2404.08448

# Construction of low regularity strong solutions to the viscous surface wave equations **Authors**: Guilong Gui111School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China. Email:.Yancan Li222School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China. Email:. ## Abstract We construct in the paper the low-regularity strong solutions to the viscous surface wave equations in anisotropic Sobolev spaces. Here we use the Lagrangian structure of the system to homogenize the free boundary conditions, and establish a new iteration scheme on a known equilibrium domain to get the low-regularity strong solutions, in which no nonlinear compatibility conditions on the initial data are required. Keywords: Viscous surface waves; Lagrangian coordinates; Global well-posedness; Anisotropic Sobolev spaces AMS Subject Classification (2010): 35Q30, 35R35, 76D03 ## 1 Introduction ### 1.1 Formulation in Eulerian Coordinates We consider in this paper the local existence of time-dependent flows of an viscous incompressible fluid in a moving domain $Ω(t)$ with an upper free surface $Σ_F(t)$ and a fixed bottom $Σ_B$ $$ \begin{cases}&∂_tu+(u·∇)u+∇ p-νΔ u=-g e_1 \mbox{in} Ω(t),\\ &∇· u=0 \mbox{in} Ω(t),\\ &(p I-νD(u))n(t)=p_\mbox{ atm}n(t) \mbox{on} Σ_F(t),\\ &V(Σ_F(t))=u· n(t) \mbox{on} Σ_F(t),\\ &u|_Σ_{B}=0,\end{cases} $$ where we denote $n(t)$ the outward-pointing unit normal on $Σ_F(t)$ , $I$ the $3× 3$ identity matrix, $(Du)_ij=∂_iu^j+∂_ju^i$ twice the symmetric gradient of the velocity $u=(u^1,u^2,u^3)$ , $u^i=u^i(t,x_1,x_2,x_3)$ , $i,j=1,2,3$ , the constant $g>0$ stands for the strength of gravity, $e_1=(1,0,0)^T$ , and $ν>0$ is the constant coefficient of viscosity. We denote $V(Σ_F(t))$ the outer-normal velocity of the free surface $Σ_F(t)$ . The tensor $(p I-νD(u))$ is known as the viscous stress tensor. Equation $\eqref{VFS-eqns-1}_1$ is the conservation of momentum, where gravity is the only external force, which points in the negative $x_1$ direction (as the vertical direction); the second equation in (1.1) means the fluid is incompressible; Equation $\eqref{VFS-eqns-1}_3$ means the fluid satisfies the kinetic boundary condition on the free boundary $Σ_F(t)$ , where $p_atm$ stands for the atmospheric pressure, assumed to be constant. the kinematic boundary condition $\eqref{VFS-eqns-1}_4$ states that the free boundary $Σ_F(t)$ is moving with speed equal to the normal component of the fluid velocity; $\eqref{VFS-eqns-1}_5$ implies that the fluid is no-slip, no-penetrated on the fixed bottom boundary. Here the effect of surface tension is neglected on the free surface. For convenience, it is natural to subtract the hydrostatic pressure from $p$ in the usual way by adjusting the actual pressure $p$ according to $\widetilde{p}=p+g x_1-p_atm$ , and still denote the new pressure $\widetilde{p}$ by $p$ for simplicity, so that after substitution the gravity term in $\eqref{VFS-eqns-1}_1$ and the atmospheric pressure term in $\eqref{VFS-eqns-1}_3$ are eliminated. A gravity term appears in $\eqref{VFS-eqns-1}_3$ . The problem can be equivalently stated as follows. Given an initial domain $Ω_0⊂ℝ^3$ bounded by a bottom surface $Σ_B$ , and a top surface $Σ_F(0)$ , as well as an initial velocity field $u_0$ , where the upper boundary does not touch the bottom, we wish to find for each $t∈[0,T]$ a domain $Ω(t)$ , a velocity field $u(t,·)$ and pressure $p(t,·)$ on $Ω(t)$ , and a transformation $\bar{η}(t,·): Ω_0→ℝ^3$ so that $$ \begin{cases}&Ω(t)=\bar{η}(t,Ω_0), \bar{η}(t,Σ_B) =Σ_B,\\ &∂_t\bar{η}=u∘\bar{η},\\ &∂_tu+(u·∇)u+∇ p-νΔ u=0 in Ω(t),\\ &∇· u=0 \mbox{in} Ω(t),\\ &≤ft((p-g x_1) I-νD(u)\right)n(t)=0 \mbox{on} Σ_F(t),\\ &u|_Σ_{B}=0,\\ &u|_t=0=u_0, \bar{η}|_t=0=x\end{cases} $$ The conditions on the initial domain $Ω_0$ are as follows: Let the equilibrium domain $Ω⊂ℝ^3$ be the horizontal infinite slab $$ \begin{split}&Ω=\{x=(x_1,x_h)|-\underline{b}<x_1<0, x_h∈ ℝ^2\}\end{split} $$ with the bottom $Σ_b=\{x_1=-\underline{b}\}$ and the top surface $Σ_0=\{x_1=0\}$ , where the positive constant $\underline{b}$ will be the depth of the fluid at infinity. We assume that $Ω_0$ is the image of $Ω$ under a diffeomorphism $\overline{σ}:Ω→Ω_0$ , where $\overline{σ}(Σ_b)=Σ_B$ , $\overline{σ}(Σ_0)=Σ_F(0)$ , $\overline{σ}$ is of the form $\overline{σ}(x)=x+ξ_0(x)$ , and $ξ_0$ satisfies $ξ_0^1∈ H^s(Σ_0)$ , and $∇ξ_0, ∇_h^s-1ξ_0∈ H^1(Ω)$ with $s>2$ , $ξ_0^1$ stands for the vertical component of the vector $ξ_0$ , $∇=(∂_1,∂_2,∂_3)^T$ , and $∇_h=(∂_2,∂_3)^T$ . ### 1.2 Known results Many mathematicians have contributions in the free boundary problems of the incompressible Navier-Stokes equations. By using Lagrangian coordinates transformation, Solonnikov [20] proved the local well-posedness of the viscous surface problem in Hölder spaces in a bounded domain whose entire boundary is a free surface. In a horizontal infinite domain, Beale [6] obtained the local well-posedness of the free boundary Navier-Stokes equations without surface tension in $L^2$ -based space-time Sobolev spaces, which was extended to $L^p$ -based space-time Sobolev spaces by Abels [1]. Furthermore, Beale [7] introduced the flattening transformation to get the global well-posedness of the viscous surface problem with surface tension in the space-time Sobolev spaces, and then the decay properties of solutions to incompressible viscous surface waves with surface tension where studied by Beale-Nishida [8] and Hataya [16]. Sylvester [21] and Tani-Tanaka [22] also discussed the well-posedness of the incompressible viscous surface problem without surface tension. Bae [4] proved global well-posedness with surface tension using energy methods rather than a Beale–Solonnikov framework. For a bounded mass of fluid with surface tension, local well-posedness was proved by Coutand and Shkoller [9]. Recently, by using the flattening transformation, Guo and Tice [14, 15] constructed the local-in-time solutions of the viscous surface wave equations by using the geometric structure in the Eulerian coordinates, and then got its global well-posedness and the algebraic decay rate of the solutions by introducing the two-tier energy method. Wu [26] extended their local well-posedness result from small data to general data. Wang, Tice