Initial-boundary value problem of the Navier–Stokes equations in the half space with nonhomogeneous data

Abstract

This paper discusses the solvability (global in time) of the initial-boundary value problem of the Navier–Stokes equations in the half space when the initial data $h\in {\dot{B}}_{q \sigma }^{\alpha -\frac{2}{q}}({{\mathbb {R}}}^n_+)$ and the boundary data $g\in \dot{ B}_q^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}({{\mathbb {R}}}^{n-1}\times {\mathbb R}_+)$ with $g_n\in \dot{B}^{\frac{1}{2} \alpha }_q ({{\mathbb {R}}}_+; \dot{B}^{-\frac{1}{q}}_q ({{\mathbb {R}}}^{n-1}))\cap L^q({\mathbb R}_+;{\dot{B}}^{\alpha -\frac{1}{q}}({{\mathbb {R}}}^{n-1}))$, for any $0<\alpha <2$ and $q =\frac{n+2}{\alpha +1}$. Compatibility condition (1.3) is required for h and g.

Appendices

Appendix A: Proof of Lemma 3.5

In [32], it was determined that

$$\begin{aligned} \Vert T_1f\Vert _{{\dot{W}}^{2,1}_q({{\mathbb {R}}}^n\times {{\mathbb {R}}})}&\le c\Vert f\Vert _{L^q({{\mathbb {R}}}^n\times {{\mathbb {R}}})}. \end{aligned}$$

(A.1)

Note that $T_1^*$ is the adjoint operator of $T_1$. Hence, (A.1) implies that

$$\begin{aligned} \Vert T_1^*f\Vert _{L^p({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c\Vert f\Vert _{\dot{W}^{-2,-1}_p({{\mathbb {R}}}^n\times {{\mathbb {R}}})}. \end{aligned}$$

(A.2)

Further, note that $D^2_yT_1^*f$ and $\ D_s T_1^*f$ comprise $L^p$ Fourier multipliers as the Fourier transform of $T_1^*f$ is $\widehat{T^*_1 f}(\xi , \eta )=\frac{1}{|\xi |^2-i \eta } {\hat{f}}(\xi ,\eta )$. Hence, we have

$$\begin{aligned} \Vert T_1^*f\Vert _{\dot{W}^{2,1}_{p}({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c \Vert f\Vert _{L^p({{\mathbb {R}}}^n\times {{\mathbb {R}}})}, \quad 1<p<\infty . \end{aligned}$$

(A.3)

As $T^*_1$ is the adjoint operator of $T_1$, (A.3) implies that

$$\begin{aligned} \Vert T_1f\Vert _{L^p({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c\Vert f\Vert _{\dot{W}^{-2,-1}_p({{\mathbb {R}}}^n\times {{\mathbb {R}}})}. \end{aligned}$$

(A.4)

By applying the real interpolation theory to (A.1) and (A.4), and (A.2) and (A.3), we obtain estimates of $T_1f$ and $T_1^* f$ in $\dot{B}^{\alpha ,\frac{\alpha }{2}}_q({{\mathbb {R}}}^n\times {{\mathbb {R}}})$ for $0<\alpha <2$.

Appendix B: Proof of Lemma 3.6

First, let us derive the estimate of $T_2g$. From [32], we have the following estimate

$$\begin{aligned} \Vert T_2g\Vert _{{\dot{W}}^{2,1}_q({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}})}\le c\Vert g\Vert _{{\dot{B}}_q^{1-\frac{1}{q},\frac{1}{2}-\frac{1}{2q}}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})}. \end{aligned}$$

(B.1)

Note that the identity

$$\begin{aligned} \int _{-\infty }^\infty \int _{{{\mathbb {R}}}^n_+} T_2g(x,t) \phi (x,t) dxdt = < g, T_1^* {{\tilde{\phi }}}|_{y_n=0}> \end{aligned}$$

(B.2)

holds for $\phi \in C^\infty _0({{\mathbb {R}}}^n_+\times {{\mathbb {R}}})$, where $T^*_1 {{\tilde{\phi }}}$ is defined in Sect. 3 with zero extension ${{\tilde{\phi }}}$ of $\phi$ and $<\cdot ,\cdot>$ is the duality pairing between ${\dot{B}}^{-1-\frac{1}{q},-\frac{1}{2}-\frac{1}{2q}}_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})$ and ${\dot{B}}^{1+\frac{1}{q},\frac{1}{2}+\frac{1}{2q}}_{q'}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}).$ Based on (4) of Proposition 2.1 and (A.3), we have

$$\begin{aligned} \Vert T_1^*\phi |_{y_n=0}\Vert _{\dot{B}^{1+\frac{1}{q}, \frac{1}{2}+\frac{1}{2q} }_{q'}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})} \le c \Vert T_1^*\phi \Vert _{\dot{W}^{2,1}_{q'}({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c \Vert \phi \Vert _{L^{q'}({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}}) }. \end{aligned}$$

