$L^1-$decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space

Abstract

The aim of this article devotes to establishing the $L^1$-decay of cubic order spatial derivatives of solutions to the Navier–Stokes equations, which is a long-time challenging problem. To solve this problem, new tools have to be found to overcome these main difficulties: $L^1-L^1$ estimate fails for the Stokes flow; the projection operator $P:\,L^1(\mathbb {R}^n_+)\rightarrow L^1_\sigma (\mathbb {R}^n_+)$ becomes unbounded; the steady Stokes’s estimates does not work any more in $L^1(\mathbb {R}^n_+)$. We first give the asymptotic behavior with weights of negative exponent for the Stokes flow and Navier–Stokes equations in $L^1(\mathbb {R}^n_+)$, and these are also independent of interest by themselves. Secondly, we decompose the convection term into two parts, and translate the unboundedness of projection operator into studying an $L^1$-estimate for an elliptic problem with homogeneous Neumann boundary conditions, which is established by using the weighted estimates of the Gaussian kernel’s convolution. Finally, a crucial new formula is given for the fundamental solution of the Laplace operator, which is employed for overcoming the strong singularity in studying the cubic order spatial derivatives in $L^1(\mathbb {R}^n_+)$.

1 Introduction and main results

In this article, we are concerned with the asymptotic behavior of solutions to the Navier–Stokes initial-value problem in the half space

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _tu-\Delta u+(u\cdot \nabla )u+\nabla p=0& \text{ in } & \mathbb {R}^n_+\times (0, \infty ),\\ \nabla \cdot u=0 & \text{ in } & \mathbb {R}^n_+\times (0, \infty ),\\ u(x, t)=0 & \text{ on } & \partial \mathbb {R}^n_+\times (0, \infty ),\\ u(x, t)\longrightarrow 0 & \text{ as } & |x|\longrightarrow \infty ,\\ u(x, 0)=u_0 & \text{ in } & \mathbb {R}^n_+,\\ \end{array} \right. \end{aligned}$$

(1.1)

where $n\ge 2$, and $\mathbb {R}^n_+=\{x=(x^\prime , x_n)\in \mathbb {R}^n\,|\,\,x_n>0 \}$ is the upper-half space of $\mathbb {R}^n$; $u=(u_1, u_2,\cdots , u_n)(x, t)$ and $p=p(x, t)$ denote unknown velocity vector and the pressure respectively, while initial datum $u_0(x)$ is assumed to satisfy a compatibility condition: $\nabla \cdot u_0=0$ in $\mathbb {R}^n_+$ and the normal component of $u_0$ equals to zero on $\partial \mathbb {R}^n_+$; and

$$\begin{aligned} \partial _t= & \frac{\partial }{\partial t}, \;\;\;\nabla =(\partial _1, \partial _2,\cdots ,\partial _n),\;\;\; \partial _j=\frac{\partial }{\partial x_j}\; (j=1,2,\cdots ,n),\\ \Delta u= & \sum \limits _{j=1}^n\partial _j\partial _ju, \;\;\;(u\cdot \nabla )u=\sum \limits _{j=1}^nu_j\partial _ju, \;\;\;\nabla \cdot u=\sum \limits _{j=1}^n\partial _ju_j. \end{aligned}$$

Throughout this paper, write

$$\begin{aligned} x=(x^\prime , x_n),\;\;x^\prime =(x_1, x_2,\cdots ,x_{n-1}), \;\;\nabla _{x^\prime }=\nabla ^\prime =(\partial _1, \partial _2,\cdots ,\partial _{n-1}), \end{aligned}$$

$C^\infty _0(\mathbb {R}^n_+)$ denotes the set of all $C^\infty$ real functions with compact support in $\mathbb {R}^n_+$, and

$$\begin{aligned} C^\infty _{0, \sigma }(\mathbb {R}^n_+)=\{\phi =(\phi _1, \cdots , \phi _n)\in C^\infty _0(\mathbb {R}^n_+);\;\nabla \cdot \phi =0\;\;\text{ in }\;\;\mathbb {R}^n_+\}; \end{aligned}$$

$L^q_\sigma (\mathbb {R}^n_+)$ ($1<q<\infty$) is the closure of $C^\infty _{0, \sigma }(\mathbb {R}^n_+)$ with respect to $\Vert \cdot \Vert _{L^q(\mathbb {R}^n_+)}$, where $L^q(\mathbb {R}^n_+)$ represents the usual Lebesgue space of vector-valued functions. The norm of $L^q(\mathbb {R}^n_+)$ is denoted by $\Vert u\Vert _{L^q(\mathbb {R}^n_+)}=(\int _{\mathbb {R}^n_+}|u(x)|^qdx)^\frac{1}{q}$ if $1\le q<\infty$; and $\Vert u\Vert _{L^\infty (\mathbb {R}^n_+)}=ess\,\sup \limits _{x\in \mathbb {R}^n_+}|u(x)|$; $\Vert \omega (x) u(t)\Vert _{L^q(\mathbb {R}^n_+)}=(\int _{\mathbb {R}^n_+}|\omega (x) u(x,t)|^qdx)^\frac{1}{q}$, $\Vert \omega (x) f(x,y)\Vert _{L^q_x(\mathbb {R}^n_+)}=(\int _{\mathbb {R}^n_+}|\omega (x) f(x,y)|^qdx)^\frac{1}{q}$, $1\le q<\infty$, $\omega (x)=x_n^\alpha ,\;|x|^\alpha$, $\alpha \ge 0$. $O(f(x))=g(x)$ means $|f(x)|\le C|g(x)|$ for some constant C. By symbol C, it means a generic positive constant which may vary from line to line.

A vector function u is called a weak solution of (1.1) if $u\in L^\infty (0, \infty ; L^2_\sigma (\mathbb {R}^n_+))\cap L^2_{loc} ([0, \infty );$ $H^1_0(\mathbb {R}^n_+))$ satisfies problem (1.1) in the sense of distributions. Moreover, the energy inequality holds for almost all $t\in [0, \infty )$ including $t=0$:

$$\begin{aligned} \Vert u(t)\Vert ^2_{L^2(\mathbb {R}^n_+)}+2\int _0^t\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}ds\le \Vert u_0\Vert ^2_{L^2(\mathbb {R}^n_+)}. \end{aligned}$$

The weak solution u of (1.1) is so far known to be unique only if u belongs to a certain class of functions which, however, does not cover the whole space $L^\infty (0, \infty ; L^2_\sigma (\mathbb {R}^n_+))\cap L^2_{loc} ([0, \infty );$ $H^1_0(\mathbb {R}^n_+))$. Furthermore if the Serrin’s condition holds: $u\in L^q(0, \infty ; L^r(\mathbb {R}^n_+))$ with $\frac{2}{q}+\frac{n}{r}\le 1$, $2\le q<\infty$, $n<r\le \infty$, then u is called a strong solution, which is smooth in $\mathbb {R}^n_+\times (0, +\infty )$.

Before stating the main results, we first recall a small-data global existence of classical solution in the half-space, see, e.g., Theorem 3.2 in [12].

Theorem 1.1

Let $u_0\in L^2_\sigma (\mathbb {R}^n_+)$ $\,(n\ge 2)$. Then there exists a number $\epsilon _0>0$ such that if $\Vert u_0\Vert _{L^n(\mathbb {R}^n_+)}\le \epsilon _0$ (smallness condition is unnecessary if $n=2$), problem (1.1) admits a unique strong solution u.

Let A denote the Stokes operator $-P\Delta$ in $\mathbb {R}^n_+$, where the Laplacian $\Delta$ in $\mathbb {R}^n_+$ is endowed with the homogeneous Dirichlet boundary condition, P is the projection operator: $L^r(\mathbb {R}^n_+)\longrightarrow L^r_\sigma (\mathbb {R}^n_+)$, $1<r<\infty$. Then the function $e^{-tA}u_0$ solves the Stokes system, that is, problem (1.1) with $u\cdot \nabla u$ deleted, with the initial datum $u_0$. The (weak or strong) solution u of problem (1.1) can be written as follows:

$$\begin{aligned} u(t)=e^{-tA}u_0-\int _0^t e^{-(t-s)A}P(u(s)\cdot \nabla ) u(s)ds. \end{aligned}$$

Note that for $t\ge s>0$, $e^{-(t-s)A}P(u(s)\cdot \nabla ) u(s)$ is also a Stokes flow with the initial data $P(u(s)\cdot \nabla ) u(s)$. That is, set $w(t)=e^{-(t-s)A}P(u(s)\cdot \nabla ) u(s),$ $t\ge s>0$, then

$$\begin{aligned} \left\{ \begin{array}{lll} \partial _tw-\Delta w+\nabla \pi =0& \text{ in }& \mathbb {R}^n_+\times (s, \infty ),\\ \nabla \cdot w=0 & \text{ in }& \mathbb {R}^n_+\times (s, \infty ),\\ w(x, t)=0 & \text{ on } & \partial \mathbb {R}^n_+\times (s, \infty ),\\ w(x, t)|_{t=s}=P(u(s)\cdot \nabla ) u(s) & \text{ in }& \mathbb {R}^n_+. \end{array} \right. \end{aligned}$$

Whence in order to establish the estimates of solutions to Navier–Stokes equations, it is necessary to investigate the linear Stokes problem.

If $u_0\in L^1_\sigma (\mathbb {R}^n_+)$ satisfies some additional assumptions, Bae and Choe [3] proved the decay rate for $t>0$: $\Vert \nabla e^{-tA}u_0\Vert _{L^q(\mathbb {R}^n_+)}\le Ct^{-1}$ with $1<q<\infty$. If the initial data $u_0$ lies in an appropriate weighted space, and satisfies the average condition: $\int _{\mathbb {R}^{n-1}}u_0(y^\prime , y_n)dy^\prime =0$ for $a.e.\;y_n>0$, Bae showed $L^1$-decay of the Stokes flow in [1], and $L^1-$time estimate in [2] for the gradient of Stokes flow, respectively. It is natural to ask whether the Stokes flow $e^{-tA}a$ belongs to $L^1(\mathbb {R}^n_+)$, $t>0$ for every $a\in L^1_\sigma (\mathbb {R}^n_+)$. The answer to this question is negative, see a specific counterexample given in the Appendix.

The first result focuses on the weighted $L^1$-time decay estimate of the Stokes flow, as far as we know, which is first considered for the negative exponent case. Of course, the following Theorem 1.2 is also very interesting by itself.

Theorem 1.2

Let $a=(a_1, a_2,\cdots , a_n)\in L^1(\mathbb {R}^n_+)$, $a_n\,|_{\partial \mathbb {R}^n_+}=0$, $\nabla \cdot a=0$ in $\mathbb {R}^n_+\,\,(n\ge 2)$. Then the Stokes flow $e^{-tA}a$ satisfies for $t>0$

$$\begin{aligned} \Vert |x^\prime |^{-\alpha }e^{-tA}a\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{\alpha }{2}}\Vert a\Vert _{L^1(\mathbb {R}^n_+)},\;\;0<\alpha <n-1; \end{aligned}$$

and

$$\begin{aligned} \Vert x_n^{-\alpha }e^{-tA}a\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{\alpha }{2}}\Vert a\Vert _{L^1(\mathbb {R}^n_+)},\;\;0<\alpha <1; \end{aligned}$$

where C depends only on $n, \alpha$.

The decay problem for solutions to the Navier–Stokes equations was first proposed by Leray [17] for the Cauchy problem. Schonbek [18, 21] attacked this problem and succeeded for the first time in showing existence of weak solutions with explicit decay rate. In [22], Schonbek first developed a very effective new method, Fourier splitting method, called also Schonbek’s method, which has been applied extensively for studying decay properties of solutions to various diffusive partial differential equations. This method does not depend on the linearized underlying equations. In a series of articles (see [18]–[23]), Schonbek completed systematic outstanding research work, made many innovative achievements on decay properties of Navier–Stokes flows in the whole space, which subsequently have been cited and generalized widely by many mathematical researchers.

All of these techniques employed in the whole space are not applicable directly to problem (1.1), because the projection operator $P:\,L^1(\mathbb {R}^n_+)\rightarrow L^1_\sigma (\mathbb {R}^n_+)$ is not bounded any more, and $P\Delta \ne \Delta P$, which causes many essential difficulties in treating the nonlinear term $(u\cdot \nabla )u$. Here we mention briefly some known results obtained on the half-space. Bae and Choe [3], Fujigaki and Miyakawa [12] studied asymptotic behavior for weak and strong solutions of (1.1) in $L^r(\mathbb {R}^n_+)$ with $1<r<\infty$. Starting from the proof of existence, Farwig, Kozono and David [10] get a weak solution satisfying $\Vert v(t)\Vert _{L^2}\longrightarrow 0$ as $t\longrightarrow \infty$, and determine an upper bound for the decay rate. Relevant topics are referred to see [4]– [9], [11, 15, 16] and the references therein.

Our second result is to establish $L^1$-decay rates for the Navier–Stokes flows with negative exponent weights. To our knowledge, few weighted $L^1$-decay estimates in such cases are available on solutions of problem (1.1).

Theorem 1.3

Assume $u_0\in L^1(\mathbb {R}^n_+)\cap L^2_\sigma (\mathbb {R}^n_+)$ $(n\ge 2)$. Then the strong solution u of (1.1) obtained in Theorem 1.1 satisfies for $t>0$

$$\begin{aligned} \Vert |x^\prime |^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le \left\{ \begin{array}{lll} Ct^{-\frac{\alpha }{2}}& \text{ if }& n\ge 3,\\ Ct^{-\frac{\alpha }{2}}\log _e(1+t)& \text{ if }& n=2,\\ \end{array} \right. \;\;\;\;0<\alpha <\min \{2, n-1\}; \end{aligned}$$

and

$$\begin{aligned} \Vert x_n^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le \left\{ \begin{array}{lll} Ct^{-\frac{\alpha }{2}}& \text{ if }& n\ge 3,\\ Ct^{-\frac{\alpha }{2}}\log _e(1+t)& \text{ if }& n=2,\\ \end{array} \right. \;\;\;\;0<\alpha <1. \end{aligned}$$

Furthermore if $x_nu_0\in L^1(\mathbb {R}^n_+)$, there holds

$$\begin{aligned} \Vert |x^\prime |^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{\alpha }{2}},\;\;\;0<\alpha <\min \{2, n-1\}; \end{aligned}$$

and

$$\begin{aligned} \Vert x_n^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{\alpha }{2}},\;\;\;0<\alpha <1. \end{aligned}$$

Remark

From the view point of mathematics, it is necessary to discuss how the classical weighted function $|x|^\alpha$ ($\alpha \in \mathbb {R}^1$) affects the large time asymptotic behavior of the Navier–Stokes flows. Due to the fact that the Stokes flow $e^{tA}u_0$ is not in $L^1$ space (see Appendix), we do not expect the solution of the Navier–Stokes system to be in $L^1$ space with weighted function $|x|^\alpha$, $\alpha \ge 0$. Therefore, it is natural to consider the case of $\alpha <0$, for which no conclusion has been found yet. The negative power $\alpha$ for the weighted function $|x|^\alpha$ implies the energy increases in the spatial direction, and simultaneously in the temporal direction the energy decreases as the power $\frac{\alpha }{2}$.

The $L^r-$asymptotic behavior of higher-order spatial derivatives of solutions of problem (1.1) is established for $1<r\le \infty$ in [14]. However, up to now, the case for $r=1$ remains still open, because the a priori estimates on the steady Stokes system are not valid any more in $L^1(\mathbb {R}^n_+)$. Applying the nonstationary Stokes’s estimates to the integral equation on the solution of (1.1), we inevitably encounter the strong singularity. Exactly speaking, under the assumptions of Theorem 1.4 below on the initial data $u_0$, together with the following Lemma 3.1, we conclude that for $t>0$

$$\begin{aligned} \displaystyle & \Vert \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le \displaystyle Ct^{-\frac{3}{2}}\int _{\mathbb {R}^n_+}|u_0(y)|dy +C\int _0^t (t-s)^{-\frac{3}{2}}\Vert P(u(s)\cdot \nabla ) u(s)\Vert _{L^1(\mathbb {R}^n_+)}ds\\ & \quad \le \displaystyle Ct^{-\frac{3}{2}}\int _{\mathbb {R}^n_+}|u_0(y)|dy+C\left( \int _0^\frac{t}{2} +\int _\frac{t}{2}^t\right) (t-s)^{-\frac{3}{2}}(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)})ds. \end{aligned}$$

In the above calculations, we made use of the Stokes’s estimate (see [14]):

Let $a=(a_1, a_2,\cdots , a_n)\in L^1(\mathbb {R}^n_+)$, $a_n\,|_{\partial \mathbb {R}^n_+}=0$, $\nabla \cdot a=0$ in $\mathbb {R}^n_+\,\,(n\ge 2)$. Then

$$\begin{aligned} \Vert \nabla ^3[e^{-tA}a]\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{3}{2}}\int _{\mathbb {R}^n_+}|a(y)|dy,\;\;\forall t>0. \end{aligned}$$

Since the strong solution u of problem (1.1) satisfies for any $0<s<t$

$$\begin{aligned} \Vert u(t)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\int _s^t\Vert \nabla u(\tau )\Vert ^2_{L^2(\mathbb {R}^n_+)}d\tau =\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}. \end{aligned}$$

A strong singularity arises from the following term for $t>0$:

$$\begin{aligned} & \displaystyle \int _\frac{t}{2}^t(t-s)^{-\frac{3}{2}}(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)})ds\\ & \quad \ge \displaystyle \int _\frac{t}{2}^t(t-s)^{-\frac{3}{2}}\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}ds\\ & \quad \ge \displaystyle \int _\frac{t}{2}^t(t-s)^{-\frac{3}{2}}ds\Vert u(t)\Vert ^2_{L^2(\mathbb {R}^n_+)}=+\infty . \end{aligned}$$

It is almost impossible to avoid this kind of difficulty by making use of such a direct method, new ideas and innovative approaches have to be found and employed. The following result (i.e. Theorem 1.4) is an attempt on such topics. Schonbek and Wiegner [23] established the decay estimates of arbitrary spatial derivatives of solutions in the whole space $\mathbb {R}^n$, and their method depends on the Fourier transform and on the commutativity between projection and differential operators, neither of which seems to be effective directly in dealing with problem (1.1). The following result is a long-time unsolved problem, which is inspired mainly by the remarkable research work by Schonbek and Wiegner in the whole space, see [23].

Theorem 1.4

Assume $u_0\in L^1(\mathbb {R}^n_+)\bigcap L^2_\sigma (\mathbb {R}^n_+)$ $(n\ge 2)$. Then the strong solution u given in Theorem 1.1 satisfies for any $t>1$

$$\begin{aligned} \Vert \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le \left\{ \begin{array}{lll} Ct^{-\frac{3}{2}}& \text{ if }& n\ge 3,\\ Ct^{-\frac{3}{2}}\log _e(1+t)& \text{ if }& n=2.\\ \end{array} \right. \end{aligned}$$

Suppose $x_nu_0\in L^1(\mathbb {R}^n_+)$, there holds for every $t>1$

$$\begin{aligned} \Vert \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{3}{2}}. \end{aligned}$$

(1.2)

Furthermore if $x_nu_0, (1+x_n)\nabla u_0\in L^2(\mathbb {R}^n_+)$, then for $0<\beta <1$

$$\begin{aligned} \Vert x_n^\beta \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{3}{2}+\frac{\beta }{2}},\;\;\;\forall \;t>1. \end{aligned}$$

(1.3)

Remarks

• (1) As seen below, the proof’s procedure of $\Vert \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}$ is technical, complicated and lengthy. However, it is still possible to establish the $L^1$-time decay of higher spatial derivatives $\nabla ^ku$ ($k\ge 4$) by using these methods employed in this article.

• (2) To our knowledge, it is the first time to show the (weighted) decay estimates of the cubic spatial derivatives in $L^1(\mathbb {R}^n_+)$. Let $x=(x^\prime , x_n), y=(y^\prime , y_n)\in \mathbb {R}^n_+$, and $0<\beta <1$. It is not sure whether similar decay results are true for $\Vert |x^\prime |^\beta \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}$. Because in the proof procedure of Theorem 1.4, we make full use of the special structure of the half space, the weight $x_n^\beta$ can be treated by $(x_n+y_n)^\beta$ in Solonnikov’s solution formula. On the other hand, since $|x^\prime |^\beta \le |x^\prime -y^\prime |^\beta +|y^\prime |^\beta$, and the strong singularity arises from the term containing the weight $|y^\prime |^\beta$ in Solonnikov’s solution formula, we readily find that the methods employed in the proof of Theorem 1.4 does not work in the case of the weight $|x^\prime |^\beta$. Up to now, the decay estimate of $\Vert |x^\prime |^\beta \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}$ is still unsolved.

This article is organized as follows: In Sect. 2, we collect some basic known results, and give the proof of Theorem 1.2. Section 3 devotes to establishing the weighted $L^1$-time decay of the strong solution of problem (1.1). By means of properties of the Gaussian kernel’s convolution, we construct some crucial weighted estimates on a class of elliptic problem relating to the convection term in problem (1.1), which will be frequently applied in the proof of Theorem 1.4. In fact, such a study is of independent interest. Together with some known decay estimates of solutions of (1.1), we establish $L^1$-time (weighted) decays of solutions of (1.1), see Theorem 1.3. In Sect. 4, we focus on studying the (weighted) decays of cubic spatial derivatives of the strong solution of (1.1). To do this, we first find an important formula on the fundamental solution of elliptic operator $-\Delta$, which is important in overcoming the strong singularity. Combining Theorem 1.3 and Solonnikov’s solution formula, we finally achieve the desired results, e.g., Theorem 1.4.

2 Weighted $L^1$-decay for the Stokes flow

In this section, we first introduce Solonnikov’s solution formula and related basic estimates.