and Kim [25] also considered the global well-posedness and decay for two layers fluid. Ren, Xiang and Zhang [18] proved the low-regularity local well-posedness of the system in Sobolev spaces in Eulerian coordinates. Wang [24] studied the anisotropic decay of the global solution by using the same method. More recently, the first author of this paper developed a mathematical approach to establish global well-posedness of the incompressible viscous surface waves based on the Lagrangian framework [13], where no nonlinear compatibility conditions on the initial data were required. Nevertheless, almost all well-posedness results in previous works were established for the initial data which has high regularity or some compatibility conditions on the initial data are needed. In the present case, a natural and important question is whether a corresponding well-posedness result can be obtained with low regularity and without compatibility conditions of the accelerated velocity on the initial data. ### 1.3 Formulation of the system in Lagrangian Coordinates Let us now, in more detail, introduce the Lagrangian coordinates in which the free boundary becomes fixed. Let $η\buildrel\hbox{ def}\over{=}\bar{η}∘\bar{σ}$ be a position of the fluid particle $x$ in the equilibrium domain $Ω$ at time $t$ so that $$ \begin{cases}&\frac{d}{dt}η(t,x)=u(t,η(t,x)), t>0, x∈Ω,\\ &η|_t=0=x+ξ_0(x), x∈Ω,\end{cases} $$ then the displacement $ξ(t,x)\buildrel\hbox{ def}\over{=}η(t,x)-x$ satisfies $$ \begin{cases}&\frac{d}{dt}ξ(t,x)=u(t,x+ξ(t,x)),\\ &ξ|_t=0=ξ_0.\end{cases} $$ We define Lagrangian quantities the velocity $v$ and the pressure $q$ in fluid as (where $x=(x_1,x_2,x_3)^T∈Ω$ ): $v(t,x)\buildrel\hbox{ def}\over{=}u(t,η(t,x))$ , $q(t,x)\buildrel\hbox{ def}\over{=}p(t,η(t,x))$ . Denote the Jacobian of the flow map $η$ by $J\buildrel\hbox{ def}\over{=}\mbox{det}(Dη)$ . Define $A\buildrel\hbox{ def}\over{=}(Dη)^-T$ , then according to definitions of the flow map $η$ and the displacement $ξ$ , we may get the identities: $$ A_i^k∂_kη^j=A_k^j∂_iη^k =δ_i^j, ∂_k(JA_i^k)=0, ∂_i η^j=δ_i^j+∂_iξ^j, A_i^j=δ_i ^j-A_i^k∂_kξ^j. $$ Set $a_ij\buildrel\hbox{ def}\over{=} J A_i^j$ . Simple computation implies that $J=1+∇·ξ+B_00+B_000$ , where $B_00:=∂_1ξ^1∇_h·ξ^h-∂_1ξ^h ·∇_hξ^1+∇_h^⊥ξ^2·∇_hξ^3$ , $B_000:=∂_1ξ^1∇_h^⊥ ξ^2·∇_ h ξ^3+∂_1ξ^2∇_h^⊥ ξ^3·∇_h ξ ^1+∂_1ξ^3∇_h^⊥ ξ^1·∇_h ξ^2$ with $ξ^h\buildrel\hbox{ def}\over{=}(ξ^2,ξ^3)^T$ , $∇_h\buildrel\hbox{ def}\over{=}(∂_2, ∂_3 )^T$ , $∇_h^⊥\buildrel\hbox{ def}\over{=}(-∂_3, ∂_2)^T$ . If the displacement $ξ$ is sufficiently small in an appropriate Sobolev space, then the flow mapping $η$ is a diffeomorphism from $Ω$ to $Ω(t)$ , which makes us to switch back and forth from Lagrangian to Eulerian coordinates. Next, we give some useful equations which we often use in what follows. Since $A(Dη)^T=I$ , differentiating it with respect to $t$ and $x$ once yields $$ \begin{split}&∂_tA_i^j=-A_k^jA_ {i}^m∂_mv^k, ∂_sA_i^j=-A_k^j A_i^m∂_m∂_sξ^k,\end{split} $$ where we used the fact $∂_tη=v$ in the first equation in (1.7). Whence differentiate the Jacobian determinant $J$ , we get $$ \begin{split}&∂_tJ=JA_i^j∂_jv^i, ∂_kJ=JA_i^j∂_j∂_kξ^i.\end{split} $$ Moreover, we may verify the following Piola identity: $$ \begin{split}&∂_j(JA_i^j)=0 ∀ i=1,2,3.\end{split} $$ Here and in what follows, the subscript notation for vectors and tensors as well as the Einstein summation convention has been adopted unless otherwise specified. Under Lagrangian coordinates, we may introduce the differential operators with their actions given by $(∇_Af)_i=A_i^j∂_jf$ , $D_A(v)=∇_Av+(∇_Av)^T$ , $Δ_Af=∇_A·∇_Af$ , so the Lagrangian version of the system (1.2) can be written on the fixed reference domain $Ω$ as $$ \begin{cases}∂_tξ=v& in Ω,\\ ∂_tv+∇_Aq-ν∇_A·D_ A(v)=0& in Ω,\\ ∇_A· v=0& in Ω,\\ (q-g ξ^1)N-νD_A(v)N=0;& on Σ_0,\\ v=0,& on Σ_b,\\ ξ|_t=0=ξ_0,v|_t=0=v_0,\end{cases} $$ where $n_0=e_1=(1,0,0)^T$ is the outward-pointing unit normal vector on the interface $Σ_0$ , $N:=JAn_0$ stands for the outward-pointing normal vector on the moving interface $Σ_F(t)$ . ### 1.4 Main result and ideas The aim of this paper is to establish the local well-posedness of he system (1.10) with low regularity data in anisotropic Sobolev spaces. We state the main result as follows, which proof will be presented in Section 4. **Theorem 1.1** *$($ Local well-posedness $)$ Let $s>2$ . Assume $(v_0,ξ_0)$ satisfies $Λ_h^s-1v_0∈{{}_0{H}^1}(Ω)$ , $∇_J_{0A_0}· v_0=0$ , and $Λ_h^s-1∇ξ_0∈ H^1(Ω)$ , $ξ_0^1∈ H^s(Σ_0)$ . There exists a positive constant $ε_0$ and $0<T≤\min\{1,δ_0(\|ξ_0^1\|_H^s{(Σ_0)}^2+\|Λ_ {h}^s-1v_0\|_H^1(Ω)^2)^-1\}$ for some $δ_0>0$ such that the system (1.10) has a unique solution $(v,ξ,q)$ depending continuously on the initial data $(v_0,ξ_0)$ which satisfies $$ \begin{split}&Λ_h^s-1∇ξ∈C([0,T];H^1(Ω)), ξ^1∈C([0,T];H^s(Σ_0)),\\ &Λ_h^s-1v∈C([0,T];{{}_0}{H}^1(Ω))∩ L^2([0, T];H^2(Ω)), Λ_h^s-1q∈ L^2([0,T];H^1(Ω))\end{split} $$ provided $$ \|Λ_h^s-1∇ξ_0\|_H^1(Ω)^2≤ε_0. $$ Moreover, for any $t∈(0,T]$ , the solution satisfies the estimate $$ \begin{split}\sup_0≤τ≤ t(\|Λ_h^s-1(∇ξ, v)\|_H^1 (Ω)^2&+\|ξ^1\|_H^s{(Σ_0)}^2)+\|Λ_h^s-1( ∇ v,q)\|^2_L_{t^2(H^1(Ω))}\\ &≤ C(\|Λ_h^s-1(∇ξ_0, v_0)\|_H^1(Ω)^2+\|ξ _0^1\|_H^s{(Σ_0)}^2).\end{split} $$* **Remark 1.1** *We may readily check that, if the maximal time $T^∗$ of the existence of the solution $(ξ, v, q)$ in Theorem 1.1 is finite: $T^∗<+∞$ , then $$ \begin{split}\lim_t≠arrow T^∗(\|Λ_h^s-1(∇ξ(t), v(t) )\|_H^1(Ω)+\|ξ^1(t)\|_H^s(Σ_0)+\|J^-1\|_L^∞( Ω))=+∞.\end{split} $$ Moreover, if in addition $\dot{Λ}^-λ_h(ξ_0^1, v_0)∈ L^2(Σ_0)× H ^1(Ω)$ for some $λ∈(0,1)$ , then $\dot{Λ}^-λ_h(ξ, v)∈C([0,T_0];L^2(Σ_0 )× H^1(Ω))$ .* **Remark 1.2** *We say the strong solution $(ξ,v)$ obtained in Theorem 1.1 low-regular, which means that the surface function $ξ^1∈ H^s(Σ_0)$ with $s>2$ has the minimal regularity guaranteeing the regular condition $∇_hξ^1∈ L^∞(Σ_0)$ .* **Remark 1.3** *The classical parabolic theory of non-homogeneous boundary conditions [17] is not enough to support us to complete the proof of Theorem 3.1 (see Sect. 3) about the well-posedness of the linear system (3.1) which is crucial in our construction of local solutions. Therefore, we can’t employ the approach in [6] to construct the solutions. In addition, due to the lack of compatibility conditions, the construction of local solutions is also different from [14, 26, 18]. And another difference from [14, 26, 18] is that our construction of local solutions is based on the Lagrange framework.