The Navier–Stokes–Vlasov–Fokker–Planck System in Bounded Domains

Abstract

This paper is concerned with the initial boundary value problem of the Vlasov–Fokker–Planck equation coupled with either the incompressible or compressible Navier–Stokes equations in a bounded domain. The global existence of unique strong solution and its exponential convergence rate to the equilibrium state are proved under the Maxwell boundary condition for the incompressible case and specular reflection boundary condition for the compressible case, respectively. For the compressible model, to overcome the lack of regularity due to the coupling with the kinetic equation in a bounded domain, an essential $L^{\frac{10}{3}}$ estimate is analyzed so that the a priori estimate can be closed by applying the ${\mathcal {S}}_{\mathcal {L}}^p$ theory developed by Guo et al. for kinetic models, [Arch Ration Mech Anal 236(3): 1389–1454 (2020)].

Appendix

In the appendix, we summarize some preliminary analysis used in deriving the $L^\infty$ estimates in Sect. 4.

1.1 Extension Beyond the Boundary

Following the approach used in [14, 23], we extend the domain $\Omega$ to the whole space preserving the hypoelliptic structure of the Fokker–Planck equation (1.19)$_3$.

By the compactness of $\partial \Omega$, there exist finitely many points $x_i^0\in \partial \Omega$, radii $r_i>0$ such that $\partial \Omega \subset \cup _i^m B(x_i^0,r_i)$. Denote $U_i=\Omega \cap B(x_i^0,r_i)$, then there exist smooth functions $\phi _i(x_1,x_2)\in C_b^3({\mathbb {R}}^2,{\mathbb {R}})$, $(i=1,2,\cdots ,m)$ such that

$$\begin{aligned} U_i=\{x\in B(x_i^0,r_i)|x_3<\phi _i(x_1,x_2)\}, \ 1\le i\le m. \end{aligned}$$

Here $B(x_i^0,r_i)$ is an open ball with center $x_i^0$ and radius $r_i>0.$ In what follows, we omit the subscript i of $\phi _i$ and $U_i$ for notational simplicity. We now change coordinates to flatten the boundary locally. To do this, for any $x\in U$, define

$$\begin{aligned} \psi ^{-1}: \left( \begin{array}{rll}y_1&{}\\ y_2&{}\\ y_3&{} \end{array}\right) \mapsto&\varrho (y_1,y_2)+y_3n(y_1,y_2)\nonumber \\ =&\left( \begin{array}{ccc}&{}y_1\\ &{}y_2\\ &{}\phi (y_1,y_2) \end{array}\right) +y_3\left( \begin{array}{cc}-\partial _1\phi &{}\\ -\partial _2\phi &{}\\ 1&{} \end{array}\right) :=\left( \begin{array}{ccc}&{}x_1\\ &{}x_2\\ &{}x_3 \end{array}\right) , \end{aligned}$$

where $\varrho (y_1,y_2)=(y_1,y_2,\phi (y_1,y_2))^T\in \partial \Omega$. Moreover, $\partial _1\varrho =(1,0,\partial _1\phi )$, $\partial _2\varrho =(0,1,\partial _2\phi )$, then the outward normal vector at the point $\varrho (y_1,y_2)\in \partial \Omega$ is chosen to be

$$\begin{aligned} n(y_1,y_2)=\partial _1\varrho \times \partial _2\varrho =(-\partial _1\phi ,-\partial _2\phi ,1)^T. \end{aligned}$$

We can choose $s>0$ small enough so that the half ball $V':=B(0,s)\cap \{y_3<0\}$ lies in $\psi (U)$. Then the phase boundary $\partial \Omega \times {\mathbb {R}}^3$ is locally flattened by defining the transformation

$$\begin{aligned} \Psi :\&{\bar{U}}\times {\mathbb {R}}^3\rightarrow {\bar{V}}_{-}\times {\mathbb {R}}^3\nonumber \\&(x,\xi )\mapsto (y,w)=(\psi (x),A\xi ), \end{aligned}$$

(5.1)

where

$$\begin{aligned} A=\left( \frac{\partial y}{\partial x}\right) =\nabla _x\psi ,\ U=U_i,\ {\bar{V}}_{-}=\psi ({\bar{U}})\supset B(0,s)\cap \{y_3<0\}. \end{aligned}$$

