, dy \, ds \nonumber \\&= \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _0^1 \partial _j \Delta {\widetilde{K}}_\Omega (x, t-\tau s) \,d\tau \int _{{\mathbb {R}}^3} s (u_j u)(y, s) \, dy \, ds \nonumber \\&\quad - \Omega \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _0^1 \partial _j {\mathbb {P}}J{\mathbb {P}}{\widetilde{K}}_\Omega (\cdot , t-\tau s) \, d\tau \int _{{\mathbb {R}}^3} s (u_j u)(y,s) \, dy \, ds \nonumber \\&=: K_1 - K_2. \end{aligned}$$

(5.45)

By (5.35) and Lemma 4.2 (2), we have

$$\begin{aligned} \Vert K_1(t) \Vert _{L^p}&\le \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _0^1 \left\| \partial _j \Delta {\widetilde{K}}_\Omega (\cdot , t-\tau s) \right\| _{L^p} \,d\tau \int _{{\mathbb {R}}^3} s |(u_j u)(y, s)| \, dy \, ds \nonumber \\&\le C \int _0^{\frac{t}{2}} \int _0^1 (t-\tau s)^{-\frac{3}{2}-\frac{3}{2}(1-\frac{2}{p})} \left\{ 1+|\Omega |(t-\tau s) \right\} ^{-(1-\frac{2}{p})} s \Vert u(s)\Vert _{L^2}^2 \, d\tau \, ds \nonumber \\&\le C t^{-\frac{3}{2}-\frac{3}{2}(1-\frac{2}{p})} \left( 1+|\Omega |t \right) ^{-(1-\frac{2}{p})} \int _0^{\frac{t}{2}} \frac{s}{(1+s)^{\frac{5}{2}}}\, ds \nonumber \\&\le C t^{-\frac{3}{2}-\frac{3}{2}(1-\frac{2}{p})} \left( 1+|\Omega |t \right) ^{-(1-\frac{2}{p})}. \end{aligned}$$

(5.46)

Hence (5.45) and (5.46) yield

$$\begin{aligned} t^{\frac{1}{2}+\frac{3}{2}(1-\frac{2}{p})} \left( 1+|\Omega |t \right) ^{1-\frac{2}{p}} \left\| J_{13}(t) + K_2(t) \right\| _{L^p}\le \frac{C}{t} \rightarrow 0 \qquad (t\rightarrow \infty ). \end{aligned}$$

(5.47)

Now, the Duhamel formula (1.20) and the decompositions (5.21), (5.34), (5.45) gives

$$\begin{aligned}&u(t) + \sum _{j=1}^3 \partial _{j}K_\Omega (\cdot ,t) \int _{{\mathbb {R}}^3} y_j u_0(y) \, dy +\sum _{j=1}^3 \partial _j {\widetilde{K}}_\Omega (\cdot , t) \int _0^\infty \int _{{\mathbb {R}}^3} (u_j u)(y,s) \, dy \, ds \nonumber \\&\qquad +\Omega \sum _{j=1}^3 \int _0^{\frac{t}{2}} \int _0^1 \partial _j {\mathbb {P}}J{\mathbb {P}}{\widetilde{K}}_\Omega (\cdot , t-\tau s) \, d\tau \int _{{\mathbb {R}}^3} s (u_j u)(y,s) \, dy \, ds \nonumber \\&\quad = \left\{ e^{-tA_\Omega }u_0 + \sum _{j=1}^3 \partial _{j}K_\Omega (\cdot ,t) \int _{{\mathbb {R}}^3} y_j u_0(y) \, dy \right\} + J_2 + J_{11} + J_{12} + (J_{13}+K_2). \end{aligned}$$

Therefore, we obtain the desired asymptotic behavior (1.25) by (3.14) in Theorem 3.4, (5.33), (5.36), (5.43) and (5.47). This completes the proof of Theorem 1.3. $\square$