* **Remark 1.4** *Motivated by [10], we establish the well-posedness of the linear system (3.1) to construct low regularity strong solutions. But there are two main difficulties to overcome. The first one is, for the linear system (3.1) in the finite depth case , we cannot deal with the non-homogeneous boundary conditions of (3.1) as directly as for the infinite case in [10]. Therefore, we need to construct another suitable divergence-free vector field to correct. On the other hand, our results allow the presence of gravity and a small perturbation in the initial domain, so we need to establish a new iteration scheme on a known equilibrium domain.* To prove Theorem 1.1, we will first linearize the surface wave equations (1.10) in Lagrangian coordinates to the system (4.1) or (3.1). To solve (3.1) with non-homogeneous boundary conditions, we will first remove the divergence of the velocity in Lagrangian coordinates and homogenize free boundary conditions in (3.1), and then solve the result system with homogeneous boundary conditions in Proposition 3.1, from which, we may obtain Theorem 3.1. With Theorem 3.1 in hand, we establish a new iteration scheme on a known equilibrium domain to get the low-regularity strong solutions (see Sect. 4). ### 1.5 Plan of the paper The rest of the paper is organized as follows. Section 2 introduces some basic estimates, which will be heavily used in this paper. In Section 3, we prove Theorem 3.1 for the linear system (3.1) in four steps, which is crucial in the proof of Theorem 1.1. Then, we use the fixed point method to prove Theorem 1.1 in Section 4. Section 5 is an appendix which gives the expressions of $B$ forms associated with the system (4.1). ### 1.6 Notations Let us end this introduction by some notations that will be used in all that follows. For operators $A,B,$ we denote $[A,B]=AB-BA$ to be the commutator of $A$ and $B.$ For $a≤sssim b$ , we mean that there is a uniform constant $C,$ which may be different on different lines, such that $a≤ Cb$ . The notation $a\thicksim b$ means both $a≤sssim b$ and $b≤sssim a$ . Throughout the paper, the subscript notation for vectors and tensors as well as the Einstein summation convention has been adopted unless otherwise specified, Einstein’s summation convention means that repeated Latin indices $i, j, k$ , etc., are summed from $1$ to $3$ , and repeated Greek indices $α, β, γ$ , etc., are summed from $2$ to $3$ . In the vector $v=(v^1,v^2,v^3)^T$ , we denote the vertical component by $v^1$ , and its horizontal component by $v^h=(v^2,v^3)^T$ . ## 2 Preliminary estimates Let’s first recall some basic estimates, which will be heavily used in the rest of the paper. **Lemma 2.1 ([3], Theorem 2.61 in[5])** *Let $f$ be a smooth function on $ℝ$ vanishing at $0 0$ , $s_1$ , $s_2$ be two positive real number, $s_1∈(0,1)$ , $s_2>0$ . If $u$ belongs to $\dot{H}^s_1(ℝ^2)∩\dot{H}^s_2(ℝ^2)∩ L^ ∞(ℝ^2)$ , then so does $f∘ u$ , and we have $$ \|f∘ u\|_\dot{H^s_i}≤ C(f^\prime,\|u\|_L^∞)\|u\|_\dot {H^s_i} for i=1,2. $$* **Lemma 2.2** *([3, 5]) Let $f$ be a smooth function such that $f^\prime(0)=0$ . Let $s>0$ . For any couple $(u,v)$ of functions in $B^s_p,r∩ L^∞$ , the function $f∘ v-f∘ u$ then belongs to $\dot{H}^s∩ L^∞$ and $$ \begin{split}&\|f∘ v-f∘ u\|_\dot{H^s}≤ C_f^\prime\prime(\|u \|_L^∞,\|v\|_L^∞)\\ & ×(\|u-v\|_\dot{H^s}\sup_τ∈[0,1]\|v+τ(u-v)\|_L^ ∞+\|u-v\|_L^∞\sup_τ∈[0,1]\|v+τ(u-v)\|_\dot{H^s}) ,\end{split} $$ where $C_f^\prime\prime(·,·)$ is a uniformly continuous function on $[0,∞)×[0,∞)$ depending only on $f^\prime\prime$ .* **Lemma 2.3 (Classical product laws in Sobolev spaces[5])** *Let $s_0>2$ , $p_1, p_2∈[2,+∞)$ , $q_1, q_2∈(2,+∞]$ , $\frac{1}{p_1}+\frac{1}{q_1}=\frac{1}{p_2}+\frac{1}{q_2}=\frac{1}{2}$ , there hold $$ \begin{split}&\|f g\|_\dot{H^s_1+s_2-1(ℝ^2)}≤sssim\|f\|_ {\dot{H}^s_1(ℝ^2)}\|g\|_\dot{H^s_2(ℝ^2)} ∀ s_1, s_2∈(-1,1), s_1+s_2>0,\\ &\|f g\|_\dot{H^s_0-1(ℝ^2)}≤sssim\|f\|_\dot{H^s_0-1( ℝ^2)}\|g\|_\dot{H^s_0-1(ℝ^2)},\\ &\|\dot{Λ}_h^σ(f g)\|_L^2(ℝ^{2_h)}≤sssim\| \dot{Λ}_h^σf\|_L^p_{1(ℝ^2_h)}\|g\|_L^q_{1( ℝ^2_h)}+\|\dot{Λ}_h^σg\|_L^p_{2(ℝ^2 _h)}\|f\|_L^q_{2(ℝ^2_h)} ∀ σ>0,\\ &\|\dot{Λ}_h^s_1(f g)\|_L^2(ℝ^{2_h)}≤sssim\|\dot {Λ}_h^s_1f\|_L^2(ℝ^{2_h)}\|\dot{Λ}_h^s_0 -1,s_2g\|_L^2(ℝ^{2_h)} ∀ s_1∈(-1,1], s_2∈ (0,1).\end{split} $$* **Lemma 2.4 (Korn-Poincaré’s inequality, Lemma 2.7 in[6])** *Let $Ω$ be defined by (1.3), then there exists a positive constant $C_korn$ , independent of $u$ , such that $$ \|u\|_H^1(Ω)≤ C_korn\|D(u)\|_L^2(Ω) $$ for all $u∈ H^1(Ω)$ with $u=0$ on $Σ_b$ .* **Lemma 2.5 (Poincaré’s inequality, Lemma A.10 in[14])** *Let $Ω$ be defined by (1.3), then there holds $$ \begin{split}&\|f\|_L^2(Ω)≤sssim \|f\|_L^2(Σ_0)+\| ∂_1f\|_L^2(Ω), \|f\|_L^∞(Ω)≤sssim \|f\| _L^∞(Σ_0)+\|∂_1f\|_L^∞(Ω).\end{split} $$* In order to extend the interface boundary forms of $∂_1v$ to the interior domain of the fluid, let us first introduce $H(f)$ as the harmonic extension of $f|_Σ_0$ into $Ω$ : $$ \begin{cases}&ΔH(f)=0 \mbox{in} Ω,\\ &H(f)|_Σ_0=f|_Σ_0, H(f)|_Σ_{b }=0.\end{cases} $$ **Lemma 2.6 (Lemma 4.5 in[13])** *Let $s∈ℝ$ , $r≥ 2$ , and $Ω$ be defined by (1.3). For the harmonic extension $H(f)$ of $f=f(t,x_1,x_h)$ (with $t∈ℝ^+$ , $(x_1,x_h)∈Ω$ ), there hold $$ \begin{split}&\|H(f)\|_H^1(Ω)≤sssim\|f\|_H^\frac{1{2} (Σ_0)}, \|\dot{Λ}_h^sH(f)\|_H^1(Ω) ≤sssim\|\dot{Λ}_h^sf\|_H^\frac{1{2}(Σ_0)},\\ &\|\dot{Λ}_h^sH(f)\|_H^2(Ω)≤sssim\|\dot{Λ }_h^sf\|_H^\frac{3{2}(Σ_0)}, \|H(f)\|_H^r( Ω)≤sssim\|f\|_H^r-\frac{1{2}(Σ_0)},\\ &\|∂_tH(f)\|_H^1(Ω)≤sssim\|∂_tf\|_H^ \frac{1{2}(Σ_0)}.