Let $a=(a_1, a_2,\cdots , a_n)\in L^1(\mathbb {R}^n_+)$, $a_n\,|_{\partial \mathbb {R}^n_+}=0$, $\nabla \cdot a=0$ in $\mathbb {R}^n_+\,\,(n\ge 2)$. Then it holds for $t>0$ (see [24])

$$\begin{aligned} {[}e^{-tA}a](x)=\int _{\mathbb {R}^n_+}\mathcal {M}(x,y,t)a(y)dy,\;\;\;x\in \mathbb {R}^n_+, \end{aligned}$$

(2.1)

where $\mathcal {M}=(M_{ij})_{i,j=1,2,\cdots ,n}$ is defined as follows

$$\begin{aligned} \displaystyle M_{ij}(x,y,t)= & \displaystyle \delta _{ij}(G_t(x-y)-G_t(x-y^*))\nonumber \\ & \displaystyle +4(1-\delta _{jn})\frac{\partial }{\partial x_j}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_i}G_t(z-y^*)dz\nonumber \\\equiv & \displaystyle \delta _{ij}(G_t(x-y)-G_t(x-y^*))+M_{ij}^*(x,y,t), \end{aligned}$$

(2.2)

$y^*=(y_1,y_2,\cdots , -y_n)$, $G_t(x)=(4\pi t)^{-\frac{n}{2}}e^{-\frac{|x|^2}{4t}}$ is the Gaussian kernel,

$$\begin{aligned} \delta _{ij}=\left\{ \begin{array}{lll} \displaystyle 1 & \text{ if }& i=j,\\ \displaystyle 0& \text{ if }& i\ne j, \end{array} \right. \qquad E(x)=\left\{ \begin{array}{lll} \displaystyle \frac{1}{n(2-n)\omega _n|x|^{n-2}} & \text{ if }& n\ge 3,\\ \displaystyle -\frac{1}{2\pi }\log _e \frac{1}{|x|}& \text{ if }& n=2, \end{array} \right. \end{aligned}$$

$\omega _n=\frac{\pi ^\frac{n}{2}}{\Gamma (\frac{n}{2}+1)}$ denotes the volume of the unit ball in $\mathbb {R}^n$, E(x) is the fundamental solution of the Laplace operator $-\Delta$. Namely, $-\Delta E(x)=\delta (x)$ in the sense of distributions, where $\delta (x)$ denotes the Dirac function concentrating at $x=0$.

Let $\vec {\ell }$ denote $(\ell _1, \ell _2, \cdots , \ell _{n-1}, \ell _n)=(\ell ^\prime , \ell _n)$. Then the estimate for $M_{ij}^*$ ($1\le i, j\le n$) in (2.2) holds for $x, y\in \mathbb {R}^n_+$, $t>0$

$$\begin{aligned} |\partial _t^s\nabla _x^{\vec {k}}\nabla _y^{\vec {m}}M_{ij}^*(x,y,t)|\le Ct^{-s-\frac{m_n}{2}}(t+x_n^2)^{-\frac{k_n}{2}}(|x-y^*|^2+t)^{-\frac{n+|k^\prime |+|m^\prime |}{2}}e^{-\frac{cy_n^2}{t}}. \end{aligned}$$

(2.3)

Proof of Theorem 1.2

Let $y=(y^\prime , y_n)\in \mathbb {R}^n_+$. Then for $t>0$

$$\begin{aligned} & \displaystyle \int _{\mathbb {R}^{n-1}}\int _0^\infty |x^\prime |^{-\alpha }(|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n}dx_ndx^\prime \nonumber \\ & \quad \le \displaystyle \frac{1}{n-1}\int _{\mathbb {R}^{n-1}}|x^\prime |^{-\alpha }(|x^\prime -y^\prime |+\sqrt{t})^{-n+1}dx^\prime \nonumber \\ & \quad =\displaystyle \frac{t^{-\frac{\alpha }{2}}}{n-1}\int _{\mathbb {R}^{n-1}}|z^\prime |^{-\alpha }(|z^\prime -t^{-\frac{1}{2}}y^\prime |+1)^{-n+1}dz^\prime \nonumber \\ & \quad \le \displaystyle \frac{t^{-\frac{\alpha }{2}}}{n-1}\int _{|z^\prime |\le 1}|z^\prime |^{-\alpha }dz^\prime +\frac{t^{-\frac{\alpha }{2}}}{n-1}\left( \int _{|z^\prime |>1}|z^\prime |^{-n}dz^\prime \right) ^\frac{\alpha }{n}\nonumber \\ & \qquad \displaystyle \times \left( \int _{|z^\prime |>1} (|z^\prime -t^{-\frac{1}{2}}y^\prime |+1)^{-\frac{n(n-1)}{n-\alpha }}dz^\prime \right) ^{1-\frac{\alpha }{n}}\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}\int _0^1s^{-\alpha +n-2}ds+Ct^{-\frac{\alpha }{2}}\left( \int _1^\infty s^{-2}ds\right) ^\frac{\alpha }{n}\nonumber \\ & \qquad \displaystyle \times \left( \int _0^\infty (s+1)^{-\frac{n(n-1)}{n-\alpha }+n-2}ds\right) ^{1-\frac{\alpha }{n}}\nonumber \\ & \quad \le \displaystyle C_1(n,\alpha )t^{-\frac{\alpha }{2}},\;\;\;0<\alpha <n-1; \end{aligned}$$

(2.4)

and

$$\begin{aligned} & \displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}x_n^{-\alpha }(|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n}dx^\prime dx_n\nonumber \\ \quad\le & \displaystyle \int _0^\infty x_n^{-\alpha }\int _{\mathbb {R}^{n-1}}(|x^\prime |+x_n+\sqrt{t})^{-n}dx^\prime dx_n\nonumber \\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}}\int _0^\infty x_n^{-\alpha }\int _{\mathbb {R}^{n-1}} (|x^\prime |+x_n+1)^{-n}dx^\prime dx_n\nonumber \\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}}\int _0^\infty x_n^{-\alpha }\int _0^\infty (s+x_n+1)^{-n}s^{n-2}dsdx_n\nonumber \\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}}\int _0^\infty x_n^{-\alpha }\int _0^\infty (s+x_n+1)^{-2}dsdx_n\nonumber \\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}}\int _0^\infty x_n^{-\alpha }(x_n+1)^{-1}dx_n\nonumber \\\le & \displaystyle Ct^{-\frac{\alpha }{2}}\left( \int _0^1 x_n^{-\alpha }dx_n+\int _1^\infty x_n^{-\alpha -1}dx_n\right) \nonumber \\ \quad\le & \displaystyle C_2(n,\alpha )t^{-\frac{\alpha }{2}},\;\;\;0<\alpha <1. \end{aligned}$$

(2.5)

Set $G_t^{(n-1)}(x^\prime )=(4\pi t)^{-\frac{n-1}{2}}e^{-\frac{|x^\prime |^2}{4t}}$, $x^\prime \in \mathbb {R}^{n-1}$; $G_t^{(1)}(x_n)=(4\pi t)^{-\frac{1}{2}}e^{-\frac{|x_n|^2}{4t}}$, $x_n\in \mathbb {R}^1$. Then $G_t(x)=G_t^{(n-1)}(x^\prime )G_t^{(1)}(x_n)$, $x=(x^\prime , x_n)$. Whence for $y=(y^\prime , y_n)\in \mathbb {R}^n_+$ and $t>0$

$$\begin{aligned} & \displaystyle \int _{\mathbb {R}^n_+}|x^\prime |^{-\alpha }|G_t(x-y)-G_t(x+y^*)|dx\nonumber \\ & \quad =\displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}|x^\prime |^{-\alpha }G_t^{(n-1)}(x^\prime -y^\prime )[G_t^{(1)}(x_n-y_n)-G_t^{(1)}(x_n+y_n)] dx^\prime dx_n\nonumber \\ & \quad \le \displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}|x^\prime |^{-\alpha }G_t^{(n-1)}(x^\prime -y^\prime )G_t^{(1)}(x_n-y_n) dx^\prime dx_n\nonumber \\ & \quad \le \displaystyle \int _{\mathbb {R}^{n-1}}|x^\prime |^{-\alpha }G_t^{(n-1)}(x^\prime -y^\prime )dx^\prime \nonumber \\ & \quad =\displaystyle t^{-\frac{\alpha }{2}}\int _{\mathbb {R}^{n-1}}|z^\prime |^{-\alpha }G_1^{(n-1)}(z^\prime -t^{-\frac{1}{2}}y^\prime )dz^\prime \nonumber \\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}\left( \int _{|z^\prime |\le 1}|z^\prime |^{-\alpha }dz^\prime +\int _{|z^\prime |>1}G_1^{(n-1)}(z^\prime -t^{-\frac{1}{2}}y^\prime ) dz^\prime \right) \nonumber \\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}\left( \int _0^1s^{-\alpha +n-2}ds+\int _{\mathbb {R}^{n-1}}G_1^{(n-1)}(z^\prime ) dz^\prime \right) \nonumber \\ & \quad \le \displaystyle C_3(n,\alpha )t^{-\frac{\alpha }{2}},\;\;\;0<\alpha <n-1; \end{aligned}$$

(2.6)

and

$$\begin{aligned} & \displaystyle \int _{\mathbb {R}^n_+}x_n^{-\alpha }|G_t(x-y)-G_t(x+y^*)|dx\nonumber \\ & \quad =\displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}x_n^{-\alpha }G_t^{(n-1)}(x^\prime -y^\prime )[G_t^{(1)}(x_n-y_n)-G_t^{(1)}(x_n+y_n)] dx^\prime dx_n\nonumber \\ & \quad \le \displaystyle \int _{\mathbb {R}^{n-1}}G_t^{(n-1)}(x^\prime -y^\prime )dx^\prime \int _0^\infty x_n^{-\alpha }G_t^{(1)}(x_n-y_n) dx_n\nonumber \\ & \quad =\displaystyle t^{-\frac{\alpha }{2}}\int _0^\infty x_n^{-\alpha }G_1^{(1)}(x_n-y_n) dx_n\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}\left( \int _0^1 x_n^{-\alpha }dx_n+\int _1^\infty G_1^{(1)}(x_n-y_n)dx_n\right) \nonumber \\ & \quad \le \displaystyle C_4(n,\alpha )t^{-\frac{\alpha }{2}},\;\;\;0<\alpha <1. \end{aligned}$$

(2.7)

Using the representation (2.1) of the Stokes flow $e^{-tA}a$, and the estimates of (2.3)–(2.7), we find for $t>0$

$$\begin{aligned} & \displaystyle \int _{\mathbb {R}^n_+}|x^\prime |^{-\alpha }|[e^{-tA}a](x)|dx\\ & \quad \le \displaystyle \int _{\mathbb {R}^n_+}\int _{\mathbb {R}^n_+}|x^\prime |^{-\alpha }|G_t(x-y)-G_t(x+y^*)||a(y)|dxdy\\ & \qquad \displaystyle +C\int _{\mathbb {R}^n_+}\int _{\mathbb {R}^n_+}|x^\prime |^{-\alpha }(|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n}|a(y)|dxdy\\ & \quad \le \displaystyle C(n,\alpha )t^{-\frac{\alpha }{2}}\Vert a\Vert _{L^1(\mathbb {R}^n_+)},\;\;\;0<\alpha <n-1; \end{aligned}$$

and

$$\begin{aligned} & \displaystyle \int _{\mathbb {R}^n_+}x_n^{-\alpha }|[e^{-tA}a](x)|dx\\ & \quad \le \displaystyle \int _{\mathbb {R}^n_+}\int _{\mathbb {R}^n_+}x_n^{-\alpha }|G_t(x-y)-G_t(x+y^*)||a(y)|dxdy\\ & \qquad \displaystyle +C\int _{\mathbb {R}^n_+}\int _{\mathbb {R}^n_+} x_n^{-\alpha }(|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n}|a(y)|dxdy\\ & \quad \le \displaystyle C(n,\alpha )t^{-\frac{\alpha }{2}}\Vert a\Vert _{L^1(\mathbb {R}^n_+)},\;\;\;0<\alpha <1. \end{aligned}$$

$\square$

3 Asymptotic behavior for the Navier–Stokes flows in $L^1(\mathbb {R}^n_+)$

Let $g=\mathcal {N}f$ denote the solution of the Neumann problem

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta g=f & \text{ in } & \mathbb {R}^n_+,\\ g(x)\longrightarrow 0 & \text{ as } & |x|\longrightarrow \infty ,\\ \partial _\nu g\,|_{\partial \mathbb {R}^n_+}=0. \end{array} \right. \end{aligned}$$

Then (see [13])

$$\begin{aligned} \mathcal {N}=\int _0^\infty F(\tau )d\tau , \end{aligned}$$

(3.1)

where the operator F(t) is defined by

$$\begin{aligned} {[}F(t)f](x)=\int _{\mathbb {R}^n_+}[G_t(x^\prime -y^\prime , x_n-y_n)+G_t(x^\prime -y^\prime , x_n+y_n)]f(y)dy, \end{aligned}$$

and $G_t(x)=(4\pi t)^{-\frac{n}{2}}e^{-\frac{|x|^2}{4t}}$ is the Gaussian kernel in $\mathbb {R}^n$.

It follows from (3.1) that the solution $g=\mathcal {N}f$ can be written as follows

$$\begin{aligned} (\mathcal {N}f)(x)=\int _0^\infty \int _{\mathbb {R}^n_+}[G_t(x^\prime -y^\prime , x_n-y_n)+G_t(x^\prime -y^\prime , x_n+y_n)]f(y)dydt. \end{aligned}$$

(3.2)

In addition, there holds for any $u, v\in L^2_\sigma (\mathbb {R}^n_+)\bigcap H^1_0(\mathbb {R}^n_+)$ (see [13])

$$\begin{aligned} P(u\cdot \nabla ) v=(u\cdot \nabla ) v+\sum \limits _{i, j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_iv_j). \end{aligned}$$

(3.3)

Lemma 3.1

Let $0<\theta <1$ and $0\le \beta \le \alpha <1$. Then for any $u, v\in C^\infty _{0,\sigma }(\mathbb {R}^n_+)$

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^\alpha \nabla \mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C(\Vert u\Vert _{L^2(\mathbb {R}^n_+)}\Vert v\Vert _{L^2(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta } u\Vert _{L^2(\mathbb {R}^n_+)}\Vert y_n^\beta v\Vert _{L^2(\mathbb {R}^n_+)}+\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^2(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^2(\mathbb {R}^n_+)}); \end{aligned}$$

(3.4)

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^\alpha \nabla ^2\mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\left( \Vert u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert v\Vert _{L^{2}(\mathbb {R}^n_+)}\right. \nonumber \\ & \qquad \left. \displaystyle +\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\right) \nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta }u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)} +\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)} \Vert y_n^\beta v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert y_n^{\alpha -\beta }\nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\big ); \end{aligned}$$

(3.5)

and

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^{-\theta }\nabla \mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C(\Vert u\Vert _{L^2(\mathbb {R}^n_+)}\Vert v\Vert _{L^2(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^2(\mathbb {R}^n_+)}). \end{aligned}$$

(3.6)

Remark

Replacing the weight $x_n^\alpha$ by $|x|^\alpha$ in (3.4) and (3.5) respectively, the following estimates hold (the proofs are similar to these of (3.4) and (3.5)) for any $u, v\in C^\infty _{0,\sigma }(\mathbb {R}^n_+)$

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^n|x|^\alpha \nabla \mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\\ & \quad \le \displaystyle C(\Vert u\Vert _{L^2(\mathbb {R}^n_+)}\Vert v\Vert _{L^2(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert |y|^{\alpha -\beta } u\Vert _{L^2(\mathbb {R}^n_+)}\Vert |y|^\beta v\Vert _{L^2(\mathbb {R}^n_+)}+\Vert |y|^{\alpha -\beta }\nabla u\Vert _{L^2(\mathbb {R}^n_+)}\Vert |y|^\beta \nabla v\Vert _{L^2(\mathbb {R}^n_+)});\\ & \displaystyle \left\| \sum \limits _{i,j=1}^n|x|^\alpha \nabla ^2\mathcal {N}\partial _i\partial _j(u_iu_j)\right\| _{L^1(\mathbb {R}^n_+)}\\ & \quad \le \displaystyle C\left( \Vert u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert v\Vert _{L^{2}(\mathbb {R}^n_+)}\right. \\ & \qquad \displaystyle +\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert |y|^{\alpha -\beta }u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert |y|^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)} +\Vert |y|^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert |y|^\beta v\Vert _{L^{2}(\mathbb {R}^n_+)}\\ & \qquad \left. \displaystyle +\Vert |y|^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert |y|^\beta \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert |y|^{\alpha -\beta }\nabla ^2u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert |y|^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\right) . \end{aligned}$$

Proof

Denote the odd and even extensions of a function f from $\mathbb {R}^n_+$ to $\mathbb {R}^n$, respectively by

$$\begin{aligned} f^*(x^\prime , x_n)=\left\{ \begin{array}{lll} \displaystyle f(x^\prime , x_n) & \text{ if } & x_n\ge 0,\\ \displaystyle -f(x^\prime , -x_n) & \text{ if } & x_n<0, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} f_*(x^\prime , x_n)=\left\{ \begin{array}{lll} \displaystyle f(x^\prime , x_n) & \text{ if } & x_n\ge 0,\\ \displaystyle f(x^\prime , -x_n) & \text{ if } & x_n<0. \end{array} \right. \end{aligned}$$

Let $u, v\in C^\infty _{0,\sigma }(\mathbb {R}^n_+)$. Then from (3.2), one has for any $1\le k\le n$

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^\alpha \partial _k\mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)} =\displaystyle \left\| \sum \limits _{i,j=1}^nx_n^\alpha \partial _k\int _0^\infty F(\tau )\partial _i\partial _j(u_iv_j)d\tau \right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad =\displaystyle C \left\| \sum \limits _{i,j=1}^n \theta (x_n)x_n^\alpha \partial _k\left( \int _0^1+\int _1^\infty \right) G_\tau *[\partial _i\partial _j(u_iv_j)]_*d\tau \right\| _{L^1(\mathbb {R}^n)} \nonumber \\ & \quad \le \displaystyle C\left\| \int _0^1\int _{\mathbb {R}^n}|x_n-y_n|^\alpha |\partial _kG_\tau (x-y)||\left[ \sum \limits _{i,j=1}^n\partial _i\partial _j(u_iv_j)\right] _*(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \qquad \displaystyle +C\left\| \int _0^1\int _{\mathbb {R}^n}|\partial _k G_\tau (x-y)||y_n|^\alpha |\left[ \sum \limits _{i,j=1}^n\partial _i\partial _j(u_iv_j)\right] _*(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \qquad \displaystyle +C\sum \limits _{i,j=1}^n\left\| \int _1^\infty \int _{\mathbb {R}^n}|x_n-y_n|^\alpha |\partial _k\partial _i\partial _j G_\tau (x-y)||w_{ij}(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \qquad \displaystyle +C\sum \limits _{i,j=1}^n\left\| \int _1^\infty \int _{\mathbb {R}^n}|\partial _k\partial _i\partial _j G_\tau (x-y)||y_n|^\alpha |w_{ij}(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \quad =\displaystyle M_1+M_2+M_3+M_4, \end{aligned}$$

(3.7)

where $\theta (x_n)=1$ if $x_n\ge 0$, $\theta (x_n)=0$ if $x_n<0$; $w_{ij}=(u_iv_j)_*$ if $1\le i,j\le n-1$ or $i=j=n$, $w_{in}=(u_iv_n)^*$ if $1\le i\le n-1$, $w_{nj}=(u_nv_j)^*$ if $1\le j\le n-1$.

Let $0\le \beta \le \alpha <1$. Then

$$\begin{aligned} \displaystyle M_1+M_2\le & \displaystyle C\int _0^1\left\| |x_n|^\alpha \partial _kG_\tau (x^\prime , x_n)\right\| _{L^1(\mathbb {R}^n)}d\tau \Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)} \nonumber \\ & \displaystyle +C\int _0^1\Vert \partial _kG_\tau \Vert _{L^1(\mathbb {R}^n)}d\tau \Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle C\int _0^1\tau ^{\frac{\alpha }{2}-\frac{1}{2}}d\tau \Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)} \nonumber \\ & \displaystyle +C\int _0^1\tau ^{-\frac{1}{2}}d\tau \Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle C\big (\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\big ); \end{aligned}$$

(3.8)

$$\begin{aligned} \displaystyle M_3+M_4\le & \displaystyle C\sum \limits _{i,j=1}^n\int _1^\infty \int _{\mathbb {R}^n}|x_n|^\alpha |\partial _k\partial _i\partial _jG_\tau (x^\prime , x_n)|dxd\tau \Vert w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \displaystyle + C\sum \limits _{i,j=1}^n\int _1^\infty \int _{\mathbb {R}^n}|\partial _k\partial _i\partial _jG_\tau (x^\prime , x_n)|dxd\tau \Vert y_n^\alpha w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle C\int _1^\infty \tau ^{\frac{\alpha }{2}-\frac{3}{2}}d\tau \Vert w_{ij}\Vert _{L^1(\mathbb {R}^n_+)} + C\sum \limits _{i,j=1}^n\int _1^\infty \tau ^{-\frac{3}{2}}d\tau \Vert y_n^\alpha w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle C\Vert u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert v\Vert _{L^{2}(\mathbb {R}^n_+)} +C\Vert y_n^{\alpha -\beta }u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta v\Vert _{L^{2}(\mathbb {R}^n_+)}. \end{aligned}$$

(3.9)

From (3.7)–(3.9), we infer that (3.4) holds.