(B.3)

By applying the estimates in (B.2)–(B.3), we have

$$\begin{aligned} \Vert T_2g \Vert _{L^q({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}})}\le c\Vert g\Vert _{\dot{B}^{-1-\frac{1}{q}, -\frac{1}{2}-\frac{1}{2q} }_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}) }. \end{aligned}$$

(B.4)

Further, by applying the real interpolation theory to (B.1) and (B.4),

we obtain the estimate of $T_2g$ in $\dot{B}^{\alpha ,\frac{\alpha }{2}}_q({{\mathbb {R}}}^n_+\times {{\mathbb {R}}})$ for $0<\alpha <2$.

Analogously, we can derive the estimate of $T_2^*g$ by observing that the identity

$$\begin{aligned} \int _{-\infty }^\infty \int _{{{\mathbb {R}}}^n_+} T_2^*g(y,s) \phi (y,s) dyds = <g, T_1 {{\tilde{\phi }}}|_{x_n=0}> \end{aligned}$$

(B.5)

holds for $\phi \in C^\infty _0({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}})$, where $T_1 {{\tilde{\phi }}}$ is defined in Sect. 3 with zero extension ${{\tilde{\phi }}}$ of $\phi$, and $<\cdot ,\cdot>$ is the duality pairing between ${\dot{B}}^{-1-\frac{1}{q},-\frac{1}{2}-\frac{1}{2q}}_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})$ and ${\dot{B}}^{1+\frac{1}{q},\frac{1}{2}+\frac{1}{2q}}_{q'}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}).$ By using the same procedure as that used for the estimate of $T_2g$, we can obtain the estimate of $T_2^*g$ as

$$\begin{aligned} \Vert T_2^*g \Vert _{L^q({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}})}\le c\Vert g\Vert _{\dot{B}^{-1-\frac{1}{q}, -\frac{1}{2}-\frac{1}{2q} }_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}) } \end{aligned}$$

(B.6)

(As the procedure is the same as that for $T_2g$, we omitted the details). As $D_sT_2^*g=T_2^*(D_sg)$, we have

$$\begin{aligned} \Vert D_sT_2^*g \Vert _{L^q({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}})}\le c\Vert D_sg\Vert _{\dot{B}^{-1-\frac{1}{q},-\frac{1}{2} -\frac{1}{2q} }_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}) }\le c\Vert g\Vert _{\dot{B}^{1-\frac{1}{q}, \frac{1}{2}-\frac{1}{2q} }_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}) }. \end{aligned}$$

(B.7)

In addition, as $\Delta _y T_2^*g=-D_sT_2^*g$ and $\frac{\partial }{\partial y_n}T_2^*g|_{y_n=0}=g$, based on the well-known elliptic theory [2, 3], we have

$$\begin{aligned} \Vert T_2^*g(s) \Vert _{ \dot{W}^{2}_q({{\mathbb {R}}}^n_+ )}\le c\Vert D_s T_2^*g(s) \Vert _{L^q({{\mathbb {R}}}^n_+ )}+c\Vert g(s)\Vert _{\dot{B}^{1-\frac{1}{q}}_q({{\mathbb {R}}}^{n-1})}. \end{aligned}$$

This implies that

$$\begin{aligned} \Vert T_2^*g \Vert _{L^q({{\mathbb {R}}};\dot{W}^{2}_q({{\mathbb {R}}}^n_+ ))}\le c\Vert g\Vert _{ \dot{B}^{1-\frac{1}{q}, \frac{1}{2}-\frac{1}{2q} }_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}) }. \end{aligned}$$

(B.8)

By combining (B.7) and (B.8), we have

$$\begin{aligned} \Vert T_2^*g \Vert _{\dot{W}^{2,1}_q({{\mathbb {R}}}^n_+ \times {{\mathbb {R}}})}\le c\Vert g\Vert _{\dot{B}^{1-\frac{1}{q}, \frac{1}{2}-\frac{1}{2q} }_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}) }. \end{aligned}$$

(B.9)

By applying the real interpolation theory to (B.6) and (B.9), we obtain the estimate of $T_2^*g$ in $\dot{B}^{\alpha ,\frac{\alpha }{2}}_q({{\mathbb {R}}}^n_+\times {{\mathbb {R}}})$ for $0<\alpha <2$. Thus, we complete the proof of Lemma 3.6.