\end{split} $$* **Lemma 2.7 (Lemma 2.8 in[6])** *Let $Ω$ be defined by (1.3), and for $f_1∈ H^r-2(Ω)$ , $f_2∈ H^r-\frac{3{2}}(Σ_0)$ , $2≤ r≤ 5$ , there is a unique solution $u∈ H^r(Ω)$ of $$ \begin{cases}Δ u=f_1&\mbox{in} Ω,\\ ∂_1u=f_2&\mbox{on} Σ_0,\\ u=0&\mbox{on} Σ_b,\\ \end{cases} $$ and the solution satisfies the estimate $$ \|u\|_H^r(Ω)≤ C(\|f_1\|_H^r-2(Ω)+\|f_2\|_H^r-\frac{ 3{2}(Σ_0)}). $$* **Lemma 2.8 (Chapter 6 in[11])** *Let $Ω$ be defined by (1.3), and for $f∈ H^r-2(Ω)$ , $r≥ 2$ , there is a unique solution $u∈ H^r(Ω)$ of $$ \begin{cases}Δ u=f&in Ω,\\ u=0&on ∂Ω,\\ \end{cases} $$ and the solution satisfies the estimate $$ \|u\|_H^r(Ω)≤ C\|f\|_H^r-2(Ω). $$* **Lemma 2.9 ([2])** *Let $Ω$ be defined by (1.3), $r≥ 2$ . Suppose $v,q$ solve $$ \begin{cases}-ν∇·D(v)+∇ q=f_1∈ H^r-2(Ω)\\ ∇· v=f_2∈ H^r-1(Ω),\\ qn_0-νD(u)n_0=f_3∈ H^r-\frac{3{2}}(Σ_0),\\ u|_Σ_{b}=0,\\ \end{cases} $$ Then there holds $$ \|∇ v\|_H^r-1(Ω)^2+\|∇ q\|_H^r-2(Ω)^2≤sssim \|f_1\|_H^r-2(Ω)^2+\|f_2\|_H^r-1(Ω)^2+\|f_3\|_H^ r-\frac{3{2}(Σ_0)}^2. $$* **Lemma 2.10 (Lions-Magenes Lemma, Lemma 1.2 on page 176 of[23])** *Let $X_0$ , $X$ and $X_1$ be three Hilbert spaces with $X_0⊂ X⊂ X_1$ . Suppose that $X_0$ is continuously embedded in $X$ and that $X$ is continuously embedded in $X_1$ , and $X_1$ is the dual space of $X_0$ . Suppose for some $T>0$ that $u∈ L^2([0,T];X_0)$ is such that $u_t∈ L^2([0,T];X_1)$ . Then $u$ is almost everywhere equal to a function continuous from $[0,T]$ into $X$ , and moreover the following equality holds in the sense of scalar distribution on $(0,T)$ : $$ \frac{1}{2}\frac{d}{dt}\|u\|_X^2=⟨ u_t,u⟩. $$* ## 3 A linearied system We are concerned with the time-dependent linear problem $$ \begin{cases}∂_tξ^1-u^1=0& on [0,T]×Σ_0 ,\\ ∂_tu-ν∇·D(u)+∇ p=F_1& in [0,T] ×Ω,\\ ∇· u=F_2& in [0,T]×Ω,\\ (p-g ξ^1) n_0-νD(u)n_0=F_3& on [0,T]× Σ_0,\\ u=0& on [0,T]×Σ_b,\\ u|_t=0=u_0& in Ω,\\ ξ^1|_t=0=ξ_0^1& on Σ_0\end{cases} $$ with $F_2:=B^1:∇ K^(1)$ , and $F_3:=B^2:∇_hK^(2)$ . For this linear system, we can get the following theorem. **Theorem 3.1** *For given $s>2$ , $T>0$ , assume that $F_1, B^i, K^(i)$ (with $i=1,2$ ) satisfy that $Λ_h^s-1(F_1, K^(i)_t)∈ L^2([0,T];L^2(Ω))$ , $Λ_h^s-1B^i∈C([0,T];H^1(Ω))$ , $Λ_h^s-1K^(i)∈C([0,T];{{}_0}H^1(Ω))$ , and $Λ_h^s-1(B^i_t, ∇ K^(i))∈ L^2([0,T];H^1(Ω))$ , and $(ξ^1_0, u_0)$ satisfies that $ξ^1_0∈ H^s(Σ_0)$ , $Λ_h^s-1u_0∈{{}_0}H^1(Ω)$ , and $∇· u_0={F_2}|_t=0$ , then there exists a unique solution $(ξ^1,u,p)$ of (3.1) satisfying $$ \begin{split}&ξ^1∈C([0,T];H^s(Σ_0)), Λ_h^ {s-1}u∈C([0,T];{H}^1(Ω))∩ L^2([0,T];H^2(Ω)),\\ &Λ_h^s-1u_t∈ L^2([0,T];L^2(Ω)), Λ_h^s-1p ∈ L^2([0,T];H^1(Ω)).\end{split} $$ Moreover, the solution satisfies the estimate $$ \begin{split}&\sup_t∈[0,T]\{\|ξ^1\|^2_H^s{(Σ_0)}+\| Λ_h^s-1u\|^2_H^1(Ω)\}+≤ft(\|Λ_h^s-1(∇ u, p)\|^2_L^2_T(H^{1(Ω))}+\|Λ_h^s-1u_t\|^2_L^2_T( L^{2(Ω))}\right)\\ &≤ C_0\bigg{(}\|ξ_0^1\|^2_H^s{(Σ_0)}+\|Λ_h^s- 1u_0\|^2_H^1(Ω)+\|Λ_h^s-1F_1\|^2_L^2_T(L^{2( Ω))}\\ &+\|Λ_h^s-1(B^1,B^2)\|^2_L^∞_T(H^{1(Ω))}(\| Λ_h^s-1(∇ K^(1),∇ K^(2))\|^2_L^2_T(H^{1(Ω ))}+\|Λ_h^s-1(K^(1)_t,K^(2)_t)\|^2_L^2_T(L^{2(Ω ))})\\ &+\|Λ_h^s-1(K^(1),K^(2))\|^2_L^∞_T(H^{1(Ω))}( \|Λ_h^s-1(B^1,B^2)\|^2_L^∞_T(H^{1(Ω))}+\| Λ_h^s-1(B^1_t,B^2_t)\|^2_L^2_T(L^{2(Ω))})\bigg{ )}.\end{split} \tag{1} $$* * Proof* The proof of Theorem 3.1 is divided into four steps. Step 1: Removing the divergence of the velocity In order to remove the divergence of the velocity in the third equation of (3.1), we look for $U=(U^1,U^2,U^3)^T$ under the form $U=∇Ψ$ solving $$ ∇· U=F_2, $$ so $Ψ$ satisfies $$ ΔΨ=F_2 in Ω. $$ We impose $Ψ$ by the boundary condition $$ Ψ|_Σ_0=0, \mbox{and} ∂_1Ψ|_Σ_{b}=0 $$ to make sure that $U^1=0$ on $Σ_b$ . Applying Lemma 2.7 to the system (3.2)-(3.3) yields $$ \|Λ_h^s-1∇Ψ\|_H^1(Ω)≤sssim\|Λ_h^s-1F_2 \|_L^2(Ω), \|Λ_h^s-1∇^2Ψ\|_H^1(Ω) ≤sssim\|Λ_h^s-1F_2\|_H^1(Ω), $$ and then $$ \|Λ_h^s-1U\|_H^1(Ω)≤sssim\|Λ_h^s-1F_2\|_L^2 (Ω), \|Λ_h^s-1U\|_H^2(Ω)≤sssim\|Λ_h^s -1F_2\|_H^1(Ω). $$ Thanks to Lemma 2.3, we have $$ \begin{split}\|Λ_h^s-1F_2\|_L^2(Ω)&≤sssim\|Λ_h^ {s-1}B^1\|_L^∞_x_{1L^2_h}\|Λ_h^s-1∇ K^(1)\|_ {L^2}≤sssim\|Λ_h^s-1B^1\|_H^1\|Λ_h^s-1∇ K^ (1)\|_L^2,\\ \|Λ_h^s-1F_2\|_H^1(Ω)&≤sssim\|Λ_h^s-1F_2\|_ {L^2(Ω)}+\|Λ_h^s-1(∇ B^1:∇ K^(1), B^1: ∇^2 K^(1))\|_L^2\\ &≤sssim\|Λ_h^s-1B^1\|_H^1\|Λ_h^s-1∇ K^(1)\| _H^1,\end{split} \tag{1} $$ which along with (3.4) implies $$ \begin{split}&\|Λ_h^s-1U\|_L^∞_T(H^{1(Ω))}+\|Λ _h^s-1U\|_L^2_T(H^{2(Ω))}\\ &≤sssim\|Λ_h^s-1B^1\|_L^∞_T(H^{1(Ω))}(\|Λ_ {h}^s-1∇ K^(1)\|_L^∞_T(L^{2(Ω))}+\|Λ_h^s-1 ∇ K^(1)\|_L^2_T(H^{1(Ω))}).\end{split} \tag{1} $$ On the other hand, since $$ \begin{split}&\|Λ_h^s-1∇Ψ_t\|^2_L^2(Ω)=-⟨ Λ_h^s-1ΔΨ_t,Λ_h^s-1Ψ_t⟩\\ &=(Λ_h^s-1(B^1K^(1)_t),Λ_h^s-1∇Ψ_t)_L^2 (Ω)-(Λ_h^s-1(B^1_t:∇ K^(1)-(∇· B^1)· K ^(1)_t),Λ_h^s-1Ψ_t)_L^2(Ω),\end{split} \tag{1} $$ one has $$ \begin{split}&\|Λ_h^s-1∇Ψ_t\|_L^2(Ω)≤\|Λ _h^s-1B^1_t\|_L^2(Ω)\|Λ_h^s-1K^(1)\|_H^1( Ω)+\|Λ_h^s-1B^1\|_H^1(Ω)\|Λ_h^s-1K^(1)_ {t}\|_L^2(Ω),\end{split} \tag{1} $$ which follows that $$ \begin{split}\|Λ_h^s-1U_t\|_L^2([0,T];L^{2(Ω))}≤sssim& \|Λ_h^s-1B^1_t\|_L^2([0,T];L^{2(Ω))}\|Λ_h^s-1 K^(1)\|_L^∞([0,T];H^{1(Ω))}\\ &+\|Λ_h^s-1B^1\|_L^∞([0,T];H^{1(Ω))}\|Λ_h^s -1K^(1)_t\|_L^2([0,T];L^{2(Ω))}.\end{split} \tag{1} $$ Hence, combining (3.5) and (3.6), we have $$ \begin{split}&\|Λ_h^s-1U\|^2_L^∞_T(H^{1(Ω))}+\| Λ_h^s-1U\|^2_L^2_T(H^{2(Ω))}+\|Λ_h^s-1U_t\| ^2_L^2_T(L^{2(Ω))}\\ &≤sssim(\|Λ_h^s-1∇ K^(1)\|^2_L^∞_T(L^{2(Ω ))}+\|Λ_h^s-1∇ K^(1)\|^2_L^2_T(H^{1(Ω))}+\| Λ_h^s-1K^(1)_t\|^2_L^2_T(L^{2(Ω))})\\ & ×\|Λ_h^s-1B^1\|^2_L^∞_T(H^{1(Ω) )}+\|Λ_h^s-1B^1_t\|^2_L^2_T(L^{2(Ω))}\|Λ_h^ {s-1}K^(1)\|^2_L^∞_T(H^{1(Ω))}.\end{split} \tag{1} $$ Similarly, for any $t_1, t_2∈[0,T]$ , we may prove $$ \begin{split}\|Λ_h^s-1(U(t_1)-U(t_2))\|_H^1&≤sssim\| Λ_h^s-1(B^1(t_1):∇ K^(1)(t_1)-B^1(t_2):∇ K^(1 )(t_2))\|_L^2\\ &≤sssim\|Λ_h^s-1(B^1(t_1)-B^1(t_2))\|_H^1\|Λ_h ^s-1∇ K^(1)(t_1)\|_L^2\\ & +\|Λ_h^s-1B^1(t_2)\|_H^1\|Λ_h^s-1∇(K^ (1)(t_1)-K^(1)(t_2))\|_L^2,\end{split} \tag{1} $$ which leads to $Λ_h^s-1U∈C([0,T];H^1(Ω)).$ With $U$ above in hand, we set $$ v:=u-U, \widetilde{F}_1:=F_1-U_t+ν∇·D(U), and \widetilde{F}_3:=F_3+νD(U)n_0, $$ then the system (3.1) can be reduced to finding $(ξ^1,v,P)$ such that $$ \begin{cases}∂_tξ^1-v^1=U^1& in [0,T]×Σ _0,\\ ∂_tv-ν∇·D(v)+∇ P=\widetilde{F}_1& in [0,T]×Ω,\\ ∇· v=0& in [0,T]×Ω,\\ (p-g ξ^1) n_0-νD(v)n_0=\widetilde{F}_3& on [0,T]×Σ_0,\\ v^1=0, v^2=-U^2, v^3=-U^3& on [0,T]×Σ_b,\\ v|_t=0=u_0-U_0& in Ω,\\ ξ^1|_t=0=ξ^1_0& on Σ_0.