Now we give the proof of (3.5). Let $0\le \beta \le \alpha <1$, then for any $1\le k\le n$

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^\alpha \nabla \partial _k\mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\left\| \int _0^1\int _{\mathbb {R}^n}|x_n-y_n|^\alpha |\partial _kG_\tau (x-y)||\nabla \left[ \sum \limits _{i,j=1}^n\partial _i\partial _j(u_iv_j)\right] _*(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \qquad \displaystyle +C\left\| \int _0^1\int _{\mathbb {R}^n}|\partial _k G_\tau (x-y)||y_n|^\alpha |\nabla \left[ \sum \limits _{i,j=1}^n\partial _i\partial _j(u_iv_j)\right] _*(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \qquad \displaystyle +C\sum \limits _{i,j=1}^n\left\| \int _1^\infty \int _{\mathbb {R}^n}|x_n-y_n|^\alpha |\partial _k\partial _i\partial _j G_\tau (x-y)||\nabla w_{ij}(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \qquad \displaystyle +C\sum \limits _{i,j=1}^n\left\| \int _1^\infty \int _{\mathbb {R}^n}|\partial _k\partial _i\partial _j G_\tau (x-y)||y_n|^\alpha |\nabla w_{ij}(y)|dyd\tau \right\| _{L^1_x(\mathbb {R}^n)}\nonumber \\ & \quad =\displaystyle N_1+N_2+N_3+N_4; \end{aligned}$$

(3.10)

$$\begin{aligned} & \displaystyle N_1+N_2\le \displaystyle C\int _0^1\Vert |x_n|^\alpha \partial _kG_\tau (x^\prime , x_n)\Vert _{L^1(\mathbb {R}^n)}d\tau \nonumber \\ & \qquad \displaystyle \times (\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)})\nonumber \\ & \qquad \displaystyle +C\int _0^1\Vert \partial _kG_\tau \Vert _{L^1(\mathbb {R}^n)}d\tau (\Vert y_n^{\alpha -\beta }\nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)})\nonumber \\ & \quad \le \displaystyle C\int _0^1\tau ^{\frac{\alpha }{2}-\frac{1}{2}}d\tau (\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)})\nonumber \\ & \qquad \displaystyle +C\int _0^1\tau ^{-\frac{1}{2}}d\tau (\Vert y_n^{\alpha -\beta }\nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)})\nonumber \\ & \quad \le \displaystyle C\big (\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)})\nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta }\nabla ^2 u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla ^2 v\Vert _{L^{2}(\mathbb {R}^n_+)}\big ); \end{aligned}$$

(3.11)

$$\begin{aligned} & \displaystyle N_3+N_4\le \displaystyle C\sum \limits _{i,j=1}^n\int _1^\infty \int _{\mathbb {R}^n}|x_n|^\alpha |\partial _k\partial _i\partial _jG_\tau (x^\prime , x_n)|dxd\tau \Vert \nabla w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \qquad \qquad \ \displaystyle + C\sum \limits _{i,j=1}^n\int _1^\infty \int _{\mathbb {R}^n}|\partial _k\partial _i\partial _jG_\tau (x^\prime , x_n)|dxd\tau \Vert y_n^\alpha \nabla w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \qquad \qquad \le \displaystyle C\sum \limits _{i,j=1}^n\left( \int _1^\infty \tau ^{\frac{\alpha }{2}-\frac{3}{2}}d\tau \Vert \nabla w_{ij}\Vert _{L^1(\mathbb {R}^n_+)} + \int _1^\infty \tau ^{-\frac{3}{2}}d\tau \Vert y_n^\alpha \nabla w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\right) \nonumber \\ & \quad \quad \quad \qquad \le \displaystyle C(\Vert u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert v\Vert _{L^{2}(\mathbb {R}^n_+)}\nonumber \\ & \quad \quad \qquad \qquad \displaystyle +\Vert y_n^{\alpha -\beta }u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert y_n^\beta \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)} +\Vert y_n^{\alpha -\beta }\nabla u\Vert _{L^{2}(\mathbb {R}^n_+)} \Vert y_n^\beta v\Vert _{L^{2}(\mathbb {R}^n_+)}).\nonumber \\ \end{aligned}$$

(3.12)

From (3.10)–(3.12), we conclude that (3.5) holds. Next we show the validity of (3.6).

Let $u, v\in C^\infty _{0,\sigma }(\mathbb {R}^n_+)$. Then from (3.2), one has for any $1\le k\le n$

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^{-\theta }\partial _k\mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)} \nonumber \\ & \quad \le \displaystyle \left\| \sum \limits _{i,j=1}^nx_n^{-\theta }\partial _k\int _0^1 G_\tau *[\partial _i\partial _j(u_iv_j)]_*d\tau \right\| _{L^1(\mathbb {R}^{n-1}\times (0,1))}\nonumber \\ & \qquad \displaystyle +\Vert \sum \limits _{i,j=1}^nx_n^{-\theta }\partial _k\int _1^\infty G_\tau *[\partial _i\partial _j(u_iv_j)]_*d\tau \Vert _{L^1(\mathbb {R}^{n-1}\times (0,1))}\nonumber \\ & \qquad \displaystyle +\left\| \sum \limits _{i,j=1}^n\partial _k\mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\sup \limits _{y\in \mathbb {R}^n}\left\| \int _0^1 x_n^{-\theta }\partial _kG_\tau (x-y)d\tau \right\| _{L^1(\mathbb {R}^{n-1}\times (0,1))} \left\| \sum \limits _{i,j=1}^n\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +C\sup \limits _{y\in \mathbb {R}^n}\sum \limits _{i,j=1}^n\left\| \int _1^\infty x_n^{-\theta }\partial _k\partial _i\partial _j G_\tau (x-y)d\tau \Vert _{L^1(\mathbb {R}^{n-1}\times (0,1))}\right\| w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\left\| \sum \limits _{i,j=1}^n\partial _k\mathcal {N}\partial _i\partial _j(u_iu_j) \right\| _{L^1(\mathbb {R}^n_+)}. \end{aligned}$$

(3.13)

Let $1\le i, j, k\le n$. $0<\theta <1$. Then for any $y=(y^\prime , y_n)\in \mathbb {R}^n$

$$\begin{aligned} & \displaystyle \left\| \int _0^1 x_n^{-\theta }\partial _kG_\tau (x-y)d\tau \right\| _{L^1(\mathbb {R}^{n-1}\times (0,1))}\nonumber \\ & \quad \le \displaystyle \int _0^1\int _0^1\int _{\mathbb {R}^{n-1}}(4\pi \tau )^{-\frac{n}{2}}\tau ^{-\frac{1}{2}}x_n^{-\theta } \frac{|x_k-y_k|}{2\sqrt{\tau }}e^{-\frac{|x-y|^2}{4\tau }}dx^\prime dx_nd\tau \nonumber \\ & \quad \le \displaystyle C\int _0^1\int _0^1\tau ^{-1}x_n^{-\theta }e^{-\frac{(x_n-y_n)^2}{8\tau }}dx_nd\tau \nonumber \\ & \quad \le \displaystyle C\int _0^1\tau ^{-1}\left( \int _0^1x_n^{-\theta (1+\eta _0)}dx_n\right) ^\frac{1}{1+\eta _0} \left( \int _0^1e^{-\frac{(1+\eta _0)(x_n-y_n)^2}{8\eta _0\tau }}dx_n\right) ^\frac{\eta _0}{1+\eta _0} d\tau \nonumber \\ & \quad \le \displaystyle C\int _0^1\tau ^{-1+\frac{\eta _0}{2(1+\eta _0)}}d\tau \le C,\;\;\;\text{ where }\;\;0<\eta _0<\frac{1}{\theta }-1; \end{aligned}$$

(3.14)

and

$$\begin{aligned} & \displaystyle \left\| \int _1^\infty x_n^{-\theta }\partial _k\partial _i\partial _jG_\tau (x-y)d\tau \right\| _{L^1(\mathbb {R}^{n-1}\times (0,1))}\nonumber \\ & \quad \le \displaystyle \int _0^1\int _1^\infty \tau ^{-\frac{3}{2}}\int _{\mathbb {R}^{n-1}}x_n^{-\theta } (4\pi \tau )^{-\frac{n}{2}}e^{-\frac{|x^\prime -y^\prime |^2+|x_n-y_n|^2}{8\tau }}dx^\prime dx_nd\tau \nonumber \\ & \quad \le \displaystyle C\int _1^\infty \int _0^1\tau ^{-2}x_n^{-\theta }e^{-\frac{(x_n-y_n)^2}{8\tau }}dx_nd\tau \nonumber \\ & \quad \le \displaystyle C\int _1^\infty \tau ^{-2}d\tau \int _0^1x_n^{-\theta }dx_n\le C. \end{aligned}$$

(3.15)

In addition, it follows from (3.4) with $\alpha =\beta =0$ that

$$\begin{aligned} & \displaystyle \left\| \sum \limits _{i,j=1}^n\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}+\sum \limits _{i,j=1}^n\Vert w_{ij}\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\big (\Vert u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\big ). \end{aligned}$$

(3.16)

From (3.13)–(3.16), we conclude that for $0<\theta <1$, $1\le k\le n$

$$\begin{aligned} \left\| \sum \limits _{i,j=1}^nx_n^{-\theta }\partial _k\mathcal {N}\partial _i\partial _j(u_iv_j)\right\| _{L^1(\mathbb {R}^n_+)}\le C\big (\Vert u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert v\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla v\Vert _{L^{2}(\mathbb {R}^n_+)}\big ), \end{aligned}$$

which is (3.6). $\square$

Lemma 3.2

[12,13,14]. Assume $u_0\in L^2_\sigma (\mathbb {R}^n_+)\bigcap L^n(\mathbb {R}^n_+)$ $(n\ge 2)$. Then the strong solution u of (1.1) given in Theorem 1.1 satisfies for $t>1$

$$\begin{aligned} \Vert \nabla ^mu(t)\Vert _{L^2(\mathbb {R}^n_+)}\le & C(1+t)^{-\frac{m}{2}-\frac{n}{4}},\;\; \;m=0,1,2,3;\\ \Vert \partial _t\nabla ^\ell u(t)\Vert _{L^2 (\mathbb {R}^n_+)}\le & Ct^{-1-\frac{\ell }{2}-\frac{n}{4}},\;\ell =0,1. \end{aligned}$$

Furthermore, if $u_0$ satisfies

$$\begin{aligned} \Vert x_nu_0\Vert _{L^2(\mathbb {R}^n_+)}+\Vert (1+x_n)\nabla u_0\Vert _{L^2(\mathbb {R}^n_+)}+\Vert (1+x_n)u_0\Vert _{L^1(\mathbb {R}^n_+)}<\infty . \end{aligned}$$

Then following estimates are true for the strong solution u with $t>1$

$$\begin{aligned} \Vert \nabla ^m u(t)\Vert _{L^2(\mathbb {R}^n_+)}\le & Ct^{-\frac{m+1}{2}-\frac{n}{4}},\;\;m=0,1,2,3;\\ \Vert \partial _t\nabla ^\ell u(t)\Vert _{L^2 (\mathbb {R}^n_+)}\le & Ct^{-\frac{3}{2}-\frac{\ell }{2}-\frac{n}{4}},\;\ell =0,1; \end{aligned}$$

and for $0<\gamma <1$ and $t>1$

$$\begin{aligned} \Vert x_n^\gamma \nabla ^\ell u(t)\Vert _{L^2(\mathbb {R}^n_+)}\le C t^{-\frac{\ell }{2}-\frac{n}{4}+\frac{\gamma }{2}},\;\ell =0,1. \end{aligned}$$

Proof of Theorem 1.3

The strong solution u of problem (1.1), which is given in Theorem 1.1, can be represented as follows

$$\begin{aligned} u(t)=e^{-tA}u_0-\int _0^t e^{-(t-s)A}Pu(s)\cdot \nabla u(s)ds. \end{aligned}$$

It follows from Theorem 1.2, Lemma 3.2 that for $0<\alpha <\min \{2, n-1\}$ and $t>0$

$$\begin{aligned} & \displaystyle \Vert |x^\prime |^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)} \le \displaystyle \Vert |x^\prime |^{-\alpha } e^{-tA}u_0\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\int _0^t\Vert |x^\prime |^{-\alpha }e^{-(t-s)A}P(u(s)\cdot \nabla ) u(s)\Vert _{L^1(\mathbb {R}^n_+)}ds\\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}\Vert u_0\Vert _{L^1(\mathbb {R}^n_+)}+C\int _0^t(t-s)^{-\frac{\alpha }{2}}\Vert P(u(s)\cdot \nabla ) u(s)\Vert _{L^1(\mathbb {R}^n_+)}ds\\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}\Vert u_0\Vert _{L^1(\mathbb {R}^n_+)} +C\int _0^t(t-s)^{-\frac{\alpha }{2}}\left( \Vert (u(s)\cdot \nabla ) u(s)\Vert _{L^1(\mathbb {R}^n_+)}\right. \\ & \qquad \left. \displaystyle +\left\| \sum \limits _{i,j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)\right\| _{L^1(\mathbb {R}^n_+)}\right) ds \\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}+C\left( \int _0^\frac{t}{2}+\int _\frac{t}{2}^t\right) (t-s)^{-\frac{\alpha }{2}} \big (\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}\big )ds\\ & \quad \le \displaystyle Ct^{-\frac{\alpha }{2}}+Ct^{-\frac{\alpha }{2}}\int _0^\frac{t}{2}(1+s)^{-\frac{n}{2}}ds +C\int _\frac{t}{2}^t(t-s)^{-\frac{\alpha }{2}}(1+s)^{-\frac{n}{2}}ds\\ & \quad \le \displaystyle \left\{ \begin{array}{lll} Ct^{-\frac{\alpha }{2}}& \text{ if }& n\ge 3,\\ Ct^{-\frac{\alpha }{2}}\log _e(1+t)& \text{ if }& n=2.\\ \end{array} \right. \end{aligned}$$

Similarly, we have for $0<\alpha <1$ and $t>0$

$$\begin{aligned} \displaystyle \Vert x_n^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle \Vert x_n^{-\alpha }e^{-tA}u_0\Vert _{L^1(\mathbb {R}^n_+)}\\ & \displaystyle +\int _0^t\Vert x_n^{-\alpha }e^{-(t-s)A}P(u(s)\cdot \nabla ) u(s)\Vert _{L^1(\mathbb {R}^n_+)}ds\\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}}+Ct^{-\frac{\alpha }{2}}\int _0^\frac{t}{2}(1+s)^{-\frac{n}{2}}ds +C\int _\frac{t}{2}^t(t-s)^{-\frac{\alpha }{2}}(1+s)^{-\frac{n}{2}}ds\\ \quad\le & \displaystyle \left\{ \begin{array}{lll} Ct^{-\frac{\alpha }{2}}& \text{ if }& n\ge 3,\\ Ct^{-\frac{\alpha }{2}}\log _e(1+t)& \text{ if }& n=2.\\ \end{array} \right. \end{aligned}$$

Now suppose $\Vert x_nu_0\Vert _{L^1(\mathbb {R}^n_+)}<\infty$. Using Theorem 1.2, Lemma 3.2, we derive for $t>0$

$$\begin{aligned} \displaystyle \Vert |x^\prime |^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle Ct^{-\frac{\alpha }{2}}+C\int _0^t(t-s)^{-\frac{\alpha }{2}} \big (\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}\big )ds\\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}}+Ct^{-\frac{\alpha }{2}}\int _0^\frac{t}{2}(1+s)^{-1-\frac{n}{2}}ds +C\int _\frac{t}{2}^t(t-s)^{-\frac{\alpha }{2}}(1+s)^{-1-\frac{n}{2}}ds\\ \quad\le & \displaystyle Ct^{-\frac{\alpha }{2}},\;\;\;0<\alpha <\min \{2, n-1\}; \end{aligned}$$

and similarly

$$\begin{aligned} \Vert x_n^{-\alpha }u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-\frac{\alpha }{2}},\;\;\;0<\alpha <1. \end{aligned}$$

$\square$

4 $L^1-$behavior for cubic spatial derivatives of Navier–Stokes flows

The first preliminary result is an important identity, which plays a crucial role in avoiding the strong singularity in the proof of Theorem 1.4.

Set $\mathscr {S}(\mathbb {R}^n)=\{f\in C^\infty (\mathbb {R}^n)\;|\lim \limits _{|x|\rightarrow \infty }|x^\eta \nabla ^\gamma f(x)|=0\;\;\text{ for } \text{ any } \text{ multi } \text{ index }\;\;\eta , \gamma \}$, where $x^\eta =x_1^{\eta _1}x_2^{\eta _2}\cdots x_n^{\eta _n}$, $\nabla ^\gamma =\partial _1^{\gamma _1}\partial _2^{\gamma _2}\cdots \partial _n^{\gamma _n}$, $\eta =(\eta _1, \eta _2,\cdots ,\eta _n)$, $\gamma =(\gamma _1, \gamma _2,\cdots ,\gamma _n)$ are multi indexes.

Lemma 4.1

Let $x=(x_1, x_2,\cdots ,x_n)\in \mathbb {R}^n_+$ ($n\ge 2$). Then

$$\begin{aligned} \sum \limits _{i=1}^n\frac{\partial }{\partial x_i}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}g(y)\frac{\partial E(x-y)}{\partial x_i}dy=\frac{1}{2}g(x),\;\;\;\;\forall g\in \mathscr {S}(\mathbb {R}^n), \end{aligned}$$

where E is the fundamental solution of the elliptic operator $-\Delta$ in $\mathbb {R}^n$, its specific expression is given in Sect. 2.

Proof

Observe that for every $h\in \mathscr {S}(\mathbb {R}^n)$ and $x=(x^\prime , x_n)\in \mathbb {R}^n_+$,

$$\begin{aligned} \int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-y)}{\partial x_i}\frac{\partial h(y)}{\partial y_n}dy=\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial y_i}\frac{\partial h(x-y)}{\partial x_n}dy,\;\;1\le i\le n; \nonumber \\ \end{aligned}$$

(4.1)

and $0<\epsilon <x_n$,

$$\begin{aligned} \displaystyle \int _\epsilon ^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial y_n}\frac{\partial h(x-y)}{\partial x_n}dy= & \displaystyle -\int _\epsilon ^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial y_n}\frac{\partial h(x-y)}{\partial y_n}dy \nonumber \\= & \displaystyle \int _\epsilon ^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial ^2 E(y)}{\partial y_n^2}h(x-y)dy\nonumber \\ & \displaystyle -\int _{\mathbb {R}^{n-1}}\frac{\partial E(y^\prime , x_n)}{\partial x_n}h(x^\prime -y^\prime ,0)dy^\prime \nonumber \\ & \displaystyle +\int _{\mathbb {R}^{n-1}}\frac{\partial E(y^\prime ,\epsilon )}{\partial y_n}h(x^\prime -y^\prime ,x_n-\epsilon )dy^\prime . \end{aligned}$$

(4.2)

In addition, it holds for $\epsilon >0$ and $n\ge 2$,

$$\begin{aligned} \frac{\partial E(y^\prime ,\epsilon )}{\partial y_n}=\frac{\partial E(y^\prime ,y_n)}{\partial y_n}\Big |_{y_n=\epsilon }=\frac{1}{n\omega _n}\frac{\epsilon }{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}. \end{aligned}$$

(4.3)

Let $\varphi \in \mathscr {S}(\mathbb {R}^{n-1})$ and $\eta >0$ be small enough. Then

$$\begin{aligned} \varphi (y^\prime )=\varphi (0^\prime )+y^\prime \cdot \nabla ^\prime \varphi (0^\prime )+O(|y^\prime |^2),\;\;\;\;\forall |y^\prime |<\eta . \end{aligned}$$

Whence for $\epsilon >0$ and $n\ge 2$,

$$\begin{aligned} \displaystyle \int _{\mathbb {R}^{n-1}}\frac{\epsilon \varphi (y^\prime )}{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}dy^\prime= & \displaystyle \int _{|y^\prime |\ge \eta }\frac{\epsilon \varphi (y^\prime )}{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}dy^\prime +\int _{|y^\prime |<\eta }\frac{\epsilon \varphi (0^\prime )}{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}dy^\prime \nonumber \\ & \displaystyle +\int _{|y^\prime |<\eta }\frac{\epsilon y^\prime \cdot \nabla ^\prime \varphi (0^\prime )}{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}dy^\prime +O(1)\int _{|y^\prime |<\eta }\frac{\epsilon |y^\prime |^2}{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}dy^\prime \nonumber \\= & \displaystyle \sum \limits _{j=1}^4I_j(\epsilon ). \end{aligned}$$

(4.4)

Now we calculate and estimate each term $I_j(\epsilon )$, $j=1,2,3,4$.

$$\begin{aligned} |I_1(\epsilon )|\le & \eta ^{-n}\epsilon \int _{\mathbb {R}^{n-1}}|\varphi (y^\prime )|dy^\prime \longrightarrow 0\;\;\;\text{ as }\;\;\;\epsilon \longrightarrow 0; \end{aligned}$$

(4.5)

$$\begin{aligned} \displaystyle |I_3(\epsilon )|\le & \displaystyle \epsilon |\nabla ^\prime \varphi (0^\prime )|\int _{|y^\prime |<\eta }\frac{|y^\prime |dy^\prime }{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}\nonumber \\\le & \displaystyle C\epsilon |\nabla ^\prime \varphi (0^\prime )|\int _0^\eta \frac{s^{n-1}ds}{(s^2+\epsilon ^2)^{\frac{n}{2}}}\nonumber \\\le & \displaystyle C\epsilon |\nabla ^\prime \varphi (0^\prime )|\int _0^\eta (s^2+\epsilon ^2)^{-\frac{1}{2}}ds\nonumber \\\le & \displaystyle C|\nabla ^\prime \varphi (0^\prime )|\epsilon \log _e(1+\epsilon ^{-1}\eta ) \longrightarrow 0\;\;\;\text{ as }\;\;\;\epsilon \longrightarrow 0; \end{aligned}$$

(4.6)

$$\begin{aligned} \displaystyle |I_4(\epsilon )|\le & \displaystyle C\epsilon |\int _{|y^\prime |<\eta }\frac{|y^\prime |^2dy^\prime }{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}\nonumber \\\le & \displaystyle C\epsilon \int _0^\eta \frac{s^nds}{(s^2+\epsilon ^2)^{\frac{n}{2}}}\nonumber \\\le & \displaystyle C\eta \epsilon \longrightarrow 0\;\;\;\text{ as }\;\;\;\epsilon \longrightarrow 0; \end{aligned}$$