Appendix C: Proof of Lemma 3.7

From [32], the following estimate is known:

$$\begin{aligned} \Vert \Gamma _t*h\Vert _{{\dot{W}}^{2,1}_q({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)}\le c\Vert h\Vert _{{\dot{B}}_q^{2-\frac{2}{q}}({{\mathbb {R}}}^n)}. \end{aligned}$$

(C.1)

Let us consider the case where $h\in \dot{B}^{-\frac{2}{q}}_q({{\mathbb {R}}}^n)$. Note that the identity $\int ^\infty _{0}\int _{{{\mathbb {R}}}^n}\Gamma _t*h(x,t)\phi (x,t)dx dt=<h,T_1^*\phi |_{s=0}>$ holds for $\phi \in C_0^\infty ({{\mathbb {R}}}^n\times {{\mathbb {R}}})$, where $T_1^*\phi (y,s)=\int ^\infty _{s}\int _{{{\mathbb {R}}}^n}\Gamma (x-y,t-s)\phi (x,t)dxdt,$, and $<\cdot ,\cdot>$ is the duality pairing between ${\dot{B}}^{-\frac{2}{q}}_q({{\mathbb {R}}}^n)$ and ${\dot{B}}^{\frac{2}{q}}_{q'}({{\mathbb {R}}}^n).$ From (A.3), we have

$$\begin{aligned} \Vert T_1^*\phi \Vert _{\dot{W}^{2,1}_{q'}({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c \Vert \phi \Vert _{L^{q'}({{\mathbb {R}}}^n\times {{\mathbb {R}}}) }. \end{aligned}$$

By using (5) of Proposition 2.1, this implies that

$$\begin{aligned} \Vert T_1^*\phi |_{s=0}\Vert _{{\dot{B}}_{q'}^{2-\frac{2}{q'}}({{\mathbb {R}}}^n)}\le c\Vert T^*_1\phi \Vert _{\dot{W}^{2,1}_{q'}({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c \Vert \phi \Vert _{L^{q'}({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+) }. \end{aligned}$$

(See [41] and [43].) Hence, we have

$$\begin{aligned} <h,T_1^*\phi |_{s=0}>\le c\Vert h\Vert _{{\dot{B}}_q^{-\frac{2}{q}}({{\mathbb {R}}}^n)}\Vert T_1^*\phi \Vert _{{\dot{B}}^{2-\frac{2}{q'}}({{\mathbb {R}}}^n)}\le c\Vert h\Vert _{{\dot{B}}_q^{-\frac{2}{q}}({{\mathbb {R}}}^n)}\Vert \phi \Vert _{L^{q'}({{\mathbb {R}}}^n\times {{\mathbb {R}}}) }. \end{aligned}$$

Again, this leads to the following conclusion

$$\begin{aligned} \Vert \Gamma _t*h\Vert _{L^q({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)}\le c\Vert h\Vert _{{\dot{B}}^{-\frac{2}{q}}_q({{\mathbb {R}}}^n)}. \end{aligned}$$

(C.2)

By interpolating (C.1) and (C.2), we have

$$\begin{aligned} \Vert \Gamma _t*h\Vert _{B^{\alpha ,\frac{\alpha }{2}}_q({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)}\le c \Vert h\Vert _{\dot{B}^{\alpha -\frac{2}{q}}_q({{\mathbb {R}}}^n)}, \qquad 0< \alpha < 2. \end{aligned}$$

(C.3)

Now, we will derive the estimate of $\Gamma _t*h|_{x_n=0}$.

• (1) Let $\alpha >\frac{1}{q}$. Then by (5) of Proposition 2.1, $\Gamma _t*h\in \dot{B}^{\alpha ,\frac{\alpha }{2}}_q({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)$ implies that $\Gamma _t*h|_{x_n=0}\in \dot{B}^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)$ with

  $$\begin{aligned} \Vert \Gamma _t*h|_{x_n=0}\Vert _{\dot{B}^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_q({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)}&\le c\Vert \Gamma _t*h\Vert _{\dot{B}^{\alpha ,\frac{\alpha }{2}}_q({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)} \le c \Vert h\Vert _{\dot{B}_q^{\alpha -\frac{2}{q}}({{\mathbb {R}}}^n)}. \end{aligned}$$

• (2) Let $0<\alpha < \frac{1}{q}.$ In this case, usual trace theorem does not hold any more. For $h\in {\dot{B}}^{\alpha -\frac{2}{q}}_q({{\mathbb {R}}}^n)$ the following identity holds:

  $$\begin{aligned}<\Gamma _t*h\Big |_{x_n=0},\phi>=<h, T_2^*\phi |_{s=0}>, \end{aligned}$$