\end{cases} $$ Step 2: Homogenizing the upper boundary conditions We denote $Ξ A(t) (∀ t∈ℝ)$ the extension of a function $A(t) (∀ t∈[0,T])$ by $$ Ξ A(t)\buildrel\hbox{ def}\over{=}\begin{cases}(3A(-t)-2A(-2t)) χ_0(t)& if t<0,\\ A(t)& if 0≤ t≤ T,\\ (3A(2T-t)-2A(3T-2t))χ_T(t)& if t>T,\\ \end{cases} $$ where $χ_0(t)∈ C_c^∞((-∞,0])$ and $χ_T(t)∈ C_c^∞([T,∞))$ are defined by $$ χ_0(t)\buildrel\hbox{ def}\over{=}\begin{cases}1& if t∈[-\frac{1}{4}T,0],\\ ∈[0,1]& if t∈[-\frac{1}{2}T,-\frac{1}{4}T],\\ 0& if t∈(-∞,-\frac{1}{2}T],\end{cases} and χ_T(t) \buildrel\hbox{ def}\over{=}\begin{cases}1& if t∈[T, \frac{5}{4}T],\\ ∈[0,1]& if t∈[\frac{5}{4}T,\frac{3}{2}T],\\ 0& if t∈[\frac{3}{2}T,+∞)\end{cases} $$ respectively. We apply the extension operator $Ξ$ to $B^1,K^(1),B^2,K^(2),Ψ,U$ , and keep the same notation for the extension of $\widetilde{F}_3(t)=B^2:∇_hK^(2)+νD(U)n_0$ ( $∀ t∈ℝ$ ). Denote two smooth cut-off function by $$ χ_b(x_1)\buildrel\hbox{ def}\over{=}\begin{cases}1,& if x_1∈[-\underline{b},-\frac{2}{3}\underline{b});\\ ∈[0,1],& if x∈(-\frac{2}{3}\underline{b},-\frac{1}{3}\underline{b}] ;\\ 0,& if x_1∈(-\frac{1}{3}\underline{b},0],\end{cases} χ_f(x )\buildrel\hbox{ def}\over{=}\begin{cases}1,& if x∈(- \frac{2}{3}\underline{b},0];\\ ∈[0,1],& if x∈(-\frac{5}{6}\underline{b},-\frac{2}{3}\underline{b}] ;\\ 0,& if x∈[-\underline{b},-\frac{5}{6}\underline{b}),\end{cases} $$ with $|\frac{d^n}{dx_1^n}(χ_b(x_1),χ_f(x_1))|≤sssim\underline{ b}^-n$ for any $n∈ℕ$ . Notice that $χ_f(Supp{χ^\prime_b})≡ 1$ . We also set $Ω_b:=Ω χ_b$ with the measure $χ_b dx$ and $Ω_f:=Ω χ_f$ with the measure $χ_f dx$ , and define $φ_1,b:=φ_1χ_b$ and $φ_1,f:=φ_1χ_f$ . Our next goal is to construct $(V,P_1)$ satisfying $$ \begin{cases}∇· V=0& in ℝ×Ω,\\ P_1n_0-νD(V)n_0=\widetilde{F}_3& on ℝ ×Σ_0,\\ V^1=0,V^2=-U^2,V^3=-U^3& on ℝ×Σ_b, \end{cases} $$ which is equivalent to $$ \begin{cases}∇· V=0& in ℝ×Ω,\\ -ν(∂_1V^β+∂_βV^1)=\widetilde{F}_3^β ( \mbox{for} β=2,3)& on ℝ×Σ_0,\\ P_1=\widetilde{F}_3^1+2ν∂_1V^1& on ℝ ×Σ_0,\\ V^1=0,V^2=-U^2,V^3=-U^3& on ℝ×Σ_b. \end{cases} $$ And our construction and estimates of $V(=V_1+V_2)$ is divided into three parts, the first two parts show the construction $V_1$ and $V_2$ and the estimates of $\|Λ_h^s-1V_t\|_L^2_T(L^{2(Ω))}$ and $\|Λ_h^s-1V\|_L^2_T(H^{2(Ω))}$ , and the third part shows the estimates of $\|Λ_h^s-1V\|_L^∞_T(H^{1(Ω))}$ . Step 2.1:The construction of $V_1$ and the estimates of $\|Λ_h^s-1(∂_tV_1,∇ V_1,∇^2V_1)\|_L^2_ T(L^{2(Ω))}$ Motivated by [10], we first look for $V_1=(V_1^1,V_1^2,V_1^3)^T$ under the form $V_1=∇×φ$ with $φ=(0,φ^2,φ^3)^T$ satisfying $$ \begin{cases}∂_tφ-Δφ=0& in ℝ ×Ω,\\ ∂_1∂_1φ^3-∂_2∂_2φ^2+ ∂_2∂_3φ^2=\frac{1}{ν}\widetilde{F}_3^2& on ℝ×Σ_0,\\ ∂_1∂_1φ^2-∂_3∂_3φ^2+ ∂_3∂_2φ^3=-\frac{1}{ν}\widetilde{F}_3^3& on ℝ×Σ_0.\end{cases} $$ Denoting the trace of $φ$ on $ℝ×Σ_0$ by $Φ=(0,Φ^2,Φ^3)^T$ and combining with (3.11), we have $$ \begin{cases}∂_tΦ^2-Δ_hΦ^2-∂_3∂_3 Φ^2+∂_3∂_2Φ^3=\frac{1}{ν}\widetilde{F}_3^3& on ℝ×Σ_0,\\ ∂_tΦ^3-Δ_hΦ^3-∂_2∂_2Φ^3+ ∂_2∂_3Φ^2=-\frac{1}{ν}\widetilde{F}_3^2&on ℝ×Σ_0.\end{cases} $$ Set $Φ_h:=(Φ^2,Φ^3)^T$ . According to the Helmholtz-Hodge decomposition in $H^s(Σ_0)$ , we have $$ Φ_h=ℙ_0Φ_h+∇_hπ on ℝ× Σ_0, $$ and then $$ ∇_h·Φ_h=Δ_hπ on ℝ×Σ_ 0, $$ where $ℙ_0$ is Leray’s projection operator in $L^2(Σ_0)$ . Applying $-∇_h(-Δ_h)^-1∇_h·$ to both sides of the linear system (3.12), we find $$ ∂_t(∇_hπ)-Δ_h(∇_hπ)=-\frac{1}{ν}∇_h (-Δ_h)^-1∇_h·\widetilde{F}_3^⊥ on ℝ×Σ_0, $$ where $\widetilde{F}_3^⊥:=(\widetilde{F}_3^3,-\widetilde{F}_3^2)^T$ . Thanks to (3.13), one finds $$ \begin{cases}∂_2(∂_2Φ^2+∂_3Φ^3)=∂_ {2}Δ_hπ on ℝ×Σ_0,\\ ∂_3(∂_2Φ^2+∂_3Φ^3)=∂_3Δ_h π on ℝ×Σ_0,\\ \end{cases} $$ which along with (3.12) implies $$ ∂_tΦ_h-2Δ_hΦ_h=\frac{1}{ν}\widetilde{F}_3^⊥ -Δ_h∇_hπ on ℝ×Σ_0. $$ The classical energy estimate of (3.15) shows that $$ \|(∇_hΦ,∇_h^2Φ)\|_L^2(ℝ;H^{s-\frac{1{2}}( Σ_0))}^2+\|Φ_t\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0) )}^2≤sssim\|(\widetilde{F}_3^⊥,Δ_h∇_hπ)\|_L^2( ℝ;H^{s-\frac{1{2}}(Σ_0))}^2. $$ While the energy estimate of (3.14) gives $$ \|Δ_h∇_hπ\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0))}^ {2}≤sssim\|\widetilde{F}_3^⊥\|_L^2(ℝ;H^{s-\frac{1{2}}( Σ_0))}^2, $$ which along with (3.16) gives rise to $$ \|(∇_hΦ,∇_h^2Φ)\|_L^2(ℝ;H^{s-\frac{1{2}}( Σ_0))}^2+\|Φ_t\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0) )}^2≤sssim\|\widetilde{F}_3\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ _0))}^2. $$ Substituting $V_1=∇×φ$ with $φ=(0,φ^2,φ^3)^T$ into the forth equation in (3.10) gives rise to $$ ∂_1φ^2=-U^3, ∂_1φ^3=U^2 on ℝ×Σ_b. $$ Hence, the function $φ$ obeys the system $$ \begin{cases}∂_tφ-Δφ=0 &on ℝ ×Ω,\\ φ=Φ &on ℝ×Σ_0,\\ ∂_1φ=-n× U, &on ℝ×Σ_b. \end{cases} $$ Denote the Fourier transform in terms of $(t,x_h)$ by $\widehat{}$ or $F=F_(t,x_{h)→(τ,ς_h)}$ , and then apply the Fourier transform $F$ to the problem (3.18), so we get the second-order ODE for $\widehat{φ}$ : $$ \begin{cases}-\frac{d^2}{d{x}_1^2}\widehat{φ}+(i τ+|ς_ {h}|^2)\widehat{φ}=0 in (-b,0),\\ \widehat{φ}|_x_{1=0}=\widehat{Φ},\\ \frac{d}{d{x}_1}\widehat{φ}|_x_{1=-b}=-n×\widehat{U}|_x_{1=-b }.\end{cases} $$ Solving the ODEs (3.19), we get the explicit expression of the solution $\widehat{φ}(τ,x_1,ς_h)$ (with $τ∈ℝ$ , $ς_h∈ℝ^2$ ) that $$ \widehat{φ}(τ,x_1,ς_h)=\widehat{φ}_1+\widehat{ φ}_2 $$ with $$ \begin{split}&\widehat{φ}_1:=\frac{e^-rx_1+e^r( 2b+x_1)}{e^2rb+1}(0,\widehat{Φ}^2,\widehat{Φ}^3)^T, \\ &\widehat{φ}_2:=\frac{e^rx_1-e^-rx_1}{r( e^rb+e^-rb)}(0,-iς_3\widehat{Ψ}|_ x_{1=-b},iς_2\widehat{Ψ}|_x_{1=-b})^T,\end{split} $$ where $r$ is defined by the equation $r^2=iτ+|ς_h|^2$ so that $\arg r∈[-\frac{π}{4},\frac{π}{4}]$ , which means $$ r=(\frac{√{τ^2+|ς_h|^4}+|ς_h|^2}{2})^\frac{1 {2}}+i sgn(τ)(\frac{√{τ^2+|ς_h|^4}-| ς_h|^2}{2})^\frac{1{2}}. $$ Let’s now get estimates of $φ$ by using the form (3.20) of $φ$ . We first deal with the estimate of $\|Λ_h^s-1(∇^2φ,∇^2∇_hφ,∂_t φ,∂_t∇_hφ,∂_t∇φ_2,∇^3 φ_2)\|_L^2(ℝ×Ω)^2$ . Thanks to Plancherel’s identity, one has $$ \begin{split}&\|Λ_h^s-1(∂_t∇_hφ,∇^2 ∇_hφ)\|_L^2(ℝ×Ω)\\ &≤sssim\big{\|}(χ_\{Re r≤ 1\}+1-χ_\{Re r≤ 1 \})|r|^2|ς_h|⟨ς_h⟩^s-1(\widehat{φ}_ {1}+\widehat{φ}_2)\big{\|}_L^2(ℝ×Ω).\end{split} $$ On the one hand, due to $\|χ_\{Re r≤ 1\}(|ς_h|^\frac{1{2}}|e^- rx_1+e^r(2b+x_1)|)|e^2rb+1|^-1\|_L^∞( ℝ×Ω)≤sssim 1$ , we get $$ \begin{split}\big{\|}χ_\{Re r≤ 1\}|r|^2|ς_h| ⟨ς_h⟩^s-1\widehat{φ}_1\big{\|}_L^2( R×Ω)≤sssim b^\frac{1{2}}\||r|^2⟨ς_h⟩^ s-\frac{1{2}}\hat{Φ}\|_L_{τ^2(L^2_h)}.