We now change variables and extend u and f to the upper half space

$$\begin{aligned} {\tilde{u}}(y)=\left\{ \begin{array}{rll} &{}u(\psi ^{-1}(y)),\ \ y_3<0,\\ &{}u(\psi ^{-1}(Ry)),\ \ y_3\ge 0, \end{array}\right. \ {\tilde{f}}(\tau ,y,w)=\left\{ \begin{array}{rll} &{}f(\tau ,\psi ^{-1}(y),A^{-1}(y)w),\ \ y_3<0,\\ &{}f(\tau ,\psi ^{-1}(Ry),A^{-1}(Ry)Rw),\ \ y_3\ge 0, \end{array}\right. \end{aligned}$$

(5.2)

where $R=\text {diag}(1,1,-1)$. Note that

$$\begin{aligned} {\tilde{f}}(\tau ,y,w)={\tilde{f}}(\tau ,y,Rw), \ \text {on}\ y_3=0. \end{aligned}$$

Denote $V_+=\{y: Ry\in V_-\}\supset B(0,s)\cap \{y_3>0\}$ and set $V={\bar{V}}_+\cup V_-.$ To extend in the $\tau$ variable, define

$$\begin{aligned} {\bar{u}}(\tau )=\left\{ \begin{array}{rll} &{}{\tilde{u}}(\tau ,y),\ \ \tau>0,\\ &{}{\tilde{u}}(0,y),\ \ -1<\tau \le 0, \end{array}\right. \ \ \tilde{{\tilde{f}}}(\tau )=\left\{ \begin{array}{rll} &{}{\tilde{f}}(\tau ,y,w),\ \ \tau >0,\\ &{}{\tilde{f}}(0,y,w),\ \ -1<\tau \le 0. \end{array}\right. \end{aligned}$$

(5.3)

Moreover, let us define $U_0\subset \Omega$ such that $\cup _{i=0}^m U_i\supset \Omega$, and then denote

$$\begin{aligned} {\tilde{\Omega }}_1=U_0\cup _{i=1}^m V_{-,i},\ {\tilde{\Omega }}_2=U_0\cup _{i=1}^m V_{i}, \end{aligned}$$

(5.4)

where $V_{-,i}$ and $V_i$ are induced by $U_i$ in the way as (5.1) and (5.2), respectively.

Let

$$\begin{aligned}&\chi (y)\ \text {be any fixed cutoff function in }\ C^\infty _0 (V),\nonumber \\&\text {and } \ {\bar{\chi }}(\tau )\ \text {be characteristic function on }\ [0,t], \end{aligned}$$

(5.5)

and denote ${\bar{f}}=\tilde{{\tilde{f}}}\chi {\bar{\chi }}$ for brevity. We now derive the equation that ${\bar{f}}(\tau ,y,w)$ satisfies. Actually, in view of (1.19)$_3$, (5.1), (5.2) and (5.3), one has

$$\begin{aligned} \partial _\tau {\bar{f}}+w\cdot \nabla _y{\bar{f}}+{\mathcal {L}}_0{\bar{f}}={\mathcal {S}},\ (\tau ,y,w)\in {\mathbb {R}}^{6+1}, \end{aligned}$$

(5.6)

where

$$\begin{aligned} {\mathcal {L}}_0{\bar{f}}=-\chi _{y_3\le 0}\nabla _w\cdot (AA^T\nabla _w{\bar{f}})-\chi _{y_3>0}\nabla _w\cdot (R{\bar{A}}{\bar{A}}^TR^T\nabla _w{\bar{f}}), \end{aligned}$$

(5.7)