(4.7)

$$\begin{aligned} \displaystyle I_2(\epsilon )= & \displaystyle \varphi (0^\prime )\int _{|y^\prime |<\eta }\frac{ \epsilon dy^\prime }{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}\nonumber \\= & \displaystyle \varphi (0^\prime )\int _{|y^\prime |<\eta }\frac{ \epsilon ^{1-n} dy^\prime }{(|\epsilon ^{-1}y^\prime |^2+1)^{\frac{n}{2}}}\nonumber \\= & \displaystyle \varphi (0^\prime )\int _{|z^\prime |<\frac{\eta }{\epsilon }}\frac{dz^\prime }{(|z^\prime |^2+1)^{\frac{n}{2}}}\nonumber \\\longrightarrow & \displaystyle \varphi (0^\prime )\int _{\mathbb {R}^{n-1}}\frac{dz^\prime }{(|z^\prime |^2+1)^{\frac{n}{2}}}\;\;\;\text{ as }\;\;\;\epsilon \longrightarrow 0. \end{aligned}$$

(4.8)

Note that the volume $\omega _m$ of the unit ball in $\mathbb {R}^m$ ($m\ge 1$) is expressed by $\omega _m=\frac{\pi ^\frac{m}{2}}{\Gamma (1+\frac{m}{2})}$. Set $m=n-1$, then

$$\begin{aligned} \displaystyle \int _{\mathbb {R}^{n-1}}\frac{dz^\prime }{(|z^\prime |^2+1)^{\frac{n}{2}}}= & \displaystyle (n-1)\omega _{n-1}\int _0^\infty \frac{s^{n-2}ds}{(s^2+1)^{\frac{n}{2}}}\nonumber \\= & \displaystyle \frac{(n-1)\pi ^\frac{n-1}{2}}{\Gamma \left( 1+\frac{n-1}{2}\right) }\int _0^\frac{\pi }{2}(\sin \theta )^{n-2}d\theta \nonumber \\= & \displaystyle \frac{(n-1)\pi ^\frac{n-1}{2}}{\Gamma \left( \frac{n+1}{2}\right) }\frac{\pi ^\frac{1}{2}\Gamma \left( \frac{n-1}{2}\right) }{2\Gamma \left( \frac{n}{2}\right) }\nonumber \\= & \displaystyle \frac{\pi ^\frac{n}{2}}{\Gamma \left( \frac{n}{2}\right) }. \end{aligned}$$

(4.9)

In the proof of (4.9), we used the known result: let $m\ge 0$ be an integer, then

$$\begin{aligned} \int _0^\frac{\pi }{2}(\cos \theta )^md\theta =\int _0^\frac{\pi }{2}(\sin \theta )^md\theta =\frac{\pi ^\frac{1}{2}\Gamma \left( \frac{m+1}{2}\right) }{2\Gamma \left( \frac{m+2}{2}\right) }. \end{aligned}$$

From (4.4)-(4.9), we find

$$\begin{aligned} \lim \limits _{\epsilon \rightarrow 0}\int _{\mathbb {R}^{n-1}}\frac{\epsilon \varphi (y^\prime )}{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}dy^\prime =\left\langle \frac{\pi ^\frac{n}{2}}{\Gamma \left( \frac{n}{2}\right) }\delta (y^\prime ), \varphi (y^\prime )\right\rangle , \end{aligned}$$

which, together with (4.3) and $\omega _n=\frac{\pi ^\frac{n}{2}}{\Gamma (1+\frac{n}{2})}$ implies

$$\begin{aligned} \displaystyle \lim \limits _{\epsilon \rightarrow 0}\frac{\partial E(y^\prime ,\epsilon )}{\partial y_n}= & \displaystyle \lim \limits _{\epsilon \rightarrow 0}\frac{1}{n\omega _n}\frac{\epsilon }{(|y^\prime |^2+\epsilon ^2)^{\frac{n}{2}}}\nonumber \\= & \displaystyle \frac{\Gamma \left( 1+\frac{n}{2}\right) }{n\pi ^\frac{n}{2}}\frac{\pi ^\frac{n}{2}}{\Gamma \left( \frac{n}{2}\right) }\delta (y^\prime )\nonumber \\= & \displaystyle \frac{1}{2}\delta (y^\prime )\;\;\;\text{ in } \text{ the } \text{ sense } \text{ of } \text{ the } \text{ distribution }. \end{aligned}$$

(4.10)

In addition, there holds that for any $x=(x^\prime , x_n)\in \mathbb {R}^n_+$, $x_n>\epsilon$,

$$\begin{aligned} \int _\epsilon ^{x_n}\int _{\mathbb {R}^{n-1}}h(x-y)\frac{\partial ^2 E(y)}{\partial y_n^2}dy=-\int _\epsilon ^{x_n}\int _{\mathbb {R}^{n-1}}h(x-y)\sum \limits _{j=1}^{n-1}\frac{\partial ^2 E(y)}{\partial y_j^2}dy. \end{aligned}$$

(4.11)

Combining (4.2), (4.10) and (4.11) yields

$$\begin{aligned} & \displaystyle \int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial y_n}\frac{\partial h(x-y)}{\partial x_n}dy\nonumber \\ & \quad =\displaystyle \lim \limits _{\epsilon \rightarrow 0}\int _\epsilon ^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial y_n}\frac{\partial h(x-y)}{\partial x_n}dy\nonumber \\ & \quad =\displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j^2}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}E(y)h(x-y)dy\nonumber \\ & \qquad \displaystyle -\int _{\mathbb {R}^{n-1}}\frac{\partial E(y^\prime , x_n)}{\partial x_n}h(x^\prime -y^\prime ,0)dy^\prime +\frac{1}{2}h(x^\prime ,x_n). \end{aligned}$$

(4.12)

Differentiating with respect to $x_n>0$ in (4.12), we get

$$\begin{aligned} & \displaystyle \frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial y_n}\frac{\partial h(x-y)}{\partial x_n}dy\nonumber \\ & \quad =\displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j^2}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}E(y)\frac{\partial h(x-y)}{\partial x_n}dy+\frac{1}{2}\frac{\partial h(x^\prime ,x_n)}{\partial x_n}\nonumber \\ & \qquad \displaystyle -\int _{\mathbb {R}^{n-1}}h(x^\prime -y^\prime , 0)\left( \sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial y_j^2}E(y^\prime , x_n)+\frac{\partial ^2 E(y^\prime , x_n)}{\partial x_n^2}\right) dy^\prime \nonumber \\ & \quad =\displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j^2}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}E(y)\frac{\partial h(x-y)}{\partial x_n}dy+\frac{1}{2}\frac{\partial h(x^\prime ,x_n)}{\partial x_n}. \end{aligned}$$

(4.13)

Combining (4.1) and (4.13) yields

$$\begin{aligned} & \displaystyle \sum \limits _{i=1}^n\frac{\partial }{\partial x_i}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-y)}{\partial x_i}\frac{\partial h(y)}{\partial y_n}dy\nonumber \\ & \quad =\displaystyle \sum \limits _{i=1}^{n-1}\frac{\partial ^2}{\partial x_i^2}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}E(y)\frac{\partial h(x-y)}{\partial x_n}dy+\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(y)}{\partial x_n}\frac{\partial h(x-y)}{\partial x_n}dy\nonumber \\ & \quad =\displaystyle \frac{1}{2}\frac{\partial h(x^\prime ,x_n)}{\partial x_n}. \end{aligned}$$

(4.14)

Let $g\in \mathscr {S}(\mathbb {R}^n)$, take $h(x)=\int _0^{x_n}g(x^\prime , t)dt$ in (4.14), we complete the proof of Lemma 4.1. $\square$

Lemma 4.2

Let $1\le j,\ell \le n-1$, $1\le k\le n$, $n\ge 2$. Then for any $t>0$,

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_k}\int _{\mathbb {R}^{n-1}}G_t(x^\prime -z^\prime , x_n)\nabla E(z^\prime , z_n)dz^\prime \Big |\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{1+\delta _{kn}}{2}}\big (|x^\prime |+x_n+z_n+\sqrt{t}\big )^{-n-1+\delta _{kn}}e^{-\frac{x_n^2}{64}},\;\;\;\forall x^\prime \in \mathbb {R}^{n-1},\;\;x_n,\;z_n>0; \nonumber \\ \end{aligned}$$

(4.15)

and

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}\int _{\mathbb {R}^{n-1}}G_t(x^\prime -z^\prime , x_n)\nabla E(z^\prime , z_n)dz^\prime \Big |\nonumber \\ & \quad \le \displaystyle Ct^{-1}\big (|x^\prime |+x_n+z_n+\sqrt{t}\big )^{-n-1}e^{-\frac{x_n^2}{64}},\;\;\;\forall x^\prime \in \mathbb {R}^{n-1},\;\;x_n,\;z_n>0. \end{aligned}$$

(4.16)

Proof

Let $x=(x^\prime , x_n)\in \mathbb {R}^n_+$ ($n\ge 2$), $z_n>0$ and set

$$\begin{aligned} J_i(x, z_n,t)=\int _{\mathbb {R}^{n-1}}G_t(x^\prime -z^\prime , x_n)\frac{\partial E(z^\prime , z_n)}{\partial z_i}dz^\prime ,\;\;\;1\le i\le n. \end{aligned}$$

Observe that for any $\lambda , t>0$

$$\begin{aligned} \displaystyle J_i(\lambda x, \lambda z_n,\lambda ^2 t)= & \displaystyle \int _{\mathbb {R}^{n-1}}G_{\lambda ^2t}(\lambda x^\prime -z^\prime , \lambda x_n)\frac{\partial E(z^\prime , \lambda z_n)}{\partial z_i}dz^\prime \nonumber \\= & \displaystyle \lambda ^{-n}\int _{\mathbb {R}^{n-1}}G_{t}(x^\prime -z^\prime , x_n)\frac{\partial E(z^\prime , z_n)}{\partial z_i}dz^\prime \nonumber \\= & \displaystyle \lambda ^{-n}J_i(x, z_n,t),\;\;\;1\le i\le n. \end{aligned}$$

(4.17)

A direct calculation shows that for $z_n>0$,

$$\begin{aligned} \int _{|z^\prime |< 1}\big |E(z^\prime , z_n)\big |dz^\prime \le C,\;\;\; \text{ where } \;\;C \;\;\text{ is } \text{ independent } \text{ of }\;\;z_n. \end{aligned}$$

Whence for $1\le i, j\le n-1$, $1\le k\le n$,

$$\begin{aligned} \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_i(x, z_n,1)\Big |= & \displaystyle \Big |\frac{\partial ^3}{\partial x_i\partial x_j\partial x_k}\int _{\mathbb {R}^{n-1}}G_1(x^\prime -z^\prime , x_n)E(z)dz^\prime \Big |\nonumber \\\le & \displaystyle \int _{\mathbb {R}^{n-1}}\Big |\frac{\partial ^3}{\partial x_i\partial x_j\partial x_k}G_{1}(x^\prime -z^\prime , x_n)\Big |\big |E(z^\prime , z_n)\big |dz^\prime \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}}\int _{\mathbb {R}^{n-1}}e^{-\frac{|x^\prime -z^\prime |^2}{8}}\big |E(z^\prime , z_n)\big |dz^\prime \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}}\left( \int _{|z^\prime |< 1}\big |E(z^\prime , z_n)\big |dz^\prime +\int _{|z^\prime |\ge 1}e^{-\frac{|x^\prime -z^\prime |^2}{8}}dz^\prime \right) \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}},\;\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\; z_n>0; \end{aligned}$$

(4.18)

and

$$\begin{aligned} \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_n(x, z_n,1)\Big |\le & \displaystyle \int _{\mathbb {R}^{n-1}}\Big |\frac{\partial ^2}{\partial x_j\partial x_k}G_{1}(x^\prime -z^\prime , x_n)\Big |\Big |\frac{\partial E(z^\prime , z_n)}{\partial z_n}\Big |dz^\prime \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}}\int _{\mathbb {R}^{n-1}}e^{-\frac{|x^\prime -z^\prime |^2}{8}}\Big |\frac{\partial E(z^\prime , z_n)}{\partial z_n}\Big |dz^\prime \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}}\left( \int _{|z^\prime |< 1}\Big |\frac{\partial E(z^\prime , z_n)}{\partial z_n}\Big |dz^\prime +\int _{|z^\prime |\ge 1}e^{-\frac{|x^\prime -z^\prime |^2}{8}}dz^\prime \right) \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}}\Big (z_n\int _0^1(s+z_n)^{-n}s^{n-2}ds+\int _{\mathbb {R}^{n-1}}e^{-\frac{|z^\prime |^2}{8}}dz^\prime \Big )\nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{8}}. \end{aligned}$$

(4.19)

Combining (4.18) and (4.19) yields for $1\le j\le n-1$, $1\le i, k\le n$

$$\begin{aligned} \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_i(x, z_n,1)\Big |\le Ce^{-\frac{x_n^2}{8}},\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0. \end{aligned}$$

(4.20)

Similar to the proofs of (4.18) and (4.19), we find for $1\le j,\ell \le n-1$, $1\le i\le n$

$$\begin{aligned} \Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}J_i(x, z_n,1)\Big |\le Ce^{-\frac{x_n^2}{16}},\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0. \end{aligned}$$

(4.21)

Let $x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0$, and set

$$\begin{aligned} \Pi _\rho (x)=\big \{z^\prime \in \mathbb {R}^{n-1}:\;|x^\prime -z^\prime |^2+x_n^2\le \frac{\rho ^2}{4}\big \}, \;\;\text{ where }\;\;\rho =\sqrt{|x^\prime |^2+(x_n+z_n)^2}. \end{aligned}$$

Using the triangle inequality yields for any $z^\prime \in \Pi _\rho (x)$, $x\in \mathbb {R}^n_+$, $z_n>0$,

$$\begin{aligned} \rho =\sqrt{|x^\prime |^2+(x_n+z_n)^2}\le \sqrt{|x^\prime -z^\prime |^2+x_n^2}+\sqrt{|z^\prime |^2+z_n^2}\le \frac{\rho }{2}+|z| \Longrightarrow |z|\ge \frac{\rho }{2}. \end{aligned}$$

Then for $1\le j, k\le n-1$ and $x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0$,

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_n(x, z_n,1)\Big |\nonumber \\ & \quad =\displaystyle \Big |\int _{\mathbb {R}^{n-1}}G_1(x^\prime -z^\prime , x_n)\frac{\partial ^3 E(z)}{\partial z_j\partial z_k\partial z_n}dz^\prime \Big |\nonumber \\ & \quad =\displaystyle \Big |\int _{\Pi _\rho (x)}G_1(x^\prime -z^\prime , x_n)\frac{\partial ^3 E(z)}{\partial z_j\partial z_k\partial z_n}dz^\prime \nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}G_1(x^\prime -z^\prime , x_n)\frac{\partial ^3 E(z)}{\partial z_j\partial z_k\partial z_n}\Big |\nonumber \\ & \quad \le \displaystyle \int _{\Pi _\rho (x)}G_1(x^\prime -z^\prime , x_n)\Big |\frac{\partial ^3 E(z)}{\partial z_j\partial z_k\partial z_n}\Big |dz^\prime \nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}\Big |\frac{\partial ^2G_1(x^\prime -z^\prime , x_n)}{\partial x_j\partial x_k}\Big |\Big |\frac{\partial E(z)}{\partial z_n}\Big |dz^\prime \nonumber \\ & \qquad \displaystyle +\int _{\partial \Pi _\rho (x)}G_1(x^\prime -z^\prime , x_n)\Big |\frac{\partial ^2 E(z)}{\partial z_k\partial z_n}\Big |d S_{z^\prime }\nonumber \\ & \qquad \displaystyle +\int _{\partial \Pi _\rho (x)}\Big |\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_j}\Big |\Big |\frac{\partial E(z)}{\partial z_n}\Big |d S_{z^\prime }\nonumber \\ & \quad \le \displaystyle C\left( \int _{\Pi _\rho (x)}|z|^{-n-1}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{4}}dz^\prime \right. \nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{8}}dz^\prime \nonumber \\ & \qquad \left. \displaystyle +e^{-\frac{\rho ^2}{16}}\int _{\partial \Pi _\rho (x)}|z|^{-n}d S_{z^\prime } +e^{-\frac{\rho ^2}{32}}\int _{\partial \Pi _\rho (x)}|z|^{-n+1}d S_{z^\prime }\right) \nonumber \\ & \quad \le \displaystyle C\Big (\rho ^{-n-1}e^{-\frac{x_n^2}{4}}\int _{\mathbb {R}^{n-1}}e^{-\frac{|x^\prime -z^\prime |^2}{4}}dz^\prime \nonumber \\ & \qquad \displaystyle +e^{-\frac{\rho ^2}{64}}\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{16}}dz^\prime \nonumber \\ & \qquad \displaystyle +e^{-\frac{\rho ^2}{32}}\rho ^{n-2}(\rho ^{-n}+\rho ^{-n+1})\Big )\nonumber \\ & \quad \le \displaystyle C\big (\rho ^{-n-1}e^{-\frac{x_n^2}{4}}+e^{-\frac{\rho ^2}{32}}(\rho ^{-2}+\rho ^{-1})\big )\nonumber \\ & \qquad \displaystyle +Ce^{-\frac{\rho ^2}{64}}\left( \int _{\int _{|z^\prime |< 1}}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}dz^\prime +\int _{|z^\prime |\ge 1}e^{-\frac{|x^\prime -z^\prime |^2}{16}}dz^\prime \right) \nonumber \\ & \quad \le \displaystyle C\left( \rho ^{-n-1}e^{-\frac{x_n^2}{4}}+e^{-\frac{\rho ^2}{32}}(\rho ^{-2}+\rho ^{-1})+e^{-\frac{\rho ^2}{64}}\right) \nonumber \\ & \quad \le \displaystyle Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n-1}); \end{aligned}$$

(4.22)

and for $1\le j\le n-1$

$$\begin{aligned} \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_n}J_n(x, z_n,1)\Big |= & \displaystyle \Big |\int _{\mathbb {R}^{n-1}}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\frac{\partial ^2 E(z)}{\partial z_j\partial z_n}dz^\prime \Big |\nonumber \\= & \displaystyle \Big |\int _{\Pi _\rho (x)}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\frac{\partial ^2 E(z)}{\partial z_j\partial z_n}dz^\prime \nonumber \\ & \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\frac{\partial ^2 E(z)}{\partial z_j\partial z_n}\Big |\nonumber \\\le & \displaystyle \int _{\Pi _\rho (x)}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\Big |\frac{\partial ^2 E(z)}{\partial z_j\partial z_n}\Big |dz^\prime \nonumber \\ & \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}\Big |\frac{\partial ^2G_1(x^\prime -z^\prime , x_n)}{\partial x_j\partial x_n}\Big |\Big |\frac{\partial E(z)}{\partial z_n}\Big |dz^\prime \nonumber \\ & \displaystyle +\int _{\partial \Pi _\rho (x)}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\Big |\frac{\partial E(z)}{\partial z_n}\Big |d S_{z^\prime }\nonumber \\\le & \displaystyle C\left( \int _{\Pi _\rho (x)}|z|^{-n}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{8}}dz^\prime \right. \nonumber \\ & \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{8}}dz^\prime \nonumber \\ & \left. \displaystyle +e^{-\frac{\rho ^2}{32}}\int _{\partial \Pi _\rho (x)}|z|^{-n+1}d S_{z^\prime }\right) \nonumber \\\le & \displaystyle C\Big (\rho ^{-n}e^{-\frac{x_n^2}{8}}\int _{\mathbb {R}^{n-1}}e^{-\frac{|x^\prime -z^\prime |^2}{8}}dz^\prime \nonumber \\ & \displaystyle +e^{-\frac{\rho ^2}{64}}\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{16}}dz^\prime \nonumber \\ & \displaystyle +e^{-\frac{\rho ^2}{32}}\rho ^{n-2}\rho ^{-n+1}\Big )\nonumber \\\le & \displaystyle C\left( \rho ^{-n}e^{-\frac{x_n^2}{8}}+e^{-\frac{\rho ^2}{32}}\rho ^{-1}+e^{-\frac{\rho ^2}{64}}\right) \nonumber \\\le & \displaystyle Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n}). \end{aligned}$$

(4.23)