\end{split} $$ Similarly, one obtains $$ \begin{split}\big{\|}(1-χ_\{Re r≤ 1\})|r|^2|ς_h| ⟨ς_h⟩^s-1\widehat{φ}_2\big{\|}_L^2( R×Ω)≤sssim b^\frac{1{2}}\||r|^2⟨ς_h⟩^ s-\frac{1{2}}\hat{Ψ}|_x_{1=-b}\|_L_{τ^2(L^2_h)},\end{split} $$ and $$ \begin{split}\big{\|}(1-χ_\{Re r≤ 1\})|r|^3⟨ ς_h⟩^s-1\widehat{φ}_2\big{\|}_L^2(ℝ ×Ω)≤sssim b^\frac{1{2}}\||r|^2⟨ς_h⟩^s+ \frac{1{2}}\hat{Ψ}|_x_{1=-b}\|_L_{τ^2(L^2_h)}.\end{split} $$ On the other hand, according to (3.21), we have $$ \begin{split}&\big{\|}(1-χ_\{Re r≤ 1\})|r|^2|ς_h |⟨ς_h⟩^s-1\widehat{φ}_1\big{\|}_L^2(\mathbb {R×Ω)}\\ &≤sssim\|(1-χ_\{Re r≤ 1\})|ς_h|^\frac{1{2}} | e^2rb+1|^-1\|e^-rx_1+e^r(2b+x_1)\|_L_{ x_1^2}\|_L^∞_τ,h\||r|^2⟨ς_h⟩^s- \frac{1{2}}\hat{Φ}\|_L_{τ,h^2}\\ &≤sssim\||r|^2⟨ς_h⟩^s-\frac{1{2}}\hat{Φ}\|_L_{ τ^2(L^2_h)}.\end{split} $$ Along the same lines, we may prove that $$ \begin{split}&\big{\|}χ_\{Re r≤ 1\}|r|^2|ς_h| ⟨ς_h⟩^s-1\widehat{φ}_2\big{\|}_L^2( R×Ω)≤sssim\||r|^2⟨ς_h⟩^s-\frac{1{2}} \hat{Ψ}|_x_{1=-b}\|_L_{τ^2(L^2_h)},\\ &\big{\|}χ_\{Re r≤ 1\}|r|^3⟨ς_h⟩^s- 1\widehat{φ}_2\big{\|}_L^2(ℝ×Ω)≤sssim\||r|^ 2⟨ς_h⟩^s-\frac{1{2}}\hat{Ψ}|_x_{1=-b}\|_L_{τ ^2(L^2_h)}.\end{split} $$ Plugging (3.24),(3.25),(3.27),(3.28) into (3.23) implies $$ \begin{split}\|Λ_h^s-1(∂_t∇_hφ,∇^2 ∇_hφ)\|^2_L^2(ℝ×Ω)&≤sssim\|∇_h^ {2}(Φ,Ψ)\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0))}^2+\|(Φ _t,Ψ_t)\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0))}^2.\end{split} $$ Combining (3.26) and (3.28), we obtain $$ \begin{split}\|Λ_h^s-1(∂_t∇φ_2,∇^3 φ_2)\|^2_L^2(ℝ×Ω)&≤sssim\|∇_h^2( Φ,Ψ)\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0))}^2+\|(Φ_t ,Ψ_t)\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_0))}^2.\end{split} $$ Similarly, we have $$ \begin{split}\|Λ_h^s-1(∂_tφ,∇^2φ)\|^2_ {L^2(ℝ×Ω)}≤sssim\|∇_h^2(Φ,Ψ)\|_L^2( ℝ;H^{s-\frac{1{2}}(Σ_0))}^2+\|(Φ_t,Ψ_t)\|_L^2( ℝ;H^{s-\frac{1{2}}(Σ_0))}^2.\end{split} $$ Hence, thanks to (3.17), (3.20), and Plancherel’s identity, we get $$ \begin{split}&\|Λ_h^s-1(∇^2φ,∇^2∇_h φ,∂_tφ,∂_t∇_hφ,∂_t∇ φ_2,∇^3φ_2)\|_L^2(ℝ×Ω)^2\\ &≤sssim\|Λ^s-1_h(\widetilde{F}_3,F_2)\|_L^2_T(H^{1( Ω))}^2+\|Λ_h^s-1B^(1)_t\|^2_L^2_T(L^{2(Ω))} \|Λ_h^s-1K^(1)\|^2_L^∞_T(H^{1(Ω))}\\ & +\|Λ_h^s-1B^(1)\|^2_L^ ∞_T(H^{1(Ω))}\|Λ_h^s-1K^(1)_t\|^2_L^2_T(L^{ 2(Ω))}.\end{split} \tag{1} $$ In order to get the full regularity of $φ$ , we need to get the estimates of $Λ_h^s-1∂_t∂_1φ_1$ and $Λ_h^s-1∂_1∂_1∇φ_1$ . Due to $∂_1∂_1∇φ_1=∂_t∇φ_1-Δ _h∇φ_1$ , we just need to recover the estimate of $Λ_h^s-1∂_t∂_1φ_1$ . Due to (3.20), we can get the estimate of $∂_t∂_1φ_1$ near the bottom as follows $$ \begin{split}\|Λ_h^s-1∂_t∂_1φ_1,b\|_L^2 (ℝ×Ω)^2≤sssim&\|χ_bΛ_h^s-1∂_t ∂_1φ_1\|_L^2(ℝ×Ω)^2+\|χ^\prime_ {b}Λ_h^s-1∂_tφ_1\|_L^2(ℝ×Ω)^ {2}\\ ≤sssim&\|\widetilde{F}_3^⊥\|_L^2_T(H^{s-\frac{1{2}}(Σ_0 ))}^2,\end{split} $$ which implies $$ \begin{split}\|Λ_h^s-1∂_t∂_1φ_1\|_L^2( ℝ×Ω_b)^2≤sssim\|\widetilde{F}_3^⊥\|_L^2_ T(H^{s-\frac{1{2}}(Σ_0))}^2.\end{split} $$ We are now in a position to estimate $∂_t∂_1φ_1$ near the upper boundary. Let’s first consider the function $ψ=(0,ψ^2,ψ^3)^T$ with $$ \begin{cases}ψ^2:=∂_tφ_1^2-Δ_hφ_1^2- ∂_3∂_3φ_1^2+∂_3∂_2φ_1^3 -\frac{1}{ν}\widetilde{F}_3^3,\\ ψ^3:=∂_tφ^3-Δ_hφ_1^3-∂_2 ∂_2φ_1^3+∂_2∂_3φ_1^2+\frac{1}{ ν}\widetilde{F}_3^2.\end{cases} $$ which implies that $ψ_f:=ψχ_f$ satisfies $$ \begin{cases}(∂_t-Δ)ψ_f=\mathfrak{F}_1+∇· \mathfrak{F}_2 (∀ (t,x)∈ℝ×Ω),\\ ψ_f|_x_{1=0}=0,\\ ψ_f|_x_{1=-b}=0,\end{cases} $$ where $$ \mathfrak{F}_1=(0,\Re^2-\frac{1}{ν}(B^2,3_t:∇_hK^(2)- ∂_αB^2,3_α,i(K^(2))^i_t)χ_f,\Re^3+\frac{1} {ν}(B^2,2_t:∇_hK^(2)-∂_αB^2,2_α,i(K^(2) )^i_t)χ_f)^T \tag{2} $$ and $\mathfrak{F}_2$ is a $3× 3$ matrix defined by $$ \mathfrak{F}_2=\frac{1}{ν}≤ft(\begin{array}[]{cccc}0&0&0\\ -∂_1(\widetilde{F}_3^3χ_f)&-(∂_2\widetilde{F}_3^ 3+(B^2,3_2,i(K^(2))_t^i)χ_f&-(2ν U^1_t+∂_3 \widetilde{F}_3^3+B^2,3_3,i(K^(2))_t^i)χ_f\\ ∂_1(\widetilde{F}_3^2χ_f)&(2ν U^1_t+∂_2 \widetilde{F}_3^2+B^2,2_2,i(K^(2))_t^i)χ_f&(∂_3 \widetilde{F}_3^2+(B^2,2_3,i(K^(2))_t^i)χ_f\end{array} \right), \tag{2} $$ and $\Re=(0,\Re^2,\Re^3)^T$ with $$ \begin{split}\Re^2:=-(2∂_1φ_1,t^2χ^\prime_f+ φ_1,t^2χ^\prime\prime_f)-(∂_2^2+2∂_3^2) (2∂_1φ_1^2χ_f^\prime+φ_1^2χ_f^\prime \prime)+∂_3∂_2(2∂_1φ_1^3χ_f^\prime +φ_1^3χ_f^\prime\prime),\\ \Re^3:=-(2∂_1φ_1,t^3χ^\prime_f+φ_1,t^3 χ^\prime\prime_f)-(∂_3^2+2∂_2^2)(2∂_1 φ_1^3χ_f^\prime+φ_1^3χ_f^\prime\prime)+ ∂_2∂_3(2∂_1φ_1^2χ_f^\prime+φ _1^2χ_f^\prime\prime).\end{split} $$ By acting the Fourier transform $F_t→τ$ with respect to time variable $t$ to the system (3.32) , we obtain $$ -iτ\widehat{ψ_f}-Δ\widehat{ψ_f}=\widehat{\mathfrak{ F}_1}+∇·\widehat{\mathfrak{F}_2}. $$ Then, we multiply the $i$ th component of the equations of (3.33) by the conjugate of $\widehat{ψ_f}^i$ , sum over $i$ , and integrate over $Ω$ to find $$ \begin{split}-∫_Ωiτ|\widehat{ψ_f}|^2 dx+∫_ Ω|∇\widehat{ψ_f}|^2 dx=(\widehat{\mathfrak{F}_1},\widehat {ψ_f})_L^2+(\widehat{\mathfrak{F}_2},∇\widehat{ψ_f})_L^ 2\end{split} $$ Keeping the real part and integrating both sides with respect to $τ$ over $ℝ$ , we have $$ \begin{split}\|∇ψ_f\|_L^2(ℝ;L^{2(Ω))}^2&=∫_ ℝRe(\widehat{\mathfrak{F}_1},\widehat{ψ_f})_L^2 dτ+∫_ℝRe(\widehat{\mathfrak{F}_2},∇\widehat {ψ_f})_L^2 dτ\\ &≤sssim\|\widehat{\mathfrak{F}_1}\|_L^2(ℝ;L^{2(Ω))}\| \widehat{ψ_f}\|_L^2(ℝ;L^{2(Ω))}+\|\widehat{\mathfrak{F} _2}\|_L^2(ℝ;L^{2(Ω))}\|∇\widehat{ψ_f}\|_L^2( ℝ;L^{2(Ω))}\\ &≤sssim\|\mathfrak{F}_1\|_L^2(ℝ;L^{2(Ω))}\|ψ_f\|_L ^2(ℝ;L^{2(Ω))}+\|\mathfrak{F}_2\|_L^2(ℝ;L^{2( Ω))}\|∇ψ_f\|_L^2(ℝ;L^{2(Ω))},\end{split} $$ which along with Poincaré inequality leads to $$ \|∇ψ_f\|^2_L^2(ℝ;L^{2(Ω))}≤sssim\|\mathfrak{F} _1\|^2_L^2(ℝ,L^{2(Ω))}+\|\mathfrak{F}_2\|^2_L^2( ℝ;L^{2(Ω))}. $$ Along the same lines, we obtain $$ \begin{split}&\|Λ_h^s-1∇ψ_f\|_L^2(ℝ,L^{2( Ω))}^2≤sssim\|Λ_h^s-1\mathfrak{F}_1\|_L^2(ℝ;L ^{2(Ω))}^2+\|Λ_h^s-1\mathfrak{F}_2\|_L^2(ℝ;L^{ 2(Ω))}^2,\end{split} $$ where $$ \begin{split}&\|Λ_h^s-1\mathfrak{F}_1\|_L^2(ℝ;L^{2( Ω))}^2+\|Λ_h^s-1\mathfrak{F}_2\|_L^2(ℝ;L^{2( Ω))}^2\\ &≤sssim\|Λ_h^s-1\widetilde{F}_3\|_L^2(ℝ;H^{1(Ω ))}^2+\|Λ_h^s-1U_t\|_L^2(ℝ;L^{2(Ω))}^2+\| Λ_h^s-1B^2_t\|_L^2(ℝ;L^{2(Ω))}^2\|Λ_h ^s-1∇_hK^(2)\|_L^∞(ℝ;L^{2(Ω))}^2\\ &+\|Λ_h^s-1B^2\|_L^∞(ℝ;H^{1(Ω))}^2\| Λ_h^s-1K^(2)_t\|_L^2(ℝ;L^{2(Ω))}^2.