and

$$\begin{aligned} {\mathcal {S}}=&- ABw\nabla _w{\bar{f}}\chi _{y_3\le 0}-R{\tilde{A}}{\tilde{B}}Rw\nabla _w{\bar{f}}\chi _{y_3>0}- A{\bar{u}}\nabla _w{\bar{f}}\chi _{y_3\le 0} - R{\tilde{A}}{\bar{u}}\nabla _w{\bar{f}}\chi _{y_3>0} \nonumber \\ {}&+\left( -\frac{|A^{-1}w|^2}{4}{\bar{f}}+\frac{3}{2}{\bar{f}}\right) \chi _{y_3\le 0} +\left( -\frac{|{\tilde{A}}^{-1}Rw|^2}{4}{\bar{f}}+\frac{3}{2}{\bar{f}}\right) \chi _{y_3>0}\nonumber \\&+\chi {\bar{\chi }}(A^{-1}w)^T{\bar{u}}\tilde{{\tilde{M}}}^{\frac{1}{2}}\chi _{y_3\le 0} +{\tilde{\chi }}{\bar{\chi }}({\tilde{A}}^{-1}Rw)^T{\bar{u}}\tilde{{\tilde{M}}}^{\frac{1}{2}}\chi _{y_3>0} \nonumber \\ {}&+\frac{1}{2}(A^{-1}w)^T{\bar{u}}{\bar{f}}\chi _{y_3\le 0} +\frac{1}{2}({\tilde{A}}^{-1}Rw)^T{\bar{u}}{\bar{f}}\chi _{y_3>0} +\chi \tilde{{\tilde{f}}}(\tau )\delta (\tau )+\chi \tilde{{\tilde{f}}}(\tau )\delta (\tau -t)\nonumber \\ {}&+{\bar{\chi }}\chi _{y_3\le 0}\tilde{{\tilde{f}}}w\cdot \nabla _y\chi +{\bar{\chi }}\chi _{y_3>0}\tilde{{\tilde{f}}}Rw\cdot \nabla _y{\tilde{\chi }}+{\bar{\chi }}\chi \chi _{t\le 0}\{w\cdot \nabla _y+{\mathcal {L}}_0\}{\tilde{f}}(0,y,w), \end{aligned}$$

(5.8)

where $\tilde{{\tilde{M}}}$ is defined as (5.3) and $\delta (\tau )$ is the Dirac function, and

$$\begin{aligned} A^{-1}&=\left( \frac{\partial x}{\partial y}\right) =\left( \nabla _x\psi \right) ^{-1}=\left( \begin{array} {ccc} 1-y_3\partial _{11}\phi \ \ \ &{}-y_3\partial _{12}\phi \ \ \ &{}-\partial _1\phi \\ -y_3\partial _{12}\phi \ \ \ &{}1-y_3\partial _{22}\phi \ \ \ &{}-\partial _2\phi \\ \partial _1\phi \ \ \ &{}\partial _2\phi \ \ \ &{}1 \end{array}\right) , \end{aligned}$$

(5.9)

$$\begin{aligned} B&=\left( \frac{\partial \xi }{\partial y}\right) =\left( \frac{\partial (A^{-1}w)}{\partial y}\right) \nonumber \\&=\begin{pmatrix} - y_3 \partial _{111}\phi w_1 \!-\! y_3 \partial _{112}\phi w_2 &{}\;\; - y_3 \partial _{112}\phi w_1 \!-\! y_3 \partial _{122}\phi w_2 &{}\;\; - \partial _{11}\phi w_1 \!-\! \partial _{12}\phi w_2 \\ [-2pt] - \partial _{11}\phi w_3 &{} - \partial _{12}\phi w_3 &{} \\[3pt] - y_3 \partial _{112}\phi w_1 \!-\! y_3 \partial _{122}\phi w_2 &{}\;\; - y_3 \partial _{122}\phi w_1 \!-\! y_3 \partial _{222}\phi w_2 &{}\;\; - \partial _{12}\phi w_1 \!-\! \partial _{22}\phi w_2 \\ [-2pt] - \partial _{12}\phi w_3 &{} - \partial _{22}\phi w_3 &{} \\[3pt] \partial _{11}\phi w_1 + \partial _{12}\phi w_2 &{} \partial _{12}\phi w_1 + \partial _{22}\phi w_2 &{} 0 \end{pmatrix}, \end{aligned}$$

(5.10)

$$\begin{aligned} {\tilde{A}}(y)&=A(Ry),\ {\tilde{B}}(y)=B(Ry),\ {\tilde{\chi }}(y)=\chi (Ry), \end{aligned}$$

(5.11)

and $\chi _{\tau >0}$, $\chi _{y_3\le 0}$ as well as $\chi _{y_3>0}$ are also characteristic functions.

Note that all the elements in $\chi A$ and $\chi B$ are uniformly bounded.