In addition, for $1\le j, \ell \le n-1$

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}J_n(x, z_n,1)\Big |\nonumber \\ & \quad =\displaystyle \Big |\int _{\mathbb {R}^{n-1}}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\frac{\partial ^3 E(z)}{\partial z_j\partial z_\ell \partial z_n}dz^\prime \Big |\nonumber \\ & \quad =\displaystyle \Big |\int _{\Pi _\rho (x)}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\frac{\partial ^3 E(z)}{\partial z_j\partial z_\ell \partial z_n}dz^\prime \nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\frac{\partial ^3 E(z)}{\partial z_j\partial z_\ell \partial z_n}\Big |\nonumber \\ & \quad \le \displaystyle \int _{\Pi _\rho (x)}\Big |\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\Big |\Big |\frac{\partial ^3 E(z)}{\partial z_j\partial z_\ell \partial z_n}\Big |dz^\prime \nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}\Big |\frac{\partial ^3G_1(x^\prime -z^\prime , x_n)}{\partial x_j\partial x_\ell \partial x_n}\Big |\Big |\frac{\partial E(z)}{\partial z_n}\Big |dz^\prime \nonumber \\ & \qquad \displaystyle +\int _{\partial \Pi _\rho (x)}\Big |\frac{\partial G_1(x^\prime -z^\prime , x_n)}{\partial x_n}\Big |\Big |\frac{\partial ^2 E(z)}{\partial z_\ell \partial z_n}\Big |d S_{z^\prime }\nonumber \\ & \qquad \displaystyle +\int _{\partial \Pi _\rho (x)}\Big |\frac{\partial ^2 G_1(x^\prime -z^\prime , x_n)}{\partial x_j\partial x_n}\Big |\Big |\frac{\partial E(z)}{\partial z_n}\Big |d S_{z^\prime }\nonumber \\ & \quad \le \displaystyle C\left( \int _{\Pi _\rho (x)}|z|^{-n-1}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{8}}dz^\prime \right. \nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{8}}dz^\prime \nonumber \\ & \qquad \left. \displaystyle +e^{-\frac{\rho ^2}{32}}\int _{\partial \Pi _\rho (x)}|z|^{-n}d S_{z^\prime } +e^{-\frac{\rho ^2}{32}}\int _{\partial \Pi _\rho (x)}|z|^{-n+1}d S_{z^\prime }\right) \nonumber \\ & \quad \le \displaystyle C\Big (\rho ^{-n-1}e^{-\frac{x_n^2}{8}}\int _{\mathbb {R}^{n-1}}e^{-\frac{|x^\prime -z^\prime |^2}{8}}dz^\prime \nonumber \\ & \qquad \displaystyle +e^{-\frac{\rho ^2}{64}}\int _{\mathbb {R}^{n-1}\backslash \Pi _\rho (x)}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}e^{-\frac{|x^\prime -z^\prime |^2+ x_n^2}{16}}dz^\prime \nonumber \\ & \qquad \displaystyle +e^{-\frac{\rho ^2}{32}}\rho ^{n-2}(\rho ^{-n}+\rho ^{-n+1})\Big )\nonumber \\ & \quad \le \displaystyle C\big (\rho ^{-n-1}e^{-\frac{x_n^2}{8}}+e^{-\frac{\rho ^2}{32}}(\rho ^{-2}+\rho ^{-1})\big )\nonumber \\ & \qquad \displaystyle +Ce^{-\frac{\rho ^2}{64}}\left( \int _{\int _{|z^\prime |< 1}}z_n(|z^\prime |^2+z_n^2)^{-\frac{n}{2}}dz^\prime +\int _{|z^\prime |\ge 1}e^{-\frac{|x^\prime -z^\prime |^2}{16}}dz^\prime \right) \nonumber \\ & \quad \le \displaystyle C\left( \rho ^{-n-1}e^{-\frac{x_n^2}{8}}+e^{-\frac{\rho ^2}{32}}(\rho ^{-2}+\rho ^{-1})+e^{-\frac{\rho ^2}{64}}\right) \nonumber \\ & \quad \le \displaystyle Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n-1}); \end{aligned}$$

(4.24)

From (4.20), (4.22) and (4.23), we have for $1\le j\le n-1$, $1\le k\le n$

$$\begin{aligned} \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_n(x, z_n,1)\Big |\le Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n-1+\delta _{kn}}),\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0.\nonumber \\ \end{aligned}$$

(4.25)

Combining (4.20) and (4.25) yields for $1\le j\le n-1$, $1\le i, k\le n$

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_i(x, z_n,1)\Big |\le \displaystyle Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n-1+\delta _{kn}})\nonumber \\ & \quad \le \displaystyle Ce^{-\frac{x_n^2}{64}}(1+|x^\prime |+x_n+z_n)^{-n-1+\delta _{kn}},\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0. \end{aligned}$$

(4.26)

From (4.17) and (4.26), we find for $1\le j\le n-1$, $1\le i, k\le n$ and $t>0$

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_i(x, z_n,t)\Big |=\displaystyle t^{-\frac{n}{2}}\Big |\frac{\partial ^2}{\partial x_j\partial x_k}J_i(t^{-\frac{1}{2}}x, t^{-\frac{1}{2}}z_n,1)\Big |\\ & \quad \le \displaystyle Ce^{-\frac{x_n^2}{64}}t^{-\frac{n}{2}-1}(1+t^{-\frac{1}{2}}\big (|x^\prime |+x_n+z_n)\big )^{-n-1+\delta _{kn}}\\ & \quad =\displaystyle Ce^{-\frac{x_n^2}{64}}t^{-\frac{1+\delta _{kn}}{2}}\big (|x^\prime |+x_n+z_n+\sqrt{t}\big )^{-n-1+\delta _{kn}},\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0, \end{aligned}$$

which implies that (4.15) holds.

From (4.21) and (4.24), we derive for $1\le j,\ell \le n-1$

$$\begin{aligned} \Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}J_n(x, z_n,1)\Big |\le Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n-1}),\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0.\nonumber \\ \end{aligned}$$

(4.27)

Combining (4.21) and (4.27), we conclude for $1\le j,\ell \le n-1$ and $1\le i\le n$

$$\begin{aligned} \Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}J_i(x, z_n,1)\Big |\le Ce^{-\frac{x_n^2}{64}}(1+\rho ^{-n-1}),\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0.\nonumber \\ \end{aligned}$$

(4.28)

From (4.17) and (4.28), we find for $1\le j,\ell \le n-1$, $1\le i\le n$ and $t>0$

$$\begin{aligned} & \displaystyle \Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}J_i(x, z_n,t)\Big |=\displaystyle t^{-\frac{n}{2}}\Big |\frac{\partial ^3}{\partial x_j\partial x_\ell \partial x_n}J_i(t^{-\frac{1}{2}}x, t^{-\frac{1}{2}}z_n,1)\Big |\\ & \quad \le \displaystyle Ce^{-\frac{x_n^2}{64}}t^{-\frac{n}{2}-\frac{3}{2}}(1+t^{-\frac{1}{2}}\big (|x^\prime |+x_n+z_n)\big )^{-n-1}\\ & \quad =\displaystyle Ce^{-\frac{x_n^2}{64}}t^{-1}\big (|x^\prime |+x_n+z_n+\sqrt{t}\big )^{-n-1},\;\;\;\forall x=(x^\prime , x_n)\in \mathbb {R}^n_+,\;\;z_n>0, \end{aligned}$$

which implies that (4.16) holds. $\square$

Proof of Theorem 1.4

Let u be the strong solution of problem (1.1) given in Theorem 1.1. Then u can be represented as follows for $t>0$ (see [24])

$$\begin{aligned} u(x,t)=\int _{\mathbb {R}^n_+}\mathcal {M}\left( x,y,\frac{t}{2}\right) u\left( y,\frac{t}{2}\right) dy-\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\mathcal {M}(x,y,t-s)P(u\cdot \nabla ) u(y,s)dyds,\nonumber \\ \end{aligned}$$

(4.29)

where the definition of $\mathcal {M}=(M_{ij})_{i,j=1,2,\cdots ,n}$ is given in Sect. 2.

Note that $M^*_{kn}=0$, $\;\forall 1\le k\le n$. Then for $1\le k\le n$

$$\begin{aligned} \displaystyle \widetilde{w}_k(x,t)&:=\displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{j=1}^nM_{kj}(x,y,t-s)\big (P(u\cdot \nabla ) u(y,s)\big )_jdyds\nonumber \\&=\displaystyle \sum \limits _{j=1}^n\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}(G_{t-s}(x^\prime -y^\prime ,x_n-y_n)-G_{t-s}(x^\prime -y^\prime ,x_n+y_n)\big )\nonumber \\&\quad \displaystyle \times \delta _{kj} \big (P(u\cdot \nabla ) u(y,s)\big )_jdyds\nonumber \\&\quad \displaystyle +\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{j=1}^{n-1}M^*_{kj}(x,y,t-s)\big (P(u\cdot \nabla ) u(y,s)\big )_jdyds\nonumber \\&=\displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}(G_{t-s}(x^\prime -y^\prime ,x_n-y_n)-G_{t-s}(x^\prime -y^\prime ,x_n+y_n)\big )\nonumber \\&\quad \displaystyle \times \big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\nonumber \\&\quad \displaystyle +\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{j=1}^{n-1}M^*_{kj}(x,y,t-s)\big ((u\cdot \nabla ) u_j(y,s)\nonumber \\&\quad \displaystyle +\sum \limits _{i, \ell =1}^n\partial _{y_j}\mathcal {N}\partial _i\partial _\ell (u_iu_\ell )(y,s)\big )dyds\nonumber \\&=\displaystyle \widetilde{I}_k(x, t)+\widetilde{J}_k(x, t). \end{aligned}$$

(4.30)

Using the heat equation yields for any $(x^\prime ,x_n)\in \mathbb {R}^n$ and $t>0$,

$$\begin{aligned} \partial _{x_n}^2G_t(x^\prime ,x_n)=\left( \partial _t-\sum \limits _{j=1}^{n-1}\partial ^2_{x_j}\right) G_t(x^\prime ,x_n), \end{aligned}$$

and

$$\begin{aligned} \lim \limits _{t\rightarrow 0^+}G_t(x^\prime ,x_n)=\delta (x^\prime ,x_n)\;\;\text{ in } \text{ the } \text{ sense } \text{ of } \text{ the } \text{ distribution }. \end{aligned}$$

Whence we have for $x=(x^\prime ,x_n)\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} \displaystyle \nabla _{x^\prime }\partial _{x_n}\widetilde{I}_k(x, t)= & \displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\nabla _{x^\prime }\partial _{x_n}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds \nonumber \\= & \displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds; \end{aligned}$$

(4.31)

and

$$\begin{aligned} \displaystyle \partial _{x_n}^2\widetilde{I}_k(x, t)= & \displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\{\partial _{x_n}^2[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\}\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds \nonumber \\= & \displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}(-\partial _s)[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds \nonumber \\ & \displaystyle -\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{j=1}^{n-1}\partial ^2_{x_j}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\nonumber \\= & \displaystyle -\big (P(u\cdot \nabla u)(x,t)\big )_k +\int _{\mathbb {R}^n_+}[G_\frac{t}{2}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_\frac{t}{2}(x^\prime -y^\prime , x_n+y_n)]\left( P(u\cdot \nabla ) u\left( y,\frac{t}{2}\right) \right) _kdy\nonumber \\ & \displaystyle +\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)-G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nonumber \\ & \displaystyle \times \partial _s\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\nonumber \\ & \displaystyle -\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)-G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nonumber \\ & \displaystyle \times \sum \limits _{j=1}^{n-1}\partial ^2_{y_j}\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds. \end{aligned}$$

(4.32)

Whence, using (4.31), (4.32) and Lemma 3.1, we conclude that for $1\le k\le n$ and $t>1$

$$\begin{aligned} & \displaystyle \Vert \partial _{x_n}^3\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla _{x^\prime }\partial _{x_n}^2\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla _{x^\prime }^2\partial _{x_n}\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \quad \le \displaystyle \Vert \nabla \big (P(u\cdot \nabla ) u(\cdot ,t)\big )_k\Vert _{L^1(\mathbb {R}^n_+)} +\int _{\mathbb {R}^n_+}\Vert \nabla \left[ G_\frac{t}{2}(x^\prime -y^\prime , x_n-y_n)\right. \\ & \qquad \left. \displaystyle -G_\frac{t}{2}(x^\prime -y^\prime , x_n+y_n)\right] \Vert _{L^1_x(\mathbb {R}^n_+)}|\left( P(u\cdot \nabla ) u\left( y,\frac{t}{2}\right) \right) _k|dy\\ & \qquad \displaystyle +\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert \nabla [G_{t-s}(x^\prime -y^\prime , x_n-y_n)-G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\Vert _{L^1_x(\mathbb {R}^n_+)}\\ & \qquad \displaystyle \big (|\partial _s\big (P(u\cdot \nabla ) u(y,s)\big )_k|+|\nabla ^2_{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_k|\big )dyds\\ & \qquad \displaystyle +\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert \nabla [G_{t-s}(x^\prime -y^\prime , x_n-y_n)\\ & \qquad \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\Vert _{L^1_x(\mathbb {R}^n_+)}\sum \limits _{j=1}^{n-1}|\partial ^2_{y_j}\big (P(u\cdot \nabla ) u(y,s)\big )_k|dyds\\ & \quad \le \displaystyle \Vert \nabla \big (u\cdot \nabla ) u(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla \sum \limits _{i, j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle + Ct^{-\frac{1}{2}}\big (\Vert \big (u\cdot \nabla ) u_k\left( \cdot ,\frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)} +\Vert \sum \limits _{i, j=1}^n\partial _k\mathcal {N}\partial _i\partial _j(u_iu_j)\left( \cdot ,\frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}\big )\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}} \big (\Vert \partial _s\big (u\cdot \nabla ) u_k(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \sum \limits _{i, j=1}^n\partial _s\partial _k\mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\big )ds\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}} \big (\Vert \sum \limits _{\ell =1}^{n-1}\partial ^2_\ell \big (u\cdot \nabla ) u_k(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \sum \limits _{\ell =1}^{n-1}\sum \limits _{i, j=1}^n\partial ^2_\ell \partial _k\mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\big )ds\\ & \quad \le \displaystyle C\big (\Vert u(t)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(t)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(t)\Vert ^2_{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert u(t)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla u(t)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(t)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(t)\Vert _{L^2(\mathbb {R}^n_+)}\big ) \\ & \qquad \displaystyle + Ct^{-\frac{1}{2}}\big (\Vert u(t)\Vert ^2_{L^2(\mathbb {R}^n_+)} +\Vert \nabla u(t)\Vert ^2_{L^2(\mathbb {R}^n_+)}\big ) \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}} \big (\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s\nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \partial _s u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s u(s)\Vert _{L^2(\mathbb {R}^n_+)} \\ & \qquad \displaystyle +\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s\nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)} \big )ds\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}} \big (\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^3 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^3 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\big )ds. \end{aligned}$$

Using Lemma 3.2 yields for $1\le k\le n$ and $t>1$

$$\begin{aligned} & \displaystyle \Vert \partial _{x_n}^3\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla _{x^\prime }\partial _{x_n}^2\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla _{x^\prime }^2\partial _{x_n}\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{1}{2}-\frac{n}{2}}+C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}}s^{-1-\frac{n}{2}}ds\nonumber \\ & \quad \le \displaystyle \widetilde{C}t^{-\frac{1}{2}-\frac{n}{2}}. \end{aligned}$$

(4.33)

Furthermore suppose $\Vert x_nu_0\Vert _{L^1(\mathbb {R}^n_+)}<\infty$, there holds for $1\le k\le n$ and $t>1$

Set

$$\begin{aligned} N_{ij}(x,y,t)=\frac{\partial }{\partial x_j}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_i}G_t(z-y^*)dz,\;\;\;1\le i, j\le n, \end{aligned}$$

and

$$\begin{aligned} b_j(y,s)=(u\cdot \nabla ) u_j(y,s)+\sum \limits _{i,\ell =1}^n\partial _{y_j}\mathcal {N}\partial _i\partial _\ell (u_iu_\ell )(y,s). \end{aligned}$$

Then

$$\begin{aligned} M_{ij}^*(x,y,t)=(1-\delta _{jn})N_{ij}(x,y,t), \end{aligned}$$

and

$$\begin{aligned} \widetilde{J}_k(x, t)=4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{j=1}^{n-1}N_{kj}(x,y,t-s)b_j(y,s)dyds. \end{aligned}$$

(4.34)

Using Lemma 4.1, we get for $x, y\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} \displaystyle N_{nn}(x,y,t)= & \displaystyle \frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_n}G_t(z-y^*)dz^\prime dz_n\nonumber \\= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial }{\partial x_j}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_j}G_t(z-y^*)dz^\prime dz_n+\frac{1}{2}G_t(x-y^*)\nonumber \\= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial }{\partial x_j}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(z)}{\partial x_j}G_t(x-y^*-z)dz^\prime dz_n+\frac{1}{2}G_t(x-y^*)\nonumber \\= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial }{\partial x_j}\int _0^{x_n}J_j(x^\prime -y^\prime , x_n+y_n-z_n, z_n,t)dz_n+\frac{1}{2}G_t(x-y^*), \nonumber \\ \end{aligned}$$

(4.35)

where

$$\begin{aligned} J_i(x^\prime , x_n, z_n,t)=\int _{\mathbb {R}^{n-1}}G_t(x^\prime -z^\prime , x_n)\frac{\partial E(z^\prime , z_n)}{\partial z_i}dz^\prime ,\;\;\;1\le i\le n. \end{aligned}$$

Let $1\le i\le n-1$. Then for $x=(x^\prime , x_n), y=(y^\prime , y_n)\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} \displaystyle N_{in}(x,y,t)= & \displaystyle \frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_i}G_t(z-y^*)dz\nonumber \\= & \displaystyle \frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}G_t(x-y^*-z)\frac{\partial E(z)}{\partial z_i}dz\nonumber \\= & \displaystyle \int _0^{x_n}\frac{\partial }{\partial x_n}\int _{\mathbb {R}^{n-1}}G_t(x-y^*-z)\frac{\partial E(z)}{\partial z_i}dz\nonumber \\ & \displaystyle +\int _{\mathbb {R}^{n-1}}G_t(x^\prime -y^\prime -z^\prime , y_n)\partial _{z_i}E(z^\prime , x_n)dz^\prime \nonumber \\= & \displaystyle \int _0^{x_n}\frac{\partial }{\partial x_n} J_i(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)dz_n+J_i(x^\prime -y^\prime , y_n,x_n,t); \nonumber \\ \end{aligned}$$

(4.36)

and

$$\begin{aligned} \displaystyle N_{ni}(x,y,t)= & \displaystyle \frac{\partial }{\partial x_i}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_n}G_t(z-y^*)dz\nonumber \\= & \displaystyle \int _0^{x_n}\frac{\partial }{\partial x_n}\int _{\mathbb {R}^{n-1}}G_t(x-y^*-z)\frac{\partial E(z)}{\partial z_i}dz\nonumber \\= & \displaystyle \int _0^{x_n}\frac{\partial }{\partial x_n} J_i(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)dz_n. \end{aligned}$$

(4.37)

Combining (4.36) and (4.37), together with Lemma 4.2 yields for $1\le i\le n-1$, $x=(x^\prime , x_n), y=(y^\prime , y_n)\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} & \displaystyle |\nabla _{x^\prime }^2N_{ni}(x,y,t)|+|\nabla _{x^\prime }^2N_{in}(x,y,t)|\nonumber \\ & \quad \le \displaystyle 2\int _0^{x_n}|\frac{\partial }{\partial x_n} \nabla _{x^\prime }^2J_i(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)|dz_n+|\nabla _{x^\prime }^2J_i(x^\prime -y^\prime , y_n,x_n,t)|\nonumber \\ & \quad \le \displaystyle Ct^{-1} (|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n-1}\int _0^{x_n}e^{-\frac{(x_n+y_n-z_n)^2}{64t}}dz_n\nonumber \\ & \qquad \displaystyle +Ct^{-\frac{1}{2}}(|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n-1}e^{-\frac{y_n^2}{64t}}\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{1}{2}} (|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n-1}. \end{aligned}$$

(4.38)

Using Lemma 4.1, we find for $1\le k\le n-1$, $x=(x^\prime , x_n), y=(y^\prime , y_n)\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_n^3}N_{nk}(x,y,t)= & \displaystyle \frac{\partial ^3}{\partial x_k\partial x_n^2}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_n}G_t(z-y^*)dz\nonumber \\ \quad= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial ^2}{\partial x_n^2}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_j}G_t(z-y^*)dz\nonumber \\ & \qquad \displaystyle +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_n^2}G_t(x-y^*)\nonumber \\ \quad= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_j\partial x_k}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_n}G_t(z-y^*)dz\nonumber \\ & \qquad \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_j\partial x_k}\frac{\partial }{\partial x_n}\int _{\mathbb {R}^{n-1}}E(x^\prime -z^\prime , 0)G_t(z^\prime -y^\prime , x_n+y_n)dz^\prime \nonumber \\ & \qquad \displaystyle +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_n^2}G_t(x-y^*)\nonumber \\ \quad= & \displaystyle \sum \limits _{j,\ell =1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_j\partial x_k}\frac{\partial }{\partial x_\ell }\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_\ell }G_t(z-y^*)dz\nonumber \\ & \qquad \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial }{\partial x_n}J_j(x^\prime -y^\prime , x_n+y_n, 0,t)\nonumber \\ & \qquad \displaystyle -\frac{1}{2}\sum \limits _{j=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_j\partial x_k}G_t(x-y^*)+\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_n^2}G_t(x-y^*); \end{aligned}$$

(4.39)

and for $1\le k, m,q\le n-1$

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_m\partial x_n^2}N_{nk}(x,y,t)= & \displaystyle \frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_n}G_t(z-y^*)dz\nonumber \\ \quad= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_k\partial x_m}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_j}G_t(z-y^*)dz\nonumber \\ & \qquad \displaystyle +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}G_t(x-y^*)\nonumber \\ \quad= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial }{\partial x_m}N_{jn}(x,y,t)+\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}G_t(x-y^*); \nonumber \\ \end{aligned}$$

(4.40)

$$\begin{aligned} & \displaystyle \frac{\partial ^3}{\partial x_m\partial x_q\partial x_n}N_{nk}(x,y,t)\nonumber \\ \quad= & \displaystyle \frac{\partial ^3}{\partial x_k\partial x_m\partial x_q}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_n}G_t(z-y^*)dz\nonumber \\ \quad= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_k\partial x_m}\frac{\partial }{\partial x_q}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_j}G_t(z-y^*)dz\nonumber \\ & \qquad \displaystyle +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_q}G_t(x-y^*)\nonumber \\ \quad= & \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial }{\partial x_m}N_{jq}(x,y,t)+\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_q}G_t(x-y^*);\nonumber \\ \end{aligned}$$

(4.41)

Using (4.36) and Lemma 4.1 yields for $1\le k, m\le n-1$, $x, y\in \mathbb {R}^n_+$ and $t>0$