\end{split} \tag{2} $$ Combining (3.29) with (3.31) yields $$ \begin{split}&\|Λ_h^s-1∂_t∂_1φ_1\|_L^2( ℝ;L^{2(Ω_f))}^2≤sssim\|Λ_h^s-1(\widetilde{F}_3 ,F_2)\|_L^2(ℝ;H^{1(Ω))}^2\\ & +\|Λ_h^s-1B^2_t\|_L^2_T(L^{2(Ω))}^2\|Λ_ {h}^s-1∇_hK^(2)\|_L^∞_T(L^{2(Ω))}^2+\|Λ_h ^s-1B^2\|_L^∞_T(H^{1(Ω))}^2\|Λ_h^s-1K^(2)_t \|_L^2_T(L^{2(Ω))}^2\\ & +\|Λ_h^s-1B^1_t\|^2_L^2_T(L^{2(Ω))}\|Λ_ {h}^s-1K^(1)\|^2_L^∞_T(H^{1(Ω))}+\|Λ_h^s-1B^1 \|^2_L^∞_T(H^{1(Ω))}\|Λ_h^s-1K^(1)_t\|^2_L ^2_T(L^{2(Ω))}.\end{split} \tag{2} $$ which along with (3.30) leads to $$ \begin{split}&\|Λ_h^s-1∂_t∂_1φ_1\|_L^2( ℝ;L^{2(Ω))}^2+\|Λ_h^s-1∂_1∂_1 ∇φ_1\|_L^2(ℝ;L^{2(Ω))}^2\\ &≤sssim\|Λ_h^s-1(\widetilde{F}_3,F_2)\|_L^2(ℝ;H^{1 (Ω))}^2+\|Λ_h^s-1(B^1_t,B^2_t)\|^2_L^2_T(L^{2 (Ω))}\|Λ_h^s-1(K^(1),K^(2))\|^2_L^∞_T(H^{1( Ω))}\\ & +\|Λ_h^s-1(B^1,B^2)\|^2_L^∞_T(H^{1( Ω))}\|Λ_h^s-1(K^(1)_t,K^(2)_t)\|^2_L^2_T(L^{2( Ω))}.\end{split} \tag{1} $$ So far, we get the sufficient regularity of $φ$ , and then, by the construction of $V_1$ , we have $$ \begin{split}F(V_1)(τ,x_1,ς_h)=F(∇ ×φ)=(iς_2\hat{φ}^3-iς_ 3\hat{φ}^2,-∂_1\hat{φ}^3,∂_1\hat{φ}^ 2)^T.\end{split} $$ which along with (3.29) and (3.34) implies $$ \begin{split}&\|Λ_h^s-1∂_tV_1\|_L^2(ℝ;L^{2( Ω))}^2+\|Λ_h^s-1∇ V_1\|_L^2(ℝ;H^{1(Ω ))}^2\\ &≤sssim\|Λ_h^s-1(\widetilde{F}_3,F_2)\|_L^2(ℝ;H^{1 (Ω))}^2+\|Λ_h^s-1(B^1_t,B^2_t)\|^2_L^2_T(L^{2 (Ω))}\|Λ_h^s-1(K^(1),K^(2))\|^2_L^∞_T(H^{1( Ω))}\\ & +\|Λ_h^s-1(B^1,B^2)\|^2_L^∞_T(H^{1( Ω))}\|Λ_h^s-1(K^(1)_t,K^(2)_t)\|^2_L^2_T(L^{2( Ω))}.\end{split} \tag{1} $$ Step 2.2:The construction of $V_2$ and the estimate of $\|Λ_h^s-1(∂_tV_2,∇ V_2,∇^2V_2)\|_L^2_ T(L^{2(Ω))}$ However, we find that $V_1$ might not satisfy $V_1^1=0$ on $ℝ×Σ_b$ , so we need construct a suitable divergence-free vector field $V_2=(V_2^1,V_2^2,V_2^3)^T$ satisfying $$ \begin{cases}∇· V_2=0&in ℝ×Ω,\\ ∂_1V_2^β+∂_βV_2^1=0&on ℝ ×Σ_0 (β=2,3),\\ V_2^β=0&on ℝ×Σ_b,\\ V_2^1=-V_1^1&on ℝ×Σ_b.\end{cases} $$ Take the Fourier transformation $\widehat{·}$ in terms of $(t,x_h)$ to (3.36), and define $$ \begin{split}\widehat{V}^1_2\buildrel\hbox{ def}\over{=}-χ _b\widehat{V}_1^1,\end{split} $$ we have $$ \begin{cases}iς_2\widehat{V}_2^2+iς_3\widehat{V}_2^ 3=-∂_1\widehat{V}^1_2&in ℝ×Ω,\\ ∂_1\widehat{V}_2^β=-iς_β\widehat{V}_2^1=0& on ℝ×Σ_0 (β=2,3),\\ V_2^β=0&on ℝ×Σ_b.\end{cases} $$ Let $V_2^β\buildrel\hbox{ def}\over{=}∂_βφ-(- Δ_h)^-1∂_β∂_1^2φ$ , that is, $$ \begin{split}\widehat{V}^β_2=iς_β\widehat{φ}-\frac{i ς_β}{|ς_h|^2}∂_1^2\widehat{φ}, ( β=2,3),\end{split} $$ where $φ$ satisfy $$ \begin{cases}\frac{d^2}{dx_1^2}\widehat{φ}-| ς_h|^2\widehat{φ}=-∂_1\widehat{V}^1_2&in ℝ×Ω,\\ ∂_1\widehat{φ}=0&on ℝ×Σ_0,\\ \widehat{φ}=0&on ℝ×Σ_b.\end{cases} $$ By solving (3.39), we can get $$ \begin{split}\widehat{φ}=&\frac{e^|ς_h|(x_1+b)- e^|ς_h|(-x_1+b)}{2|ς_h|(e^| ς_h|b+e^-|ς_h|b)}∫_-b^0∂_1 \widehat{V}_2^1(e^|ς_h|\bar{x_1}+e^-| ς_h|\bar{x_1})d\bar{x}_1\\ &+\frac{1}{2|ς_h|}∫_-b^x_1∂_1\widehat{V}_2^1( e^|ς_h|(\bar{x_1-x_1)}+e^-|ς_h|( \bar{x_1-x_1)})d\bar{x}_1\end{split} $$ then we can verify that $\widehat{V}^β$ $(β=2,3)$ defined by (3.38) satisfies (3.37), and combining with $$ \begin{split}\widehat{V}_2^1=-χ_b\widehat{V}_1^1=-χ_b\bigg{( }\frac{e^-rx_1+e^r(2b+x_1)}{e^2rb+1}(i ς_2\widehat{Φ}^3-iς_3\widehat{Φ}^2)-\frac{| ς_h|^2(e^rx_1-e^-rx_1)}{r(e^rb +e^-rb)}\widehat{Ψ}|_x_{1=-b}\bigg{)},\end{split} $$ we can get $$ \begin{split}&\|Λ_h^s-1∂_tV_2\|_L^2_T(L^{2(Ω)) }^2+\|Λ_h^s-1∇ V_2\|_L^2_T(H^{1(Ω))}^2\\ &≤sssim\|∇_h(Φ,Ψ)\|_L^2(ℝ;H^{s+\frac{1{2}}(Σ_ {0}))}^2+\|(Φ_t,Ψ_t)\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_ {0}))}^2.\end{split} $$ Moreover, we can also obtain $$ \begin{split}&\|Λ_h^s-1V_2(0)\|^2_H^1(Ω)+\|Λ_h^ {s-1}∇ V_2\|^2_L^2(Ω;L^{∞_t)}\\ &≤sssim\|∇_h(Φ,Ψ)\|_L^2(ℝ;H^{s+\frac{1{2}}(Σ_ {0}))}^2+\|(Φ_t,Ψ_t)\|_L^2(ℝ;H^{s-\frac{1{2}}(Σ_ {0}))}^2,\end{split} \tag{0} $$ and according to Lebesgue dominated convergence theorem, we have $\|Λ_h^s-1∇ V_2(t)\|_H^1(Ω)$ is continuous on $[0,T]$ . Taking $V=V_1+V_2$ and $P_1=2ν∂_1V^1+\widetilde{F}_3^1$ , we can see that $(V,P_1)$ satisfies (3.10) and $$ \begin{split}&\|Λ_h^s-1∂_tV\|_L^2_T(L^{2(Ω))}^2 +\|Λ_h^s-1∇ V\|_L^2_T(H^{1(Ω))}^2+\|Λ_h^ s-1P_1\|_L^2_T(H^{1(Ω))}^2\\ &≤sssim\|Λ_h^s-1(\widetilde{F}_3,U)\|_L^2_T(H^{1(Ω))} ^2+\|Λ_h^s-1B^2_t\|_L^2(L^{2(Ω))}^2\|Λ_h^s -1∇_hK^(2)\|_L^∞_T(L^{2(Ω))}^2\\ & +\|Λ_h^s-1B^2\|_L^∞_T(H^{1(Ω))}^2\|Λ _h^s-1K^(2)_t\|_L^2_T(L^{2(Ω))}^2,\end{split} \tag{2} $$ which along with Sobolev’s embedding theorem lead to $Λ_h^s-1V∈C([0,T];L^2(Ω))$ . Step 2.3:The estimates of $\|Λ_h^s-1V\|_L^∞_T(H^{1(Ω))}$ Next, we want to consider the continuity of $\|Λ_h^s-1∇ V(t)\|_L^2(Ω)$ in $[0,T]$ . However, by using (3.20) and (3.35) we can just get the estimate near the bottom as follow $$ \begin{split}&\|Λ_h^s-1∇ V_1(0)\|_L^2(Ω_\frac{1{2b })}+\|Λ_h^s-1∇ V_1\|_L^2(Ω_\frac{1{2b};{L}^∞ _t)}\\ &≤sssim\bigg{\|}|ς_h|^2⟨ς_h⟩^s-1\frac{ e^-rx_1+e^r(2b+x_1)}{e^2rb+1}\widehat{ Φ}χ_\{-b≤ x_{1≤-\frac{1}{2}b\}}\bigg{\|}_L^2_h(L^{2_x_{1 }(L^1_τ))}\\ &≤sssim\bigg{\|}\frac{e^Re r(2b+x_1)}{e^2 Re rb+1}χ_\{-b≤ x_{1≤-\frac{1}{2}b\}}\bigg{\|}_L^ ∞_h(L^{2_x_1(L^2_τ))}\||ς_h|^2⟨ς _h⟩^s-1\hat{Φ}\|_L^2_h(L^{2_τ)}≤sssim\|\widetilde{F} _3^⊥\|_L_{τ^2(H^s-\frac{1{2}}(Σ_0))},\end{split} \tag{0} $$ where $Ω_\frac{1{2}b}:=\{x∈Ω|-b≤ x_1≤-\frac{1}{2}b\}$ . Let’s now find the information about $Λ_h^s-1∇ V_1(t)$ near the free upper bound. On the one hand, we find that $V_1$ satisfies $$ \begin{split}\frac{d}{dt}\|Λ_h^s-1∇_hV_1 \|_L^2(Ω_f)^2&=2⟨Λ_h^s-1∇_h∂_tV_ 1,Λ_h^s-1∇_hV_1χ_f⟩\\ &=-2(Λ_h^s-1∂_tV_1,Λ_h^s-1Δ_hV_1χ_ f)_L^2(Ω)\end{split} $$ and $$ Λ_h^s-1∂_1V_1^1=-Λ_h^s-1∇_h· V_1, h. $$ According to Lemma 2.10, we know that $\|Λ_h^s-1(∂_1V_1^1,∇_hV_1)(t)\|_L^2(Ω_ {f)}$ is continuous on $[0,T]$ . Moreover, we have $$ \sup_-∞<t≤ T\|Λ_h^s-1(∂_1V_1^1,∇_hV_1 )\|^2_L^2(Ω_f)≤ 2\|Λ_h^s-1∂_tV_1\|_L^2 (-∞,T;L^{2(Ω_f))}\|Λ_h^s-1∇ V_1\|_L^2(- ∞,T;L^{2(Ω_f))} $$ On the other hand, inspired by the ”good unknowns” in [13], we define $G^β$ as $$ G^β:=∂_1V_1,f^β+∂_βV_1,f^1- \frac{1}{ν}\widetilde{F}_3,f^β (\mbox{with} β=2,3) $$ which along with the second equation in (3.10) implies $G^β|_x_{1=0}=0$ . Then, we find that $V^β$ $(β=2,3)$ satisfies $$ \begin{split}&∫_ΩΛ_h^s-1∂_tV_1,f^β· Λ_h^s-1∂_1∂_1V_1,f^βdx=∫_ ΩΛ_h^s-1∂_tV_1,f^β·Λ_h^s-1 ∂_1(G^β-∂_βV_1,f^1+\frac{1}{ν} \widetilde{F_3}^β)dx\\ &=-⟨Λ_h^s-1∂_t∂_1V^β_1,f,Λ_h ^s-1G^β⟩-∫_ΩΛ_h^s-1∂_tV_ 1,f^β·Λ_h^s-1(∂_1∂_βV_1,f^1- \frac{1}{ν}∂_1\widetilde{F}_3,f^β)dx\\ &=-\frac{1}{2}\frac{d}{dt}\|Λ_h^s-1∂_1V_ 