Remark 4.1

In the above extension, it is crucial that across the boundary the extended equation (5.6) is satisfied by ${\bar{f}}$ is weak sense. Namely, ${\bar{f}}$ should satisfy the following weak formation of equation (5.6) in the whole space

$$\begin{aligned}&\iint _{{\mathbb {R}}^3\!\times {\mathbb {R}}^3} \left[ ({\bar{f}}\Phi )(t)-({\bar{f}}\Phi )(0) \right] dydw +\int _0^t\int _{{\mathbb {R}}^3\times {\mathbb {R}}^3}{\bar{f}}{\mathcal {L}}_0\Phi dydwd\tau \\&\quad = \int _0^t\!\iint _{{\mathbb {R}}^3\!\times {\mathbb {R}}^3} \Big \{{\bar{f}} \big [(\partial _\tau +w\!\cdot \!\nabla _{\!y})\Phi \big ] + {\mathcal {S}}\Phi \Big \} dydwd\tau , \end{aligned}$$

where $\Phi \in C_0^\infty ({\mathbb {R}}^{6+1})$ and the boundary term vanishes due to

$$\begin{aligned} {\bar{f}}(t ,y_1,y_2,0- ,w) = {\bar{f}}(t ,y_1,y_2,0+ ,w). \end{aligned}$$

1.2 ${\mathcal {S}}_{\mathcal {L}}^P$ Theory

For general interest, we summarize in the following properties of the operator ${\mathcal {L}}$ defined in (4.33). First of all, the following assumptions on the coefficients of ${\mathcal {L}}$ are imposed:

[H.1] $a_{ij}=a_{ji}\in L^\infty ({\mathbb {R}}^{6+1})$, there exists $\Lambda >0$ such that

$$\begin{aligned} \Lambda ^{-1}|\xi |^2\le \sum \limits _{i,j}^3a_{ij}\xi _i\xi _j\le \Lambda |\xi |^2. \end{aligned}$$

[H.2] The matrix ${\mathscr {B}}=(b_{ij})_{i,j=1,2,\cdots ,6}=$ has the following form

$$\begin{aligned} \begin{aligned}&\left( \begin{array} {ccccc} 0 \ \ \ &{} {\mathscr {B}}_1\ \ \ &{} 0 \ \ \ &{} \cdots \ \ \ &{} 0 \\[3mm] 0 \ \ \ &{} 0 \ \ \ &{} {\mathscr {B}}_2 \ \ \ &{} \cdots \ \ \ &{} 0 \\[3mm] \vdots \ \ \ &{} \vdots \ \ \ &{} \vdots \ \ \ &{} \ddots \ \ \ &{} \vdots \\[3mm] 0 \ \ \ &{} 0\ \ \ &{} 0 \ \ \ &{} \cdots \ \ \ &{} {\mathscr {B}}_r \\[3mm] 0 \ \ \ &{} 0\ \ \ &{} 0 \ \ \ &{} \cdots \ \ \ &{} 0 \end{array} \right) , \end{aligned} \end{aligned}$$

where each ${\mathscr {B}}_j$ is a $m_{j-1}\times m_j$ block matrix of rank $m_j$, with $j=1,2,\cdots ,r$ and $m_0\ge m_1\ge \cdots \ge m_r\ge 1$ and $m_0+m_1+\cdots +m_r=6.$

[H.3] The coefficients

$$\begin{aligned} a_{ij}({\mathbf {z}}) := a_{ij}(t,y,w)\,\in \, \mathrm{VMO}_{{\mathcal {L}}}\big ({\mathbb {R}}_t\!\times \!{\mathbb {R}}_y^3\!\times \!{\mathbb {R}}_w^3\big ), \end{aligned}$$

where the $\mathrm{VMO}_{{\mathcal {L}}}$ space is defined as

$$\begin{aligned} \mathrm{BMO}_{{\mathcal {L}}}\big ({\mathbb {R}}^{6+1}\big )&\,:=\, \Big \{f\in L^1_\mathrm{loc}\big ({\mathbb {R}}^{6+1}\big ): \Vert f\Vert _{*}<+\infty \Big \} \,,\\ \mathrm{VMO}_{{\mathcal {L}}}\big ({\mathbb {R}}^{6+1}\big )&\,:=\, \Big \{f\in \mathrm{BMO}_{{\mathcal {L}}}\big ({\mathbb {R}}^{5+1}\big ): \lim _{r\rightarrow 0}\eta _f(r)=0 \Big \} \,, \end{aligned}$$