  (C.4)

  holds for any $\phi \in C^\infty _0({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})$, where $T_2^*\phi (y,s)=\int ^\infty _s\int _{{{\mathbb {R}}}^{n-1}}\Gamma (x'-y',y_n,t-s)\phi (x',t)dx'dt$ and $<\cdot ,\cdot>$ is the duality pairing between ${\dot{B}}^{\alpha -\frac{2}{q}}_q({{\mathbb {R}}}^n)$ and ${\dot{B}}^{-\alpha +\frac{2}{q}}_{q'}({{\mathbb {R}}}^n)$ . From the result of Lemma 3.6, $T_2^*\phi \in {\dot{B}}^{-\alpha +2,-\frac{\alpha }{2}+1}_{q'}({{\mathbb {R}}}^n_+\times {{\mathbb {R}}})$ with

  $$\begin{aligned} \Vert T_2^*\phi \Vert _{ {\dot{B}}^{-\alpha +2,-\frac{\alpha }{2}+1}_{q'}({{\mathbb {R}}}^n_+\times {{\mathbb {R}}})} \le c\Vert \phi \Vert _{{\dot{B}}^{-\alpha +\frac{1}{q},-\frac{\alpha }{2}+\frac{1}{2q}}_{q'}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})}. \end{aligned}$$

  By (4) of Proposition 2.1, this implies that $T_2^*\phi \Big |_{s=0}\in {\dot{B}}^{-\alpha +\frac{2}{q}}_{q'}({{\mathbb {R}}}^n_+ )$ with

  $$\begin{aligned} \Vert T_2^*\phi \Big |_{s=0}\Vert _{ {\dot{B}}^{-\alpha +\frac{2}{q}}_{q'}({{\mathbb {R}}}^n_+ )} \le c\Vert \phi \Vert _{{\dot{B}}^{-\alpha +\frac{1}{q},-\frac{\alpha }{2}+\frac{1}{2q}}_{q'}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})}. \end{aligned}$$

  Hence

  $$\begin{aligned} |<h, T_2^*\phi \Big |_{s=0}>|\le c\Vert h\Vert _{{\dot{B}}^{\alpha -\frac{2}{q}}_q({{\mathbb {R}}}^n)}\Vert \phi \Vert _{{\dot{B}}^{-\alpha +\frac{1}{q},-\frac{\alpha }{2}+\frac{1}{2q}}_{q'}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})} \end{aligned}$$

  Applying the above estimate to (C.4), $\Gamma _t*h|_{x_n=0}\in {\dot{B}}^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_{q}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})$ with

  $$\begin{aligned} \Vert \Gamma _t*h|_{x_n=0}\Vert _{{\dot{B}}^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_{q}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}})}\le c\Vert h\Vert _{{\dot{B}}^{\alpha -\frac{2}{q}}_q({{\mathbb {R}}}^n)}. \end{aligned}$$

  (C.5)

• (3) Finally let us consider the case $\alpha =\frac{1}{q}$. Using the real interpolation, we get the case of $\alpha =\frac{1}{q}$.

Appendix D: Proof of Lemma 3.8

• Let ${\tilde{f}}\in L^p({{\mathbb {R}}};{B}^{\beta }_p({{\mathbb {R}}}^n))$ be the zero extension of f to ${{\mathbb {R}}}^n\times {{\mathbb {R}}}$. Note that $D_x \Gamma * {{\tilde{f}}} = \Gamma * D_x {{\tilde{f}}}$. From (A.1), we have

  $$\begin{aligned} \Vert D_x\Gamma * {{\tilde{f}}}\Vert _{\dot{W}^{2,1}_p({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c\Vert D_x {{\tilde{f}}}\Vert _{L^p( {{\mathbb {R}}}; L^p({{\mathbb {R}}}^n))} \le c\Vert f\Vert _{L^p( {{\mathbb {R}}}_+;{\dot{W}}^1_p({{\mathbb {R}}}^n))} \end{aligned}$$

  and

  $$\begin{aligned} \Vert D_x\Gamma *{{\tilde{f}}}\Vert _{\dot{W}^{1,\frac{1}{2}}_p({{\mathbb {R}}}^n\times {{\mathbb {R}}})}\le c \Vert \Gamma *{{\tilde{f}}}\Vert _{\dot{W}^{2,1}_p({{\mathbb {R}}}^n\times {{\mathbb {R}}})} \le c\Vert f\Vert _{L^p( {{\mathbb {R}}}_+;L^p({{\mathbb {R}}}^n))}. \end{aligned}$$