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_n^3}N_{mk}(x,y,t)= & \displaystyle \frac{\partial ^3}{\partial x_k\partial x_n^2}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_m}G_t(z-y^*)dz\nonumber \\= & \displaystyle \frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_n}N_{mn}(x,y,t)\nonumber \\= & \displaystyle \frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_n}\int _0^{x_n}\frac{\partial }{\partial x_n} J_m(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)dz_n\nonumber \\ & \displaystyle +\frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_n}J_m(x^\prime -y^\prime , y_n,x_n,t)\nonumber \\= & \displaystyle \frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_n}\int _0^{x_n}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)dz_n\nonumber \\ & \displaystyle +\frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , y_n,x_n,t)\nonumber \\= & \displaystyle -\sum \limits _{\ell =1}^{n-1}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}\frac{\partial }{\partial x_\ell }\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_\ell }G_t(z-y^*)dz\nonumber \\ & \displaystyle +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}G_t(x-y^*)\nonumber \\ & \displaystyle +\frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , y_n,x_n,t); \end{aligned}$$

(4.42)

and for $1\le j,q\le n-1$

$$\begin{aligned} & \displaystyle \frac{\partial ^3}{\partial x_j\partial x_n^2}N_{mk}(x,y,t)=\displaystyle \frac{\partial ^3}{\partial x_k\partial x_j\partial x_n}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_m}G_t(z-y^*)dz\nonumber \\ & \quad =\displaystyle \frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_n}N_{mn}(x,y,t)\nonumber \\ & \quad = \displaystyle \frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_n}\int _0^{x_n}\frac{\partial }{\partial x_n} J_m(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)dz_n\nonumber \\ & \qquad \displaystyle +\frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_n} J_m(x^\prime -y^\prime , y_n,x_n,t)\nonumber \\ & \quad =\displaystyle \frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_n}\int _0^{x_n}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , x_n+y_n-z_n,z_n,t)dz_n\nonumber \\ & \qquad \displaystyle +\frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , y_n,x_n,t)\nonumber \\ & \quad = \displaystyle -\sum \limits _{\ell =1}^{n-1}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_j}\frac{\partial }{\partial x_\ell }\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_\ell }G_t(z-y^*)dz\nonumber \\ & \qquad \displaystyle +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_j}G_t(x-y^*)+\frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , y_n,x_n,t); \end{aligned}$$

(4.43)

$$\begin{aligned} & \displaystyle \frac{\partial ^3}{\partial x_j\partial x_q\partial x_n}N_{mk}(x,y,t)=\displaystyle \frac{\partial ^3}{\partial x_k\partial x_j\partial x_q}\frac{\partial }{\partial x_n}\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_m}G_t(z-y^*)dz\nonumber \\ & \quad = \displaystyle \frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_q}N_{mn}(x,y,t). \end{aligned}$$

(4.44)

It follows from (4.34) and (4.39) that for $x\in \mathbb {R}^n_+$ and $t>0$

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_n^3}\widetilde{J}_n(x, t) & =\displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^3}{\partial x_n^3}N_{nk}(x,y,t-s)b_k(y,s)dyds\nonumber \\ & = \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\left( \sum \limits _{j,\ell =1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_j\partial x_k}N_{\ell \ell }(x,y,t-s)\right. \nonumber \\ & \quad \ \displaystyle -\sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial }{\partial x_n}J_j(x^\prime -y^\prime , x_n+y_n, 0,t-s)\nonumber \\ & \quad \ \left. \displaystyle -\frac{1}{2}\sum \limits _{j=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_j\partial x_k}G_{t-s}(x-y^*)+\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_n\partial x_n}G_{t-s}(x-y^*)\right) b_k(y,s)dyds\nonumber \\ & = \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j,\ell =1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}N_{\ell \ell }(x,y,t-s)\frac{\partial }{\partial y_j}b_k(y,s)dyds\nonumber \\ & \quad \ \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_j}J_n(x^\prime -y^\prime , x_n+y_n, 0,t-s)\frac{\partial }{\partial y_k}b_k(y,s)dyds\nonumber \\ & \quad \ \displaystyle -2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}G_{t-s}(x-y^*)\frac{\partial }{\partial y_j}b_k(y,s)dyds\nonumber \\ & \quad \ \displaystyle +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_n\partial x_n}G_{t-s}(x-y^*)\frac{\partial }{\partial y_k}b_k(y,s)dyds; \end{aligned}$$

(4.45)

From (4.34), (4.40) and (4.41), we have for $1\le m, q\le n-1$, $x\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_m\partial x_n^2}\widetilde{J}_n(x, t)= & \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^3}{\partial x_m\partial x_n^2}N_{nk}(x,y,t-s)b_k(y,s)dyds\nonumber \\= & \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big ( \sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial }{\partial x_m}N_{jn}(x,y,t-s)\nonumber \\ & \displaystyle -\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}G_{t-s}(x-y^*)\Big )b_k(y,s)dyds\nonumber \\= & \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}N_{jn}(x,y,t-s)\frac{\partial }{\partial y_m}b_k(y,s)dyds\nonumber \\ & \displaystyle +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_n}G_{t-s}(x-y^*)\frac{\partial }{\partial y_m}b_k(y,s)dyds; \end{aligned}$$

(4.46)

and

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_m\partial x_q\partial x_n}\widetilde{J}_n(x, t) & =\displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^3}{\partial x_m\partial x_q\partial x_n}N_{nk}(x,y,t-s)b_k(y,s)dyds\nonumber \\ & = \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big ( \sum \limits _{j=1}^{n-1}\frac{\partial ^2}{\partial x_j\partial x_k}\frac{\partial }{\partial x_m}N_{jq}(x,y,t)\nonumber \\ & \displaystyle \quad \ -\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_q}G_t(x-y^*)\Big )b_k(y,s)dyds\nonumber \\ & = \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}N_{jq}(x,y,t-s)\frac{\partial }{\partial y_m}b_k(y,s)dyds\nonumber \\ & \displaystyle \quad \ +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1} \frac{\partial ^2}{\partial x_k\partial x_q}G_{t-s}(x-y^*)\nonumber \\ & \quad \ \times \frac{\partial }{\partial y_m}b_k(y,s)dyds. \end{aligned}$$

(4.47)

Let $1\le j, m, q\le n-1$. Using (4.34) and (4.42)–(4.44) yields for $x\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_n^3}\widetilde{J}_m(x, t) & =\displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^3}{\partial x_n^3}N_{mk}(x,y,t-s)b_k(y,s)dyds\nonumber \\ & = \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big (-\sum \limits _{\ell =1}^{n-1}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_\ell }N_{\ell n}(x,y,t-s)\nonumber \\ & \displaystyle \quad \ +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_n}G_{t-s}(x-y^*)\nonumber \\ & \displaystyle \quad \ +\frac{\partial ^2}{\partial x_k\partial x_n}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , y_n,x_n,t-s)\Big )b_k(y,s)dyds\nonumber \\ & = \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+} \sum \limits _{k,\ell =1}^{n-1}\frac{\partial ^2}{\partial x_\ell \partial x_k}N_{\ell n}(x,y,t-s)\frac{\partial }{\partial y_m}b_k(y,s)dyds\nonumber \\ & \displaystyle \quad \ +4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_m \partial x_n}J_n(x^\prime -y^\prime , y_n,x_n,t-s)\frac{\partial }{\partial y_k}b_k(y,s)dyds\nonumber \\ & \displaystyle \quad \ +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_n\partial x_m}G_{t-s}(x-y^*)\frac{\partial }{\partial y_k}b_k(y,s)dyds; \end{aligned}$$

(4.48)

$$\begin{aligned} & \displaystyle \frac{\partial ^3}{\partial x_j\partial x_n^2}\widetilde{J}_m(x, t) =\displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_n^2}N_{mk}(x,y,t-s)b_k(y,s)dyds\nonumber \\ & =\displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big (-\sum \limits _{\ell =1}^{n-1}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_j}\frac{\partial }{\partial x_\ell }\int _0^{x_n}\int _{\mathbb {R}^{n-1}}\frac{\partial E(x-z)}{\partial x_\ell }G_t(z-y^*)dz\nonumber \\ & \displaystyle \quad \ +\frac{1}{2}\frac{\partial ^3}{\partial x_k\partial x_m\partial x_j}G_{t-s}(x-y^*)\nonumber \\ & \quad \ +\frac{\partial ^2}{\partial x_k\partial x_j}\frac{\partial }{\partial x_m} J_n(x^\prime -y^\prime , y_n,x_n,t-s) \Big )b_k(y,s)dyds\nonumber \\ & = \displaystyle -4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+} \sum \limits _{k,\ell =1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}N_{\ell \ell }(x,y,t-s)\frac{\partial }{\partial y_m}b_k(y,s)dyds\nonumber \\ & \displaystyle \quad \ +4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_k \partial x_j}J_n(x^\prime -y^\prime , y_n,x_n,t-s)\frac{\partial }{\partial y_m}b_k(y,s)dyds\nonumber \\ & \displaystyle \quad \ +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}G_{t-s}(x-y^*)\frac{\partial }{\partial y_m}b_k(y,s)dyds; \end{aligned}$$

(4.49)

and

$$\begin{aligned} \displaystyle \frac{\partial ^3}{\partial x_j\partial x_q\partial x_n}\widetilde{J}_m(x, t)= & \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^3}{\partial x_j\partial x_q\partial x_n}N_{mk}(x,y,t-s)b_k(y,s)dyds\nonumber \\= & \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\frac{\partial ^2}{\partial x_k\partial x_j}N_{mn}(x,y,t-s)\frac{\partial }{\partial y_q}b_k(y,s)dyds. \nonumber \\ \end{aligned}$$

(4.50)

Using (4.38) yields for every $1\le i, k, \ell \le n-1$, $y=(y^\prime , y_n)\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} & \displaystyle \left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_\ell } N_{in}(x,y,t)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{1}{2}}\int _{\mathbb {R}^n_+}(x_n+y_n)^\epsilon (|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n-1}dx^\prime dx_n\nonumber \\ & \quad \le \displaystyle Ct^{-1+\frac{\epsilon }{2}},\;\;\;\;\epsilon \in (0,1). \end{aligned}$$

(4.51)

By Lemma 4.2, we get for every $1\le k, \ell \le n-1$, $y=(y^\prime , y_n)\in \mathbb {R}^n_+$ and $t>0$,

$$\begin{aligned} & \displaystyle \left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_\ell }J_n(x^\prime -y^\prime , y_n,x_n,t)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle Ct^{-\frac{1}{2}}\int _{\mathbb {R}^n_+}(x_n+y_n)^\epsilon (|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n-1}dx^\prime dx_n\nonumber \\ & \quad \le \displaystyle Ct^{-1+\frac{\epsilon }{2}},\;\;\;\;\epsilon \in (0,1); \end{aligned}$$

(4.52)

and

$$\begin{aligned} & \displaystyle \left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_\ell \partial x_n}J_n(x^\prime -y^\prime , y_n,x_n,t)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle Ct^{-1}\int _{\mathbb {R}^n_+}(x_n+y_n)^\epsilon (|x^\prime -y^\prime |+x_n+y_n+\sqrt{t})^{-n}e^{-\frac{(x_n+y_n)^2}{64t}}dx^\prime dx_n\nonumber \\ & \quad \le \displaystyle Ct^{-1}\int _0^\infty \int _0^\infty (s+x_n+y_n+\sqrt{t})^{-n+\epsilon +n-2}e^{-\frac{(x_n+y_n)^2}{64t}}dsdx_n\nonumber \\ & \quad \le \displaystyle Ct^{-1}\int _0^\infty (x_n+y_n+\sqrt{t})^{-1+\epsilon }e^{-\frac{(x_n+y_n)^2}{64t}}dx_n\nonumber \\ & \quad \le \displaystyle Ct^{-1+\frac{\epsilon }{2}}\int _0^\infty (\tau +1)^{-1+\epsilon }e^{-\frac{\tau ^2}{64}}d\tau \nonumber \\ & \quad \le \displaystyle Ct^{-1+\frac{\epsilon }{2}},\;\;\;\;\epsilon \in (0,1). \end{aligned}$$

(4.53)

From (4.45)–(4.47) and (4.51)–(4.53), we get for $t>0$

$$\begin{aligned} & \displaystyle \left\| \frac{\partial ^3}{\partial x_n^3}\widetilde{J}_n(x, t)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle 4\sum \limits _{k,j,\ell =1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}N_{\ell \ell }(x,y,t-s)\Vert _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times |y_n^{-\epsilon }\frac{\partial }{\partial y_j}b_k(y,s)|dyds\nonumber \\ & \qquad \displaystyle +4\sum \limits _{k,j=1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_j^2}J_n(x^\prime -y^\prime , x_n+y_n,0,t-s)\Vert _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times |y_n^{-\epsilon }\frac{\partial }{\partial y_k}b_k(y,s)|dyds\nonumber \\ & \qquad \displaystyle +2\sum \limits _{k,j=1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}G_{t-s}(x-y^*)\Vert _{L^1_x(\mathbb {R}^n_+)}|y_n^{-\epsilon }\frac{\partial }{\partial y_j}b_k(y,s)|dyds\nonumber \\ & \qquad \displaystyle +2\sum \limits _{k=1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_n^2}G_{t-s}(x-y^*)\Vert _{L^1_x(\mathbb {R}^n_+)}|y_n^{-\epsilon }\frac{\partial }{\partial y_k}b_k(y,s)|dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1); \end{aligned}$$

(4.54)

moreover, for $1\le m, q\le n-1$,

$$\begin{aligned} & \displaystyle \left\| \frac{\partial ^3}{\partial x_m\partial x_n^2}\widetilde{J}_n(x, t)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}N_{jn}(x,y,t-s)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)\right| dyds\nonumber \\ & \qquad \displaystyle +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\big \Vert (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_n}G_{t-s}(x-y^*)\big \Vert _{L^1(\mathbb {R}^n_+)}|y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)|dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1); \end{aligned}$$

(4.55)

and

$$\begin{aligned} & \displaystyle \left\| \frac{\partial ^3}{\partial x_m\partial x_q\partial x_n}\widetilde{J}_n(x, t)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}N_{jq}(x,y,t-s)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)\right| dyds\nonumber \\ & \qquad \displaystyle +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_q}G_{t-s}(x-y^*)\right\| _{L^1(\mathbb {R}^n_+)}|y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)|dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1). \end{aligned}$$

(4.56)

From (4.48)–(4.53), we derive that for $1\le j, m, q\le n-1$ and $t>0$

$$\begin{aligned} & \displaystyle \left\| \frac{\partial ^3}{\partial x_n^3}\widetilde{J}_m(x, t)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle 4\sum \limits _{k,\ell =1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+} \left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_\ell \partial x_k}N_{\ell n}(x,y,t-s)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)\right| dyds\nonumber \\ & \qquad \displaystyle +4\sum \limits _{k=1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_m \partial x_n}J_n(x^\prime -y^\prime , y_n,x_n,t-s)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \big |y_n^{-\epsilon }\frac{\partial }{\partial y_k}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +2\sum \limits _{k=1}^{n-1}\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_n\partial x_m}G_{t-s}(x-y^*)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_k}b_k(y,s)\right| dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1); \end{aligned}$$

(4.57)

$$\begin{aligned} & \displaystyle \left\| \frac{\partial ^3}{\partial x_j\partial x_n^2}\widetilde{J}_m(x, t)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+} \sum \limits _{k,\ell =1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}N_{\ell \ell }(x,y,t-s)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)\right| dyds\nonumber \\ & \qquad \displaystyle +4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k \partial x_j}J_n(x^\prime -y^\prime , y_n,x_n,t-s)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)\right| dyds\nonumber \\ & \displaystyle \qquad +2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}G_{t-s}(x-y^*)\right\| _{L^1_x(\mathbb {R}^n_+)}\left| y_n^{-\epsilon }\frac{\partial }{\partial y_m}b_k(y,s)\right| dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1); \end{aligned}$$

(4.58)

and

$$\begin{aligned} & \displaystyle \left\| \frac{\partial ^3}{\partial x_j\partial x_q\partial x_n}\widetilde{J}_m(x, t)\right\| _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle \displaystyle 4\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\left\| (x_n+y_n)^\epsilon \frac{\partial ^2}{\partial x_k\partial x_j}N_{mn}(x,y,t-s)\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \left| y_n^{-\epsilon }\frac{\partial }{\partial y_q}b_k(y,s)\right| dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1). \end{aligned}$$

(4.59)

From (4.54)–(4.59), we obtain for $t>0$

$$\begin{aligned} \big \Vert \nabla ^3\widetilde{J}(\cdot , t)\Vert _{L^1(\mathbb {R}^n_+)}\le C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}\Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;\;\epsilon \in (0,1). \end{aligned}$$

(4.60)

Recall that

$$\begin{aligned} b_j(y,s)=(u\cdot \nabla ) u_j(y,s)+\sum \limits _{i,\ell =1}^n\partial _{y_j}\mathcal {N}\partial _i\partial _\ell (u_iu_\ell )(y,s),\;\;\;1\le j\le n. \end{aligned}$$

Whence we conclude for $\epsilon \in (0,1)$,

$$\begin{aligned} \displaystyle \Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle \Vert y_n^{-\epsilon }\nabla ^\prime ((u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \displaystyle +\Vert y_n^{-\epsilon }\nabla ^\prime \left( \sum \limits _{i,\ell =1}^n\nabla \mathcal {N}\partial _i\partial _\ell (u_iu_\ell )(\cdot ,s)\right) \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle \Vert y_n^{-\epsilon }(\nabla ^\prime u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \displaystyle +\Vert y_n^{-\epsilon }(u\cdot \nabla ^\prime \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \displaystyle +\Vert y_n^{-\epsilon } \sum \limits _{i,\ell =1}^n\nabla \mathcal {N}\partial _i\partial _\ell \nabla ^\prime (u_iu_\ell )(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}. \end{aligned}$$

(4.61)

To proceed, let $0<\epsilon <1$. Then for $s>0$

$$\begin{aligned} & \displaystyle \Vert y_n^{-\epsilon }(\nabla ^\prime u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}+\Vert y_n^{-\epsilon }(u\cdot \nabla ^\prime \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\\ & \quad \le \displaystyle \int _0^1\int _{\mathbb {R}^{n-1}}y_n^{-\epsilon }|\nabla ^\prime u(y, s)||\nabla u(y, s)|dy^\prime dy_n+\int _0^1\int _{\mathbb {R}^{n-1}}y_n^{-\epsilon }|u(y, s)||\nabla ^\prime \nabla u(y, s)|dy^\prime dy_n\\ & \qquad \displaystyle +\Vert \nabla ^\prime (u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}+\Vert (u\cdot \nabla ^\prime \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\\ & \quad \le \displaystyle \Vert y_n^{-\epsilon }\nabla ^\prime u(s)\Vert _{L^2(\mathbb {R}^{n-1}\times (0,1))}\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert y_n^{-\epsilon } u(s)\Vert _{L^2(\mathbb {R}^{n-1}\times (0,1))}\Vert \nabla ^\prime \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \nabla ^\prime u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^\prime \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}. \end{aligned}$$

One-dimensional Hardy inequality yields for $s>0$ and $\epsilon \in (0,1)$,

$$\begin{aligned} & \displaystyle \Vert y_n^{-\epsilon }\nabla ^\prime u(s)\Vert ^2_{L^2(\mathbb {R}^{n-1}\times (0,1))}+\Vert y_n^{-\epsilon } u(s)\Vert ^2_{L^2(\mathbb {R}^{n-1}\times (0,1))}\\ & \quad \le \displaystyle \int _{\mathbb {R}^{n-1}}\left( \int _0^1y_n^{2-2\epsilon }\frac{|\nabla ^\prime u(y^\prime ,y_n, s)|^2}{y_n^2}dy_n+\int _0^1y_n^{2-2\epsilon }\frac{|u(y^\prime ,y_n, s)|^2}{y_n^2}dy_n\right) dy^\prime \\ & \quad \le \displaystyle \int _{\mathbb {R}^{n-1}}\left( \int _0^\infty \frac{|\nabla ^\prime u(y^\prime ,y_n, s)|^2}{y_n^2}dy_n +\int _0^\infty \frac{|u(y^\prime ,y_n, s)|^2}{y_n^2}dy_n\right) dy^\prime \\ & \quad \le \displaystyle C\int _{\mathbb {R}^{n-1}}\left( \int _0^\infty \big |\partial _n\nabla ^\prime u(y^\prime , y_n, s)\big |^2dy_n+\int _0^\infty \big |\partial _nu(y^\prime , y_n, s)\big |^2dy_n\right) dy^\prime \\ & \quad \le \displaystyle C(\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}). \end{aligned}$$

Whence we obtain for $0<\epsilon <1$ and $s>0$

$$\begin{aligned} & \displaystyle \Vert y_n^{-\epsilon }\nabla ^\prime (u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}+\Vert y_n^{-\epsilon }(u\cdot \nabla ^\prime \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}). \end{aligned}$$

(4.62)

In addition, it follows from Lemma 3.1 that for $0<\epsilon <1$ and $s>0$

$$\begin{aligned} & \displaystyle \Vert y_n^{-\epsilon } \sum \limits _{i,\ell =1}^n\nabla \mathcal {N}\partial _i\partial _\ell \nabla ^\prime (u_iu_\ell )(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}). \end{aligned}$$

(4.63)

Inserting (4.62) and (4.63) into (4.61), we find for $0<\epsilon <1$ and $s>0$

$$\begin{aligned} \Vert y_n^{-\epsilon }\nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\le C(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}). \end{aligned}$$

(4.64)

Combining (4.60) and (4.64), we obtain for $0<\epsilon <1$ and $t>1$

$$\begin{aligned} \displaystyle \big \Vert \nabla ^3\widetilde{J}(\cdot , t)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)})ds\nonumber \\\le & \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\epsilon }{2}}s^{-1-\frac{n}{2}}ds\nonumber \\\le & \displaystyle Ct^{-1-\frac{n}{2}+\frac{\epsilon }{2}}. \end{aligned}$$

(4.65)

From (4.29), (4.30), (4.33) and (4.65), together with Theorem 1.3, we conclude for $t>1$

$$\begin{aligned} & \displaystyle \Vert \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le \displaystyle \int _{\mathbb {R}^n_+}\left\| \nabla ^3_x\mathcal {M}\left( x,y,\frac{t}{2}\right) y_n^\frac{1}{2}\right\| _{L^1_x(\mathbb {R}^n_+)}\left| y_n^{-\frac{1}{2}}u\left( y,\frac{t}{2}\right) \right| dy\\ & \displaystyle +\sum \limits _{k=1}^n(\Vert \nabla ^3\widetilde{I}_k(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla ^3\widetilde{J}_k(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)})\\\le & \displaystyle Ct^{-\frac{3}{2}+\frac{1}{4}}\Vert y_n^{-\frac{1}{2}}u\left( \frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}+C\left( t^{-\frac{1}{2}-\frac{n}{2}}+t^{-1-\frac{n}{2}+\frac{\epsilon }{2}}\right) \;\;\;\text{ where }\;\;0<\epsilon <1\\\le & \displaystyle \left\{ \begin{array}{lll} Ct^{-\frac{3}{2}}& \text{ if }& n\ge 3,\\ Ct^{-\frac{3}{2}}\log _e(1+t)& \text{ if }& n=2.\\ \end{array} \right. \end{aligned}$$

In addition, suppose $\Vert x_nu_0\Vert _{L^1(\mathbb {R}^n_+)}<\infty$, together with $(4.33)^\prime$, there holds for any $t>1$

$$\begin{aligned} \displaystyle \Vert \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle \int _{\mathbb {R}^n_+}\left\| \nabla ^3_x\mathcal {M}\left( x,y,\frac{t}{2}\right) y_n^\frac{1}{2}\right\| _{L^1_x(\mathbb {R}^n_+)}\left| y_n^{-\frac{1}{2}}u\left( y,\frac{t}{2}\right) \right| dy\\ & \displaystyle +\sum \limits _{k=1}^n(\Vert \nabla ^3\widetilde{I}_k(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert \nabla ^3\widetilde{J}_k(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)})\\\le & \displaystyle Ct^{-\frac{3}{2}+\frac{1}{4}}\Vert y_n^{-\frac{1}{2}}u\left( \frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}+C\left( t^{-\frac{3}{2}-\frac{n}{2}}+t^{-1-\frac{n}{2}+\frac{\epsilon }{2}}\right) \\\le & \displaystyle C\left( t^{-\frac{3}{2}}+t^{-1-\frac{n}{2}+\frac{\epsilon }{2}}\right) \;\;\;\text{ where }\;\;0<\epsilon <1\\\le & \displaystyle \widetilde{C}t^{-\frac{3}{2}}. \end{aligned}$$

which is (1.2).