with

$$\begin{aligned} \Vert f\Vert _{*} \,:=&\, \sup _{{\bar{\rho }}>0}\frac{1}{|{\mathfrak {B}}_{{\bar{\rho }}}|}\int _{{\mathfrak {B}}_{{\bar{\rho }}}} \big |f({\mathbf {w}})-f_{{\mathfrak {B}}_{{\bar{\rho }}}}\big | d{\mathbf {w}},\ \ \eta _f(r) \,:=\, \sup _{{\bar{\rho }}\le r}\frac{1}{|{\mathfrak {B}}_{{\bar{\rho }}}|}\int _{{\mathfrak {B}}_{{\bar{\rho }}}} \big |f({\mathbf {w}})-f_{{\mathfrak {B}}_{\!{\bar{\rho }}}}\big | d{\mathbf {w}} \,,\\ f_{{\mathfrak {B}}_{\!{\bar{\rho }}}}=&\frac{1}{|{\mathfrak {B}}_{\!{\bar{\rho }}}|}\int _{{\mathfrak {B}}_{\!{\bar{\rho }}}}fd\mathbf{z}. \end{aligned}$$

Note that the quasi-distance d is chosen according to the following two definitions.

Definition 4.1

(Quasi-Norm) For any  ${\mathbf {z}}:=(t,y,w)\in {\mathbb {R}}^{6+1}\backslash \{{\mathbf {0}}\}$ , define the “quasi-norm”  $\Vert {\mathbf {z}}\Vert :={\bar{\rho }}$  to be the unique positive solution to the equation

$$\begin{aligned} \frac{t^2}{{\bar{\rho }}^4} + \frac{|y|^2}{{\bar{\rho }}^6} + \frac{|w|^2}{{\bar{\rho }}^2} =1; \end{aligned}$$

while if  ${\mathbf {z}}={\mathbf {0}}$ , we set  $\Vert {\mathbf {z}}\Vert =0$.

Based on this quasi-norm, the quasi-distance is defined as

Definition 4.2

(Quasi-Distance) Let  ${\mathbf {z}}:=(t,y,w),\, {\mathbf {w}}:=(\tau ,y',w') \in {\mathbb {R}}^{6+1}$, define

$$\begin{aligned} d({\mathbf {z}},{\mathbf {w}}) := \big \Vert {\mathbf {w}}^{-1}\!\circ {\mathbf {z}}\big \Vert \,, \end{aligned}$$

where

$$\begin{aligned} {\mathbf {w}}^{-1}\!\circ {\mathbf {z}} \,:= (\tau ,y',w')^{-1}\! \circ (t,y,w) = \big (\,t-\tau ,\, y-y'-(t-\tau )w',\, w-w'\,\big ) \,. \end{aligned}$$

Furthermore, we define also the d-ball centered at ${\mathbf {z}}$ with radius r as

$$\begin{aligned} {\mathfrak {B}}_r({\mathbf {z}}) := \left\{ {\mathbf {w}}\in {\mathbb {R}}^{6+1}: d({\mathbf {z}},{\mathbf {w}}) <r \right\} \,. \end{aligned}$$

Note that $({\mathbb {R}}_y^3\times {\mathbb {R}}_w^3,d\mathbf{z},d)$ is a metric measure space with “spatial homogeneous dimension" $Q=12.$ Moreover, the quasi-distance d naturally induces a topology on ${\mathbb {R}}_t\!\times \!{\mathbb {R}}_y^3\!\times \!{\mathbb {R}}_w^3$, which allows us to define the Hölder space $C^{\,k,\beta }_{\mathcal {L}}$ accordingly.