  By interpolating these two estimates, we can obtain

  $$\begin{aligned} \Vert D_x\Gamma *{f}\Vert _{\dot{B}^{\beta +1,\frac{\beta +1}{2}}_p({{\mathbb {R}}}^n\times {{\mathbb {R}}})}\le c\Vert { f}\Vert _{L^p({{\mathbb {R}}}_+;{\dot{B}}^{\beta }_p({{\mathbb {R}}}^n))}, \,\, 0<\beta <1. \end{aligned}$$

  (D.1)

  Further, by applying Besov imbedding (see (3) of Proposition 2.1), for $1-\alpha +\beta -(n+2)(\frac{1}{p}-\frac{1}{q})=0$ , we have

  $$\begin{aligned} \Vert D_x\Gamma *{f}\Vert _{{\dot{B}}_{q}^{\alpha ,\frac{\alpha }{2}}({{\mathbb {R}}}^n\times {{\mathbb {R}}})}\le c \Vert { f}\Vert _{L^p({{\mathbb {R}}}_+;{\dot{B}}^\beta _p({{\mathbb {R}}}^n))}. \end{aligned}$$

  (D.2)

  Note that $D_x\Gamma *{f}(x,t)=0$ for $t\le 0$. Hence, $D_x\Gamma *{f}\in {\dot{B}}_{q(0)}^{\alpha ,\frac{\alpha }{2}}({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+).$

• Now, we derive the estimate of $D_x\Gamma *{f}\Big |_{x_n=0}$.

• (1) Let $\alpha >\frac{1}{q}$. Then, according to the usual trace theorem, $D_x\Gamma *{f}\in B^{\alpha ,\frac{\alpha }{2}}_{q(0)}({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)$ implies that $D_x\Gamma *{f}|_{x_n=0}\in B^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_{q(0)}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)$ with

  $$\begin{aligned} \Vert D_x\Gamma *{f}|_{x_n=0}\Vert _{B^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_{q(0)}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)}&\le c\Vert D_x\Gamma *{f}\Vert _{B^{\alpha ,\frac{\alpha }{2}}_{q}({{\mathbb {R}}}^n\times {{\mathbb {R}}}_+)} \nonumber \\&\le c \Vert { f}\Vert _{L^p({{\mathbb {R}}}_+ ;{\dot{B}}^\beta _p({{\mathbb {R}}}^n)))}. \end{aligned}$$

  (D.3)

• (2) Let $0<\alpha \le \frac{1}{q}$. In this case, the usual trace theorem does not hold true. If $\alpha +\frac{n+1}{p}-\frac{n+2}{q}>0$, we can choose r with $p<r<q,$$\alpha +\frac{n+1}{r}-\frac{n+2}{q}>0$. Set $\gamma =\alpha +\frac{n+2}{r}-\frac{n+2}{q}$, then $\alpha -\frac{1}{q}-\frac{n+1}{q}=\gamma -\frac{1}{r}-\frac{n+1}{r}$ and $\alpha -\frac{1}{q}<\gamma -\frac{1}{r}$. Hence, by using the Besov embedding theorem,

  $$\begin{aligned} \Vert D_x\Gamma *{f}|_{x_n=0}\Vert _{B^{\alpha -\frac{1}{q},\frac{\alpha }{2}-\frac{1}{2q}}_{q(0)}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)}\le c\Vert D_x\Gamma *{f}|_{x_n=0}\Vert _{B^{\gamma -\frac{1}{r},\frac{\gamma }{2}-\frac{1}{2r}}_{r0}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)}. \end{aligned}$$

  As $\gamma >\frac{1}{r}$, the use of the usual trace theorem gives

  $$\begin{aligned}&\Vert D_x\Gamma *{f}|_{x_n=0}\Vert _{B^{\gamma -\frac{1}{r},\frac{\gamma }{2}-\frac{1}{2r}}_{r0}({{\mathbb {R}}}^{n-1}\times {{\mathbb {R}}}_+)}\\&\quad \le c\Vert D_x\Gamma *{f}\Vert _{B^{\gamma ,\frac{\gamma }{2}}_{r0}({{\mathbb {R}}}^{n-1}\times ({{\mathbb {R}}}_+ ))} \le c \Vert { f}\Vert _{L^p({{\mathbb {R}}}_+;{\dot{B}}^\beta _p({{\mathbb {R}}}^n))}. \end{aligned}$$

  Hence, the proof of Lemma 3.8 is completed.