Now suppose

$$\begin{aligned} \Vert x_nu_0\Vert _{L^2(\mathbb {R}^n_+)}+\Vert (1+x_n)\nabla u_0\Vert _{L^2(\mathbb {R}^n_+)}+\Vert x_nu_0\Vert _{L^1(\mathbb {R}^n_+)}<\infty . \end{aligned}$$

We give the proof of $\Vert x_n^\beta \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}$, where u is the strong solution of (1.1), given in Theorem 1.1.

From (4.31), we have for $x=(x^\prime ,x_n)\in \mathbb {R}^n_+$ and $t>0$

$$\begin{aligned} \displaystyle \nabla _{x^\prime }\partial _{x_n}\widetilde{I}_k(x, t)= & \displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{y_n}[-G_{t-s}(x^\prime -y^\prime , x_n-y_n)\\ & \displaystyle +G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\\ & \displaystyle -2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}G_{t-s}(x^\prime -y^\prime , x_n+y_n)\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\\= & \displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\\ & \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nabla _{y^\prime }\partial _{y_n}\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\\ & \displaystyle -2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}G_{t-s}(x^\prime -y^\prime , x_n+y_n)\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds, \end{aligned}$$

which implies

$$\begin{aligned} & \displaystyle \nabla _{x^\prime }\partial _{x_n}\partial _{x_n}\widetilde{I}_k(x, t)=\displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \quad \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nabla _{y^\prime }\partial _{y_n}\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\nonumber \\ & \qquad \displaystyle -2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}\partial _{x_n}G_{t-s}(x^\prime -y^\prime , x_n+y_n)\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds, \end{aligned}$$

(4.66)

and

$$\begin{aligned} & \displaystyle \nabla _{x^\prime }\nabla _{x^\prime }\partial _{x_n}\widetilde{I}_k(x, t)=\displaystyle \int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\nabla _{x^\prime }[G_{t-s}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \qquad \displaystyle -G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nabla _{y^\prime }\partial _{y_n}\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\nonumber \\ & \qquad \displaystyle -2\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}G_{t-s}(x^\prime -y^\prime , x_n+y_n)\nabla _{y^\prime }\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds. \end{aligned}$$

(4.67)

From (4.32), we get for any $x=(x^\prime ,x_n)\in \mathbb {R}^n_+$ and $t>0$

$$\begin{aligned} \displaystyle \partial _{x_n}^3\widetilde{I}_k(x, t)= & \displaystyle -\partial _{x_n}\big (P(u\cdot \nabla u)(x,t)\big )_k +\int _{\mathbb {R}^n_+}\partial _{x_n}[G_\frac{t}{2}(x^\prime -y^\prime , x_n-y_n)\nonumber \\ & \displaystyle -G_\frac{t}{2}(x^\prime -y^\prime , x_n+y_n)]\left( P(u\cdot \nabla ) u\left( y,\frac{t}{2}\right) \right) _kdy\nonumber \\ & \displaystyle +\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)-G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nonumber \\ & \displaystyle \times \partial _s\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds\nonumber \\ & \displaystyle -\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\partial _{x_n}[G_{t-s}(x^\prime -y^\prime , x_n-y_n)-G_{t-s}(x^\prime -y^\prime , x_n+y_n)]\nonumber \\ & \displaystyle \times \sum \limits _{j=1}^{n-1}\partial ^2_{y_j}\big (P(u\cdot \nabla ) u(y,s)\big )_kdyds. \end{aligned}$$

(4.68)

In addition, applying Lemmas 3.1, 3.2 to the strong solution u of problem (1.1), we find for any $1\le \ell , m\le n-1$ and $0\le \gamma <1$, $s>1$

$$\begin{aligned} & \displaystyle \Vert y_n^\gamma \partial _sP(u\cdot \nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}+\Vert y_n^\gamma \partial _{y_\ell }\partial _{y_m}P(u\cdot \nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \partial _{y_\ell }\partial _{y_n}P(u\cdot \nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle \Vert y_n^\gamma \sum \limits _{i, j=1}^n\partial _s\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \sum \limits _{i, j=1}^n\partial _{y_\ell }\partial _{y_m}\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \sum \limits _{i, j=1}^n\partial _{y_\ell }\partial _{y_n}\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \partial _s(u\cdot \nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert y_n^\gamma \partial _{y_\ell }\partial _{y_m}(u\cdot \nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \partial _{y_\ell }\partial _{y_n}(u\cdot \nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle \Vert y_n^\gamma \sum \limits _{i, j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_i\partial _su_j+u_j\partial _su_i)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \sum \limits _{i,j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_i\partial _{y_\ell }\partial _{y_m}u_j+u_j\partial _{y_\ell }\partial _{y_m}u_i\nonumber \\ & \qquad \displaystyle +\partial _{y_\ell }u_i\partial _{y_m}u_j +\partial _{y_\ell }u_j\partial _{y_m}u_i)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \sum \limits _{i,j=1}^n\partial _{y_n}\nabla \mathcal {N}\partial _i\partial _j(u_i\partial _{y_\ell }u_j+u_j\partial _{y_\ell }u_i)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma (\partial _su\cdot \nabla ) u+(u\cdot \partial _s\nabla ) u(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma (\partial _{y_\ell }\partial _{y_m}(u\cdot \nabla ) u+(u\cdot \partial _{y_\ell }\partial _{y_m}\nabla ) u\nonumber \\ & \qquad \displaystyle +(\partial _{y_\ell }u\cdot \partial _{y_m}\nabla ) u +(\partial _{y_m}u\cdot \partial _{y_\ell }\nabla ) u)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma ((\partial _{y_\ell }\partial _{y_n}u\cdot \nabla ) u+(u\cdot \partial _{y_\ell }\partial _{y_n}\nabla ) u\nonumber \\ & \qquad \displaystyle +(\partial _{y_\ell }u\cdot \partial _{y_n}\nabla ) u +(\partial _{y_n}u\cdot \partial _{y_\ell }\nabla ) u)(\cdot , s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\big (\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _su(s)\Vert _{L^2(\mathbb {R}^n_+)} +\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s\nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _su(s)\Vert _{L^2(\mathbb {R}^n_+)} +\Vert y_n^\gamma \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s\nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^3 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert y_n^\gamma \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^3 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert y_n^\gamma \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert y_n^\gamma \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \partial _s\nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}+\Vert y_n^\gamma \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle +\Vert y_n^\gamma u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^3 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\big )\nonumber \\ & \quad \le \displaystyle Cs^{-\frac{3}{2}-\frac{n}{2}+\frac{\gamma }{2}}. \end{aligned}$$

(4.69)

From (4.66)–(4.69), using Lemmas 3.1, 3.2, we obtain for $1\le k\le n$, $0<\beta <1$ and $t>1$

$$\begin{aligned} & \displaystyle \Vert x_n^\beta \partial _{x_n}^3\widetilde{I}_k(x,t)\Vert _{L^1_x(\mathbb {R}^n_+)} +\Vert x_n^\beta \nabla _{x^\prime }\partial _{x_n}^2\widetilde{I}_k(x,t)\Vert _{L^1_x(\mathbb {R}^n_+)} +\Vert x_n^\beta \nabla _{x^\prime }^2\partial _{x_n}\widetilde{I}_k(x,t)\Vert _{L^1_x(\mathbb {R}^n_+)}\\ & \quad \le \displaystyle \Vert x_n^\beta \nabla \big (P(u\cdot \nabla ) u(\cdot ,t)\big )_k\Vert _{L^1(\mathbb {R}^n_+)} +\int _{\mathbb {R}^n_+}\big (\Vert |x_n-y_n|^\beta \nabla _x G_\frac{t}{2}(x^\prime -y^\prime , x_n-y_n)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert (x_n+y_n)^\beta \nabla _x G_\frac{t}{2}(x^\prime -y^\prime , x_n+y_n)\Vert _{L^1(\mathbb {R}^n_+)}\big )|\big (P(u\cdot \nabla ) u\left( y,\frac{t}{2}\right) \big )_k|dy\\ & \qquad \displaystyle +\int _{\mathbb {R}^n_+}\Vert \nabla _x G_\frac{t}{2}(x^\prime -y^\prime , x_n-y_n)\Vert _{L^1(\mathbb {R}^n_+)}y_n^\beta \left| \left( P(u\cdot \nabla ) u\left( y,\frac{t}{2}\right) \right) _k\right| dy\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\big (\Vert |x_n-y_n|^\beta \nabla _x G_{t-s}(x^\prime -y^\prime , x_n-y_n)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert (x_n+y_n)^\beta \nabla _x G_{t-s}(x^\prime -y^\prime , x_n+y_n)\Vert _{L^1(\mathbb {R}^n_+)}\big )\\ & \qquad \displaystyle \times \big (|\partial _s\big (P(u\cdot \nabla ) u(y,s)\big )_k|+|(\Delta _{y^\prime }+\nabla _{y^\prime }\partial _{y_n})\big (P(u\cdot \nabla ) u(y,s)\big )_k|\big )dyds\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert \nabla _x G_{t-s}(x^\prime -y^\prime , x_n-y_n)\Vert _{L^1(\mathbb {R}^n_+)}\\ \end{aligned}$$

$$\begin{aligned} & \qquad \displaystyle \times \big (y_n^\beta |\partial _s\big (P(u\cdot \nabla ) u(y,s)\big )_k|+y_n^\beta |(\Delta _{y^\prime }+\nabla _{y^\prime }\partial _{y_n}+\nabla ^2_{y^\prime })\big (P(u\cdot \nabla ) u(y,s)\big )_k|\big )dyds\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\Vert (x_n+y_n)^\beta \nabla _x\partial _{x_n} G_{t-s}(x^\prime -y^\prime , x_n+y_n)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle \times |\nabla _{y^\prime }\big (P(u\cdot \nabla ) u(y,s)\big )_k|dyds\\ & \quad \le \displaystyle \Vert x_n^\beta \nabla \big (u\cdot \nabla ) u(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)} +\Vert x_n^\beta \nabla \sum \limits _{i, j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot ,t)\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle + Ct^{-\frac{1}{2}+\frac{\beta }{2}}\big (\Vert \big (u\cdot \nabla ) u_k\left( \cdot ,\frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)} +\Vert \sum \limits _{i, j=1}^n\partial _k\mathcal {N}\partial _i\partial _j(u_iu_j)\left( \cdot ,\frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}\big )\\ & \qquad \displaystyle + Ct^{-\frac{1}{2}}\big (\Vert y_n^\beta \big (u\cdot \nabla ) u_k\left( \cdot ,\frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)} +\Vert y_n^\beta \sum \limits _{i, j=1}^n\partial _k\mathcal {N}\partial _i\partial _j(u_iu_j)\left( \cdot ,\frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}\big )\\ \end{aligned}$$

$$\begin{aligned} & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}+\frac{\beta }{2}} s^{-\frac{3}{2}-\frac{n}{2}}ds+C\int _\frac{t}{2}^t(t-s)^{-\frac{1}{2}}s^{-\frac{3}{2}-\frac{n}{2}+\frac{\beta }{2}}ds\\ & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\big (\Vert \nabla \big ((u\cdot \nabla ) u(\cdot ,s)\big )\Vert _{L^1(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert \nabla \sum \limits _{i, j=1}^n\nabla \mathcal {N}\partial _i\partial _j(u_iu_j)(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\big )\\ & \quad \le \displaystyle Ct^{-1-\frac{n}{2}+\frac{\beta }{2}}+\Vert x_n^\beta u(t)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(t)\Vert _{L^2(\mathbb {R}^n_+)} + \Vert x_n^\beta \nabla u(t)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla u(t)\Vert _{L^2(\mathbb {R}^n_+)} \\ & \qquad \displaystyle +\Vert u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2 u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert y_n^\beta u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}+\Vert y_n^\beta \nabla u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}\Vert \nabla ^2u(t)\Vert _{L^{2}(\mathbb {R}^n_+)}\\ & \qquad \displaystyle + Ct^{-\frac{1}{2}+\frac{\beta }{2}}\left( \left\| u\left( \frac{t}{2}\right) \right\| ^2_{L^{2}(\mathbb {R}^n_+)}+\Vert \nabla u\left( \frac{t}{2}\right) \Vert ^2_{L^{2}(\mathbb {R}^n_+)}\right) \\ \end{aligned}$$

$$\begin{aligned} & \qquad \displaystyle + Ct^{-\frac{1}{2}}\left( \left\| y_n^\beta u\left( \frac{t}{2}\right) \right\| _{L^2(\mathbb {R}^n_+)}\left\| \nabla u\left( \frac{t}{2}\right) \right\| _{L^2(\mathbb {R}^n_+)} +\left\| u\left( \frac{t}{2}\right) \right\| ^2_{L^2(\mathbb {R}^n_+)}+\left\| \nabla u\left( \frac{t}{2}\right) \right\| ^2_{L^2(\mathbb {R}^n_+)}\right. \\ & \qquad \left. \displaystyle +\left\| u\left( \frac{t}{2}\right) \right\| _{L^2(\mathbb {R}^n_+)}\left\| y_n^\beta u\left( \frac{t}{2}\right) \right\| _{L^2(\mathbb {R}^n_+)} +\left\| \nabla u\left( \frac{t}{2}\right) \right\| _{L^2(\mathbb {R}^n_+)}\left\| y_n^\beta \nabla u\left( \frac{t}{2}\right) \right\| _{L^2(\mathbb {R}^n_+)}\right) \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\big ( \Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2u(s)\Vert _{L^2(\mathbb {R}^n_+)} +\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}\\ & \qquad \displaystyle +\Vert u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)} +\Vert \nabla u(s)\Vert _{L^2(\mathbb {R}^n_+)}\Vert \nabla ^2 u(s)\Vert _{L^2(\mathbb {R}^n_+)}\big )ds\\ & \quad \le \displaystyle Ct^{-1-\frac{n}{2}+\frac{\beta }{2}} +C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}s^{-\frac{3}{2}-\frac{n}{2}}ds\\ & \quad \le \displaystyle \widetilde{C}t^{-1-\frac{n}{2}+\frac{\beta }{2}}, \end{aligned}$$

which implies for $0<\beta <1$ and $t>1$

$$\begin{aligned} \Vert x_n^\beta \nabla ^3_x\widetilde{I}_k(x,t)\Vert _{L^1(\mathbb {R}^n_+)}\le Ct^{-1-\frac{n}{2}+\frac{\beta }{2}},\;\;\;k=1,2\cdots ,n. \end{aligned}$$

(4.70)

Let $0<\beta <1$, from (4.45)–(4.47), (4.51) and (4.52), we get for any $t>0$

$$\begin{aligned} & \displaystyle \Big \Vert x_n^\beta \frac{\partial ^3}{\partial x_n^3}\widetilde{J}_n(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j,\ell =1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}N_{\ell \ell }(x,y,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_j}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_j\partial x_j}J_n(x^\prime -y^\prime , x_n+y_n, 0,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \big |\frac{\partial }{\partial y_k}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}G_{t-s}(x-y^*)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_j}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_n\partial x_n}G_{t-s}(x-y^*)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_k}b_k(y,s)\big |dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds; \end{aligned}$$

(4.71)

and there holds for $1\le m, q\le n-1$,

$$\begin{aligned} & \displaystyle \Big \Vert x_n^\beta \frac{\partial ^3}{\partial x_m\partial x_n^2}\widetilde{J}_n(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}N_{jn}(x,y,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_n}G_{t-s}(x-y^*)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds; \end{aligned}$$

(4.72)

and

$$\begin{aligned} & \displaystyle \Big \Vert x_n^\beta \frac{\partial ^3}{\partial x_m\partial x_q\partial x_n}\widetilde{J}_n(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k,j=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}N_{jq}(x,y,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_q}G_{t-s}(x-y^*)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds. \end{aligned}$$

(4.73)

Combining (4.71), (4.72) and (4.73) yields for $0<\beta <1$ and $t>0$

$$\begin{aligned} \Vert x_n^\beta \nabla ^3\widetilde{J}_n(x, t)\Vert _{L^1(\mathbb {R}^n_+)}\le C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds. \end{aligned}$$

(4.74)

Let $1\le j, m, q\le n-1$. Using (4.48)–(4.53) yields $0<\beta <1$ and $t>0$

$$\begin{aligned} & \displaystyle \Big \Vert x_n^\beta \frac{\partial ^3}{\partial x_n^3}\widetilde{J}_m(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+} \sum \limits _{k,\ell =1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_\ell \partial x_k}N_{\ell n}(x,y,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_m \partial x_n}J_n(x^\prime -y^\prime , y_n,x_n,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \big |\frac{\partial }{\partial y_k}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_n\partial x_m}G_{t-s}(x-y^*)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_k}b_k(y,s)\big |dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds; \end{aligned}$$

(4.75)

$$\begin{aligned} & \displaystyle \Big \Vert x_n^\beta \frac{\partial ^3}{\partial x_j\partial x_n^2}\widetilde{J}_m(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+} \sum \limits _{k,\ell =1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}N_{\ell \ell }(x,y,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k \partial x_j}J_n(x^\prime -y^\prime , y_n,x_n,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \qquad \displaystyle \times \big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \qquad \displaystyle +C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}G_{t-s}(x-y^*)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_m}b_k(y,s)\big |dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds; \end{aligned}$$

(4.76)

and

$$\begin{aligned} & \displaystyle \Big \Vert x_n^\beta \frac{\partial ^3}{\partial x_j\partial x_q\partial x_n}\widetilde{J}_m(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t\int _{\mathbb {R}^n_+}\sum \limits _{k=1}^{n-1}\Big \Vert (x_n+y_n)^\beta \frac{\partial ^2}{\partial x_k\partial x_j}N_{mn}(x,y,t-s)\Big \Vert _{L^1_x(\mathbb {R}^n_+)}\big |\frac{\partial }{\partial y_q}b_k(y,s)\big |dyds\nonumber \\ & \quad \le \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds. \end{aligned}$$

(4.77)

From (4.74)–(4.77), we derive for $0<\beta <1$ and $t>0$

$$\begin{aligned} \Vert x_n^\beta \nabla ^3\widetilde{J}_m(x, t)\Big \Vert _{L^1(\mathbb {R}^n_+)}\le C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}\Vert \nabla _{y^\prime }b(y,s)\Vert _{L^1(\mathbb {R}^n_+)}ds,\;\;\;m=1,2,\cdots ,n.\nonumber \\ \end{aligned}$$

(4.78)

Recall the definition of $b(y,s)=(b_1(y,s), b_2(y,s),\cdots ,b_n(y,s))$:

$$\begin{aligned} b_j(y,s)=(u\cdot \nabla ) u_j(y,s)+\sum \limits _{i,\ell =1}^n\partial _{y_j}\mathcal {N}\partial _i\partial _\ell (u_iu_\ell )(y,s),\;\;s>0,\;\;1\le j\le n. \end{aligned}$$

Whence using Lemma 3.1 yields for $s>0$,

$$\begin{aligned} \displaystyle \Vert \nabla ^\prime b(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle \Vert \nabla ^\prime ((u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \displaystyle +\Vert \nabla ^\prime \left( \sum \limits _{i,\ell =1}^n\nabla \mathcal {N}\partial _i\partial _\ell (u_iu_\ell )(\cdot ,s)\right) \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle \Vert (\nabla ^\prime u\cdot \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}+\Vert (u\cdot \nabla ^\prime \nabla ) u(\cdot ,s))\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \displaystyle +\Vert \sum \limits _{i,\ell =1}^n\nabla \mathcal {N}\partial _i\partial _\ell \nabla ^\prime (u_iu_\ell )(\cdot ,s)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle C(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}). \end{aligned}$$