Definition 4.3

($C^{\,k,\beta }_{\mathcal {L}}$  Space) Assume  $\Pi$  is an open set in ${\mathbb {R}}^{6+1}$. If there exist constants  $\beta \in (0,1]$  and  $C_0>0$  such that

$$\begin{aligned} \big |f({\mathbf {z}})-f({\mathbf {w}})\big | \,\le \, C_0 \big \Vert {\mathbf {w}}^{-1}\!\circ {\mathbf {z}}\big \Vert ^{\beta }, \end{aligned}$$

for every  ${\mathbf {z}}, {\mathbf {w}} \in \Pi$ . Then we call f satisfies ${\mathcal {L}}$-Hölder condition, or has (uniform) ${\mathcal {L}}$ -Hölder continuity. In this case, f can be equipped with a semi-norm

$$\begin{aligned} |f|_{C^{\,0,\beta }_{\mathcal {L}}(\Pi )} \,:= \sup _{{\mathbf {z}}\ne {\mathbf {w}}\in \Pi } \frac{|f({\mathbf {z}})-f({\mathbf {w}})|}{\Vert {\mathbf {w}}^{-1}\!\circ {\mathbf {z}}\Vert ^{\beta }} \;. \end{aligned}$$

Additionally, if the function f  and its derivatives up to order k  are bounded on the closure of $\Pi$ (bounded), then the  ${\mathcal {L}}$ -Hölder space  $C^{\,k,\beta }_{\mathcal {L}}({\bar{\Pi }})$  is a Banach space with respect to the norm

$$\begin{aligned} \Vert f\Vert _{C^{\,k,\beta }_{\mathcal {L}}} \,:=\, \Vert f\Vert _{C^k} +\, \max _{|\alpha '|=k} \big |D^{\alpha '}\!f \big |_{C^{\,0,\beta }_{\mathcal {L}}} \;, \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{C^k(\Pi )} \,:=\, \max _{|\alpha '|\le k}\, \sup _{{\mathbf {z}}\in \Pi } \big |D^{\alpha '}\!f({\mathbf {z}}) \big | \;. \end{aligned}$$

The following lemma from [2, 34] is used in the proof of (4.31).

Lemma 4.2

Let $\Pi$ be a bounded open set in ${\mathbb {R}}_t\times {\mathbb {R}}^3_y\times {\mathbb {R}}^3_w$, and f be a solution of ${\mathcal {L}}f={\mathcal {S}}$. Suppose that the conditions [H.1]–[H.3] are satisfied in $\Pi$, and that $f\in L^p(\Pi )$, ${\mathcal {S}}\in L^{p}(\Pi )$ with $p\in (1,\infty )$. Then ${\mathcal {Y}}f\in L^{p}_\mathrm{{loc}}(\Pi )$, $\partial _{v_i v_j}f\in L^{p}_\mathrm{{loc}}(\Pi )\; (i,j=1,2,3)$. More precisely, for every open subset  $\Pi '\subset \subset \Pi$, it holds

$$\begin{aligned} \big \Vert {\mathcal {Y}}f\big \Vert _{L^{p}(\Pi ')}&\,\le \, C_1 \left( \Vert {\mathcal {S}}\Vert _{L^{p}(\Pi )} + \Vert f\Vert _{L^p(\Pi )} \right) , \\ \big \Vert D^2_{vv}f\big \Vert _{L^{p}(\Pi ')}&\,\le \, C_1 \left( \Vert {\mathcal {S}}\Vert _{L^{p}(\Pi )} + \Vert f\Vert _{L^p(\Pi )} \right) , \end{aligned}$$

where $C_1>0$ depends on p, $\mathrm{dist}(\Pi ',\Pi )$, eigenvalues (elliptic constant) of $(a_{ij}({\mathbf {z}}))$, $\alpha$ and $C_0$ related to the Hölder semi-norm of $a_{ij}({\mathbf {z}})$.

Finally, the following lemma provides us the embedding estimates of Sobolev type space ${\mathcal {S}}_{{\mathcal {L}}}^p$ defined by (4.32) and Morrey type space, cf. [2, 34].

Lemma 4.3

For any $u\in S^{p}_{{\mathcal {L}}}({\mathbb {R}}_t\!\times \!{\mathbb {R}}_y^3\!\times \!{\mathbb {R}}_w^3)$ with $p\in (1,\infty )$, if $7<p<14$, then $u\in C^{0,\beta }_{\mathcal {L}}({\mathbb {R}}_t\!\times \!{\mathbb {R}}_y^3\!\times \!{\mathbb {R}}_w^3)$  with $\beta = 2-\frac{14}{p} \in (0,1)$, such that

$$\begin{aligned} |u|_{C^{\,0,\beta }_{\mathcal {L}}} \lesssim \Vert u\Vert _{S^{p}_{{\mathcal {L}}}}. \end{aligned}$$