(4.79)

Inserting (4.79) into (4.78), using Lemma 3.2, we find for $0<\beta <1$ and $t>1$

$$\begin{aligned} \displaystyle \Vert x_n^\beta \nabla ^3\widetilde{J}_m(x, t)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}(\Vert u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)}+\Vert \nabla ^2 u(s)\Vert ^2_{L^2(\mathbb {R}^n_+)})ds\nonumber \\\le & \displaystyle C\int _\frac{t}{2}^t(t-s)^{-1+\frac{\beta }{2}}s^{-1-\frac{n}{2}}ds\nonumber \\\le & \displaystyle Ct^{-1-\frac{n}{2}+\frac{\beta }{2}},\;\;\;m=1,2,\cdots ,n. \end{aligned}$$

(4.80)

Combining (4.30), (4.70) and (4.80), we conclude for $0<\beta <1$ and $t>1$

$$\begin{aligned} \displaystyle \Vert x_n^\beta \nabla ^3\widetilde{w}_m(x, t)\Vert _{L^1(\mathbb {R}^n_+)}\le & \displaystyle \Vert x_n^\beta \nabla ^3\widetilde{I}_m(x, t)\Vert _{L^1(\mathbb {R}^n_+)}+\Vert x_n^\beta \nabla ^3\widetilde{J}_m(x, t)\Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\\le & \displaystyle Ct^{-1-\frac{n}{2}+\frac{\beta }{2}},\;\;\;m=1,2,\cdots ,n. \end{aligned}$$

(4.81)

Observe that for $t>0$, $u(y,\frac{t}{2})|_{\partial \mathbb {R}^n_+}=0$. So for $t>0$,

$$\begin{aligned} \displaystyle \int _{\mathbb {R}^n_+}\partial _{x_j}G_t(x-y)u\left( y,\frac{t}{2}\right) dy= & \displaystyle \int _{\mathbb {R}^n_+}(-\partial _{y_j})G_t(x-y)u\left( y,\frac{t}{2}\right) dy\\= & \displaystyle \int _{\mathbb {R}^n_+}G_t(x-y)\partial _{y_j}u\left( y,\frac{t}{2}\right) dy,\;\;\;1\le j\le n. \end{aligned}$$

Combining the estimate (2.3), Theorem 1.3 and Lemma 3.2, we derive for $0<\beta <1$ and $t>1$

$$\begin{aligned} & \displaystyle \left\| \int _{\mathbb {R}^n_+}x_n^\beta \nabla ^3_x\mathcal {M}\left( x,y,\frac{t}{2}\right) u\left( y,\frac{t}{2}\right) dy\right\| _{L^1_x(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle \int _{\mathbb {R}^n_+} \big (|x_n-y_n|^\beta \nabla ^2_xG_t(x-y)\Vert _{L^1_x(\mathbb {R}^n_+)}|\nabla u\left( y,\frac{t}{2}\right) |dy\nonumber \\ & \qquad +\displaystyle \int _{\mathbb {R}^n_+}\Vert \nabla ^2_xG_t(x-y)\Vert _{L^1_x(\mathbb {R}^n_+)}|y_n^\beta \nabla u\left( y,\frac{t}{2}\right) |dy\nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^n_+}\Vert (x_n+y_n)^{\beta +\frac{1}{2}}\nabla ^3_xG_t(x-y^*)\Vert _{L^1_x(\mathbb {R}^n_+)}|y_n^{-\frac{1}{2}}u\left( y,\frac{t}{2}\right) |dy\nonumber \\ & \qquad \displaystyle +\int _{\mathbb {R}^n_+} \Vert (x_n+y_n)^{\beta +\frac{1}{2}}\nabla ^3_x\mathcal {M}^*\left( x,y,\frac{t}{2}\right) \Vert _{L^1_x(\mathbb {R}^n_+)}|y_n^{-\frac{1}{2}}u\left( y,\frac{t}{2}\right) |dy\nonumber \\ & \quad \le \displaystyle Ct^{-1+\frac{\beta }{2}}\Vert \nabla u\left( \frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}+Ct^{-1}\Vert y_n^\beta \nabla u\left( \frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}+ Ct^{-1+\frac{\beta }{2}}\Vert y_n^{-\frac{1}{2}}u\left( \frac{t}{2}\right) \Vert _{L^1(\mathbb {R}^n_+)}\nonumber \\ & \quad \le \displaystyle \widetilde{C}t^{-\frac{3}{2}+\frac{\beta }{2}}. \end{aligned}$$

(4.82)

Let $0<\beta <1$. Combining (4.29), (4.81) and (4.82) yields for $t>1$

$$\begin{aligned} \Vert x_n^\beta \nabla ^3u(t)\Vert _{L^1(\mathbb {R}^n_+)}\le C\left( t^{-\frac{3}{2}+\frac{\beta }{2}}+t^{-1-\frac{n}{2}+\frac{\beta }{2}}\right) \le \widetilde{C}t^{-\frac{3}{2}+\frac{\beta }{2}}, \end{aligned}$$

which is (1.3). $\square$

Appendix

This section devotes to finding a counterexample for the Stokes flow $e^{-tA}u_0$ with initial datum $u_0\in L^1_\sigma (\mathbb {R}^n_+)$. That is,

Proposition A

There exists an initial vector function $u_0=(u_{01}, u_{02}, \cdots , u_{0n})\in L^1(\mathbb {R}^n_+)\,\,(n\ge 2)$, which satisfies $\nabla \cdot u_0=0$ in $\mathbb {R}^n_+$ and $u_{0n}\,|_{\partial \mathbb {R}^n_+}=0$, such that $e^{-tA}u_0\not \in L^1(\mathbb {R}^n_+)$ for each $t>0$.

We first give some notations and introduce some known results.

Let $\mathcal {F}_m$ ($\mathcal {F}^{-1}_m$) be the (inverse) Fourier transform in $\mathbb {R}^m$ given by

$$\begin{aligned} \mathcal {F}_m[f](\xi )=\int _{\mathbb {R}^m}e^{-i\xi \cdot x}f(x)dx \;\;\Big (\mathcal {F}^{-1}_m[f](\xi )=\frac{1}{(2\pi )^m}\int _{\mathbb {R}^m}e^{i\xi \cdot x}f(x)dx\Big ). \end{aligned}$$

A direct calculation shows that for the function $f(x)=e^{-a|x|^2}$ in $\mathbb {R}^m$ with $a>0$,

$$\begin{aligned} \mathcal {F}_m[f](y)=\left( \frac{\pi }{a}\right) ^\frac{m}{2}e^{-\frac{|y|^2}{4a}}. \end{aligned}$$

The Riesz operators $S_j\,\,(j=1,2,\cdots ,n-1)$ are defined by

$$\begin{aligned} \mathcal {F}_{n-1}[S_jf](\xi ^\prime )=\frac{i\xi _j}{|\xi ^\prime |}\mathcal {F}_{n-1}[f](\xi ^\prime ), \end{aligned}$$

where $\xi ^\prime =(\xi _1, \xi _2, \cdots , \xi _{n-1})\in \mathbb {R}^{n-1}$.

In the following arguments, for simplicity, we denote the Fourier transform $\mathcal {F}_{n-1}[g]$ in $\mathbb {R}^{n-1}$ by $\widehat{g}$. Recall $G_t(x)=(4\pi t)^{-\frac{n}{2}}e^{-\frac{|x|^2}{4t}}$, $x=(x^\prime , x_n)\in \mathbb {R}^n$ satisfies $G_t(x)=G_t^{(n-1)}(x^\prime )G_t^{(1)}(x_n)$, where

$$\begin{aligned} G_t^{(n-1)}(x^\prime )=(4\pi t)^{-\frac{n-1}{2}}e^{-\frac{|x^\prime |^2}{4t}}, \;\; G_t^{(1)}(x_n)=(4\pi t)^{-\frac{1}{2}}e^{-\frac{|x_n|^2}{4t}}. \end{aligned}$$

Define the operator E(t) by

$$\begin{aligned} \displaystyle (E(t)g)(x)= & \displaystyle \int _{\mathbb {R}^n_+}[G_t(x^\prime -y^\prime , x_n-y_n)-G_t(x^\prime -y^\prime , x_n+y_n)]g(y)dy\\= & \displaystyle \int _{\mathbb {R}^n_+}G_t^{(n-1)}(x^\prime -y^\prime )[G_t^{(1)}(x_n-y_n)-G_t^{(1)}(x_n+y_n)]g(y)dy. \end{aligned}$$

By the solution formula in [25], the Stokes flow $e^{-tA}u_0=(u^\prime , u_n)$ can be represented as

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle u_n=\displaystyle UE(t)V_1u_0,\\ \displaystyle u^\prime =\displaystyle E(t)V_2u_0-SUE(t)V_1u_0, \end{array} \right. \end{aligned}$$

(A.1)

where

$$\begin{aligned} V_1u_0= & -S\cdot u_0^\prime +u_{0n},\;\;\;\;V_2 u_0=u^\prime _0+S u_{0n},\;\;\;\;S=(S_1, S_2, \cdots ,S_{n-1}),\\ \widehat{Ug}(\xi ^\prime , x_n)= & |\xi ^\prime |\int _0^{x_n}e^{-|\xi ^\prime |(x_n-y_n)}\widehat{g}(\xi ^\prime , y_n)dy_n. \end{aligned}$$

Proof of Proposition A

Set

$$\begin{aligned} f=E(t)V_1u_0=E(t)[u_{0n}-S\cdot u_0^\prime ],\;\;\;\; h(\xi ^\prime ,s)= \left\{ \begin{array}{lll} \displaystyle |\xi ^\prime |e^{-|\xi ^\prime |s}& \text{ if } & s\ge 0,\\ \displaystyle 0& \text{ if } & s<0. \end{array} \right. \end{aligned}$$

Then $u_n=Uf$ and for $t>0$

$$\begin{aligned} \displaystyle \widehat{u_n}(\xi ^\prime , x_n, t)= & \displaystyle \widehat{Uf}(\xi ^\prime , x_n) =|\xi ^\prime |\int _0^{x_n}e^{-|\xi ^\prime |(x_n-y_n)}\widehat{f}(\xi ^\prime , y_n)dy_n\\= & \displaystyle \int _0^\infty h(\xi ^\prime ,x_n-y_n)\widehat{f}(\xi ^\prime , y_n)dy_n. \end{aligned}$$

Whence using the definition of h yields for $t>0$

$$\begin{aligned} \displaystyle \int _0^\infty \widehat{u_n}(\xi ^\prime , x_n, t)dx_n= & \displaystyle \int _0^\infty \left( \int _0^\infty h(\xi ^\prime ,x_n-y_n)dx_n\right) \widehat{f}(\xi ^\prime , y_n)dy_n\nonumber \\= & \displaystyle \int _0^\infty \left( \int _{-y_n}^\infty h(\xi ^\prime ,s)ds\right) \widehat{f}(\xi ^\prime , y_n)dy_n\nonumber \\= & \displaystyle \int _0^\infty \left( \int _0^\infty h(\xi ^\prime ,s)ds\right) \widehat{f}(\xi ^\prime , y_n)dy_n\nonumber \\= & \displaystyle \int _0^\infty \left( \int _0^\infty |\xi ^\prime |e^{-|\xi ^\prime |s}ds\right) \widehat{f}(\xi ^\prime , y_n)dy_n\nonumber \\= & \displaystyle \int _0^\infty \widehat{f}(\xi ^\prime , y_n)dy_n. \end{aligned}$$

(A.2)

Note that

$$\begin{aligned} \displaystyle \widehat{f}(\xi ^\prime , y_n)= & \widehat{E(t)[u_{0n}-S\cdot u_0^\prime ]}(\xi ^\prime , y_n)\nonumber \\= & \displaystyle \widehat{G_t^{(n-1)}}(\xi ^\prime )\int _0^\infty [G_t^{(1)}(y_n-z_n)-G_t^{(1)}(y_n+z_n)]\nonumber \\ & \displaystyle \times [\widehat{u_{0n}}(\xi ^\prime , z_n)-\frac{i\xi ^\prime }{|\xi ^\prime |}\cdot \widehat{u_{0}^\prime }(\xi ^\prime , z_n)]dz_n. \end{aligned}$$

(A.3)

Inserting (A.3) into (A.2) yields for $t>0$

$$\begin{aligned} \displaystyle \int _0^\infty \widehat{u_n}(\xi ^\prime , x_n, t)dx_n= & \displaystyle \widehat{G_t^{(n-1)}}(\xi ^\prime )\int _0^\infty \left( \int _0^\infty [G_t^{(1)}(y_n-z_n)-G_t^{(1)}(y_n+z_n)]dy_n\right) \nonumber \\ & \displaystyle \times [\widehat{u_{0n}}(\xi ^\prime , z_n)-\frac{i\xi ^\prime }{|\xi ^\prime |}\cdot \widehat{u_{0}^\prime }(\xi ^\prime , z_n)]dz_n\nonumber \\= & \displaystyle \widehat{G_t^{(n-1)}}(\xi ^\prime )\int _0^\infty \left( \int _{-z_n}^\infty G_t^{(1)}(s)ds-\int _{z_n}^\infty G_t^{(1)}(s)ds\right) \nonumber \\ & \displaystyle \times [\widehat{u_{0n}}(\xi ^\prime , z_n)-\frac{i\xi ^\prime }{|\xi ^\prime |}\cdot \widehat{u_{0}^\prime }(\xi ^\prime , z_n)]dz_n\nonumber \\= & \displaystyle 2\widehat{G_t^{(n-1)}}(\xi ^\prime )\int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) \nonumber \\ & \displaystyle \times [\widehat{u_{0n}}(\xi ^\prime , z_n)-\frac{i\xi ^\prime }{|\xi ^\prime |}\cdot \widehat{u_{0}^\prime }(\xi ^\prime , z_n)]dz_n\nonumber \\= & \displaystyle 2\widehat{G_t^{(n-1)}}(\xi ^\prime )\int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) \widehat{u_{0n}}(\xi ^\prime , z_n)dz_n\nonumber \\ & \displaystyle -2\widehat{SG_t^{(n-1)}}(\xi ^\prime )\cdot \int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) \widehat{u_{0}^\prime }(\xi ^\prime , z_n)dz_n\nonumber \\= & \displaystyle I_{1t}(\xi ^\prime )+I_{2t}(\xi ^\prime ). \end{aligned}$$

(A.4)

Set

$$\begin{aligned} u_{01}(x^\prime , x_n)= & G_1^{(n-1)}(x^\prime )(e^{-x_n}-2e^{-2x_n}),\\ u_{0n}(x^\prime , x_n)= & -\partial _{x_1}G_1^{(n-1)}(x^\prime )\int _0^{x_n}(e^{-s}-2e^{-2s})ds. \end{aligned}$$

Then $u_0=(u_{01},0,\cdots ,0,u_{0n})$ satisfies $\Vert u_0\Vert _{L^1(\mathbb {R}^n_+)}<\infty$ and

$$\begin{aligned} \nabla \cdot u_0=\partial _1u_{01}+\partial _nu_{0n}=0, \;\;\;\;u_{0n}(x^\prime , 0)=0,\;\;\forall \;x^\prime \in \mathbb {R}^{n-1}. \end{aligned}$$

For the given $u_0$, from (A.4), we have for $t>0$

$$\begin{aligned} & \displaystyle \Vert \mathcal {F}^{-1}_{n-1}[I_{1t}]\Vert _{L^1(\mathbb {R}^{n-1})}\nonumber \\ & \quad =\displaystyle 2\left\| \int _{\mathbb {R}^{n-1}}G_t^{(n-1)}(x^\prime -y^\prime )\int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) u_{0n}(y^\prime , z_n)dz_ndy^\prime \right\| _{L^1(\mathbb {R}^{n-1})}\nonumber \\ & \quad \le \displaystyle 2\Vert G_t^{(n-1)}\Vert _{L^1(\mathbb {R}^{n-1})}\int _{\mathbb {R}^{n-1}}\int _0^\infty \left( \int _0^\infty G_t^{(1)}(s)ds\right) |u_{0n}(y^\prime , z_n)|dz_ndy^\prime \nonumber \\ & \quad \le \displaystyle 2\Vert u_{0n}\Vert _{L^1(\mathbb {R}^n_+)}. \end{aligned}$$

(A.5)

In addition,

$$\begin{aligned} \displaystyle I_{2t}(\xi ^\prime )= & \displaystyle -2\widehat{S_1G_t^{(n-1)}}(\xi ^\prime ) \int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) \widehat{u_{01}}(\xi ^\prime , z_n)dz_n\\= & \displaystyle -2\widehat{S_1G_t^{(n-1)}}(\xi ^\prime )\widehat{G_1^{(n-1)}}(\xi ^\prime ) \int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) (e^{-z_n}-2e^{-2z_n})dz_n\\= & \displaystyle 2\widehat{S_1G_{t+1}^{(n-1)}}(\xi ^\prime ) \int _0^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) (2e^{-2z_n}-e^{-z_n})dz_n, \end{aligned}$$

from which, we get for $t>0$

$$\begin{aligned} \displaystyle |\mathcal {F}^{-1}_{n-1}[I_{2t}](x^\prime )|= & \displaystyle 2|[S_1G_{t+1}^{(n-1)}](x^\prime )| \left( \int _0^{\log _e2}\Big (\int _0^{z_n}G_t^{(1)}(s)ds\Big )(2e^{-2z_n}-e^{-z_n})dz_n\right. \nonumber \\ & \displaystyle \left. +\int _{\log _e2}^\infty \Big (\int _0^{z_n}G_t^{(1)}(s)ds\Big )(e^{-z_n}-2e^{-2z_n})dz_n\right) \nonumber \\\ge & \displaystyle 2|[S_1G_{t+1}^{(n-1)}](x^\prime )|\int _{\log _e2}^\infty \left( \int _0^{z_n}G_t^{(1)}(s)ds\right) (e^{-z_n}-2e^{-2z_n})dz_n\nonumber \\\ge & \displaystyle 2|[S_1G_{t+1}^{(n-1)}](x^\prime )|\int _0^{\log _e2}G_t^{(1)}(s)ds\int _{\log _e2}^\infty (e^{-z_n}-2e^{-2z_n})dz_n\nonumber \\\ge & \displaystyle C(t)|[S_1G_{t+1}^{(n-1)}](x^\prime )|, \end{aligned}$$

(A.6)

where

$$\begin{aligned} C(t)=\int _0^{\log _e2}G_t^{(1)}(s)ds\int _{\log _e2}^\infty (e^{-z_n}-2e^{-2z_n})dz_n=\frac{1}{4}\int _0^{\log _e2}G_t^{(1)}(s)ds. \end{aligned}$$

Observe that for $\xi ^\prime =(\xi _1,0,\cdots ,0)$,

$$\begin{aligned} \lim \limits _{\xi _1\rightarrow 0^+}\frac{\xi _1}{|\xi ^\prime |}e^{-(t+1)|\xi ^\prime |^2}=1, \;\;\;\lim \limits _{\xi _1\rightarrow 0^-}\frac{\xi _1}{|\xi ^\prime |}e^{-(t+1)|\xi ^\prime |^2}=-1. \end{aligned}$$

This shows that for $t>0$, the following function

$$\begin{aligned} \widehat{S_1G_{t+1}^{(n-1)}}(\xi ^\prime )=\frac{i\xi _1}{|\xi ^\prime |}e^{-(t+1)|\xi ^\prime |^2} \end{aligned}$$

is not continuous at $\xi ^\prime =0$. Whence $S_1G_{t+1}^{(n-1)}\not \in L^1(\mathbb {R}^{n-1})$. Together with (A.6), we conclude for $t>0$

$$\begin{aligned} \Vert \mathcal {F}^{-1}_{n-1}[I_{2t}]\Vert _{L^1(\mathbb {R}^{n-1})}\ge C(t)\Vert S_1G_{t+1}^{(n-1)}\Vert _{L^1(\mathbb {R}^{n-1})}=+\infty . \end{aligned}$$

(A.7)

From (A.4), (A.5) and (A.7), we derive for $t>0$

$$\begin{aligned} & \displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}|u_n(x^\prime , x_n, t)|dx^\prime dx_n\\ & \quad \ge \displaystyle \int _{\mathbb {R}^{n-1}}\Big |\int _0^\infty u_n(x^\prime , x_n, t)dx_n\Big |dx^\prime \\ & \quad \ge \displaystyle \int _{\mathbb {R}^{n-1}}|\mathcal {F}^{-1}_{n-1}[I_{2t}](x^\prime )|dx^\prime -\int _{\mathbb {R}^{n-1}}|\mathcal {F}^{-1}_{n-1}[I_{1t}](x^\prime )|dx^\prime \\ & \quad \ge \displaystyle \Vert \mathcal {F}^{-1}_{n-1}[I_{2t}]\Vert _{L^1(\mathbb {R}^{n-1})}-2\Vert u_{0n}\Vert _{L^1(\mathbb {R}^n_+)}=+\infty , \end{aligned}$$

then by (A.1)

$$\begin{aligned} \displaystyle \Vert e^{-tA}u_0\Vert _{L^1(\mathbb {R}^n_+)}=\Vert u\Vert _{L^1(\mathbb {R}^n_+)}= & \displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}\big (|u^\prime (x^\prime , x_n, t)|^2+|u_n(x^\prime , x_n, t)|^2\big )^\frac{1}{2}dx^\prime dx_n\\\ge & \displaystyle \int _0^\infty \int _{\mathbb {R}^{n-1}}|u_n(x^\prime , x_n, t)|dx^\prime dx_n=+\infty . \end{aligned}$$

$\square$