1,f^β\|_L^2(Ω)^2-⟨Λ_h^s-1∂_t ∂_1V^β_1,f,Λ_h^s-1∂_βV^1⟩+ ⟨Λ_h^s-1∂_t∂_1V^β_1,f,\frac{1}{ν} Λ_h^s-1\widetilde{F}_3,f^β⟩\\ & -∫_ΩΛ_h^s-1∂_tV_1,f^β·Λ_ h^s-1(∂_1∂_βV_1,f^1-\frac{1}{ν}∂_1 \widetilde{F}_3,f^β)dx,\end{split} $$ which lead to $$ \begin{split}&\frac{1}{2}\frac{d}{dt}≤ft(\|Λ_h^s- 1∂_1V_1,f^β\|_L^2(Ω)^2+∫_ΩΛ_h^ s-1∂_1V_1,f^β·Λ_h^s-1∂_βV_1,f^ 1dx-∫_ΩΛ_h^s-1∂_1V_1,f^β· \frac{1}{ν}Λ_h^s-1\widetilde{F}_3,f^βdx\right)\\ & =-∫_ΩΛ_h^s-1∂_1∂_1V_1,f^ β·Λ_h^s-1∂_tV_1,f^βdx-∫_ ΩΛ_h^s-1∂_1∂_βV_1,f^β· Λ_h^s-1∂_tV^1_1,fdx\\ & +⟨\frac{1}{ν}Λ_h^s-1(∂_tF_3,f^ β+ν∂_β∂_tU^1_f+ν∂_1U_t^β χ_f),Λ_h^s-1∂_1V_1,f^β⟩\\ & -∫_ΩΛ_h^s-1∂_tV_1,f^β ·Λ_h^s-1(∂_1∂_βV_1,f^1-\frac{1}{ν} ∂_1\widetilde{F}_3,f^β)dx,\end{split} $$ where $$ \begin{split}&⟨\frac{1}{ν}Λ_h^s-1(∂_tF_3,f^β +ν∂_β∂_tU^1_f),Λ_h^s-1∂_1V_1, f^β⟩\\ =&-∫_Ω\frac{1}{ν}Λ_h^s-1(B^2K^(2)_t)χ_f· Λ_h^s-1∂_1∇_hV_1,f^βdx+∫_Ω \frac{1}{ν}Λ_h^s-1((∇_h· B^2)K_t^(2))χ_f ·Λ_h^s-1∂_1V_1,f^βdx\\ &+∫_Ω\frac{1}{ν}Λ_h^s-1(B_t^2:∇_hK^(2))χ _f·Λ_h^s-1∂_1V_1,f^βdx-∫_Ω Λ_h^s-1∂_tU^1_f·Λ_h^s-1∂_β ∂_1V_1,f^βdx,\end{split} \tag{2} $$ and $$ \begin{split}&⟨Λ_h^s-1∂_1∂_tU^βχ_f ,Λ_h^s-1∂_1V_1,f^β⟩=⟨Λ_h^s-1 ∂_1∂_β∂_tΨχ_f,Λ_h^s-1∂ _1V_1,f^β⟩\\ =&⟨Λ_h^s-1∂_βU^1_tχ_f,Λ_h^s-1 ∂_1V_1,f^β⟩=-∫_ΩΛ_h^s-1U^1_t χ_f·Λ_h^s-1∂_1∂_βV_1,f^β dx.\end{split} $$ Denoting $$ E_1^β(t):=\frac{1}{2}\|Λ_h^s-1∂_1V_1,f^ {β}\|_L^2(Ω)^2-4\|Λ_h^s-1∂_βV_1,f^1 \|^2_L^2-4\|\frac{1}{ν}Λ_h^s-1\widetilde{F}_3,f^β\|^ {2}_L^2, $$ we have $$ \begin{split}\sup_-∞<t≤ TE_1^β(t)≤sssim&\| Λ_h^s-1(\widetilde{F}_3,f^β,V_t,U_t,∇ V,∇^2V )\|^2_L^2(-∞,T;L^{2(Ω))}\\ +\|Λ_h^s-1&B^2_t\|_L^2_T(L^{2)}^2\|Λ_h^s-1 ∇_hK^(2)\|_L^∞_T(L^{2)}^2+\|Λ_h^s-1B^2\|_L^ {∞_T(H^1)}^2\|Λ_h^s-1K^(2)_t\|_L^2_T(L^{2)}^2 .\end{split} \tag{2} $$ Since $\|Λ_h^s-1(∂_βV_1,f^1(t),\widetilde{F}_3,f^β (t))\|_L^2∈C([0,T])$ , so $\|Λ_h^s-1∂_1V_1,f^β(t)\|_L^2$ continuous on $[0,T]$ due to Lemma 2.10. Moreover, there is $$ \sup_0≤ t≤ T\|Λ_h^s-1∂_1V_1,f^β(t)\|^2_L ^2(Ω)≤ 2\sup_0≤ t≤ T≤ft(E_1^β(t)+4\| Λ_h^s-1∂_βV^1(t)\|^2_L^2+4\|\frac{1}{ν} Λ_h^s-1\widetilde{F}_3^β(t)\|^2_L^2\right). $$ As a whole, we know that $Λ_h^s-1V∈C([0,T];H^1(Ω))$ , and $$ \begin{split}&\|Λ_h^s-1V\|^2_L_{T^∞(H^1(Ω))} ≤sssim\|Λ_h^s-1F_1\|^2_L^2_T(L^{2(Ω))}\\ & +\|Λ_h^s-1(B^1,B^2)\|^2_L^∞_T(H^{1(Ω))}( \|Λ_h^s-1∇(K^(1),K^(2))\|^2_L^2_T(H^{1(Ω))}+\| Λ_h^s-1(K^(1)_t,K_t^(2))\|^2_L^2_T(L^{2(Ω))})\\ & +(\|Λ_h^s-1(B^1,B^2)\|^2_L^∞_T(H^{1(Ω))} +\|Λ_h^s-1(B_t^1,B_t^2)\|^2_L^2_T(L^{2(Ω))})\| Λ_h^s-1(K^(1),K^(2))\|^2_L^∞_T(H^{1(Ω))}.\end{split} \tag{1} $$ Next, setting $$ W:=u-U-V, \widetilde{f}_0:=U^1+V^1, \widetilde{f}_1:= \widetilde{F}_1-V_t+∇·D(V)-∇ P_1, $$ the initial problem reduces to solving $(W,Q)$ that satisfy $$ \begin{cases}∂_tξ^1-W^1=\widetilde{f}_0& on [0,T] ×Σ_0,\\ ∂_tW-ν∇·D(W)+∇ Q=\widetilde{f}_1& in [0,T]×Ω,\\ ∇· W=0& in [0,T]×Ω,\\ (Q-gξ^1)n_0-νD(W)n_0=0& on [0,T]×Σ_0 ,\\ W=0& on [0,T]×Σ_b,\\ W|_t=0=W_0& in Ω,\\ ξ^1|_t=0=ξ_0^1& on Σ_0\end{cases} $$ with $W_0:=u_0-U_0-V_0$ . Step 3: Solving the homogeneous boundary equations (3.40) Consider the homogeneous boundary linear system (3.40), we can get the following proposition, which will be proved at the end of this section. **Proposition 3.1** *Assume that $s>2$ , $\widetilde{f}_0∈ L^2(0,T;H^s(Σ_0))$ , $Λ_h^s-1\widetilde{f}_1∈ L^2(0,T;L^2(Ω))$ , $ξ^1_0∈ H^s(Σ_0)$ , and $Λ_h^s-1W_0∈{{}_0}H^1(Ω)$ with $∇· W_0=0$ , then there exists a unique solution $(ξ^1,W,Q)$ of (3.40) satisfying $$ \begin{split}ξ^1∈C([0,T];H^s(Σ_0)), Λ_h^s-1 Q∈ L^2(0,T;H^1(Ω)), Λ_h^s-1W∈C([0,T];{{}_0 }H^1(Ω))∩ L^2(0,T;H^2(Ω)).\end{split} $$ Moreover, there holds $$ \begin{split}&\sup_0≤ t≤ T≤ft(\|ξ^1\|^2_H^s(Σ_0)+\| Λ_h^s-1W\|^2_H^1(Ω)\right)+\|Λ_h^s-1(∇ W,Q )\|^2_L^2(0,T;H^{1(Ω))}\\ &≤ C(\|ξ^1_0\|^2_H^s(Σ_0)+\|Λ_h^s-1W_0\|^2 _H^1(Ω)+\|\widetilde{f}_0\|^2_L^2(0,T;H^{s(Σ_0))}+\| Λ_h^s-1\widetilde{f}_1\|^2_L^2(0,T;L^{2(Ω))}).\end{split} $$* Step 4: End of the proof of Theorem 3.1 Set $u:=U+V+W$ , $P=P_1+Q$ , then putting all the previous steps together, we can get (3.1) and then complete the proof of Theorem 3.1. ∎ In order to prove Proposition 3.1, thanks to [16], we get the Helmholtz decomposition in $Ω$ as follows $$ L^2(Ω)=ℙL^2(Ω)⊕\{∇ϑ∈ L^2| ϑ∈ H^1(Ω),ϑ=0 on Σ_0\}, $$ where $ℙL^2(Ω)\buildrel\hbox{ def}\over{=}\overline{\{u ∈ L^2∩C^∞(\overline{Ω}):∇· u=0 in Ω,u· n_0=0 on Σ_b\}}^L^{2(Ω)}$ . From this, we may decompose the pressure term $∇ Q$ as $∇ Q=ℙ(∇ Q)+∇ q$ , where $ℙ(∇ Q)=∇π_1+∇π_2$ , and $(π_1, π_2)$ solves the system $$ \begin{cases}Δπ_i=0,(i=1,2)& in Ω,\\ π_1=gξ^1, π_2=-2ν(∂_2W^2+∂_3W^3)& on Σ_0,\\ ∂_1π_1=∂_1π_2=0& on Σ_b.\\ \end{cases} $$ Hence, the system (3.40) can be reformulated to the following equivalent system $$ \begin{cases}∂_tξ^1-W^1=\widetilde{f}_0& on [0,T] ×Σ_0,\\ ∂_tW-νℙ∇·D(W)+∇π_1+∇π_2 =ℙ\widetilde{f}_1& in [0,T]×Ω,\\ ∇· W=0& in [0,T]×Ω,\\ ν(∂_1W^2+∂_2W^1)=0& on [0,T]×Σ_ 0,\\ ν(∂_1W^3+∂_2W^1)=0& on [0,T]×Σ_ 0,\\ W=0& on [0,T]×Σ_b,\\ W|_t=0=W_0& in Ω,\\ ξ^1|_t=0=ξ_0^1& on Σ_0.\end{cases} $$ where $π_1$ and $π_2$ are determined by (3.42). Set $G\buildrel\hbox{ def}\over{=}≤ft(\begin{array}[]{cc}0&1\ A_1&A_2\end{array}\right)$ , $D(G)\buildrel\hbox{ def}\over{=}\{(ξ^1,W)^T: ∇· W =0 (\mbox{in} Ω), (∂_1W^α+∂_αW^1 )|_Σ_0=0 (\mbox{with} α=2,3), W|_Σ_{b}=0\}$ , where $A_1ξ^1=-∇π_1$ and $A_2W=νℙ∇·D(W)-∇π_2$ . Then, the problem (3.43) is reduced to solving the following evolution equations $$ \begin{cases}∂_t≤ft(\begin{array}[]{c}ξ^1\\ W\end{array}\right)-G≤ft(\begin{array}[]{c}ξ^1\\ W\end{array}\right)=≤ft(\begin{array}[]{c}\widetilde{f}_0\\ \widetilde{f}_1\end{array}\right)\\ ≤ft.≤ft(\begin{array}[]{c}ξ^1\\ W\end{array}\right)\right|_t=0=≤ft(\begin{array}[]{c}ξ_0^1\\ W_0\end{array}\right).\end{cases} $$ For the operator $G$ , we have the following lemma. **Lemma 3.1** *$G$ is a dissipative operator in $D(G)$ $→$ $L^2(Σ_0)× L_σ^2(Ω)$ .* * Proof* For any $λ>0$ , we have $$ \begin{split}&\|(λ-G)(ξ^1,W)^T\|^2_L^2(Σ_0)× L^{2 (Ω)}=((λ-G)(ξ^1,W)^T,(λ-G)(ξ^1,W)^T)_L^2( Σ_0×Ω)\\ &=λ^2\|(ξ^1,W)^T\|^2_L^2(Σ_0)× L^{2(Ω)}+ \|G(ξ^1,W)^T\|^2_L^2(Σ_0)× L^{2(Ω)}-(λ(ξ^ {1},W)^T,G(ξ^1,W)^T)_L^2(Σ_0×Ω).\end{split} $$ Notice that $$ \begin{split}-(λ(ξ^1,W)^T,G(ξ^1,W)^T&)_L^2(Σ_0 ×Ω)=-λ(ξ^1,W^1)_L^2(Σ_0)-λ(A_1ξ^1 +A_2W,W)_L^2(Ω)\\ =&-λ(ξ^1,W^1)_L^2(Σ_0)-λ(-∇π_1+ν ℙ∇·D(W)-∇π_2,W)_L^2(Ω)\\ =&-λ(ν∇·D(W)-∇π_2,W)_L^2(Ω)=\frac{ λ}{2}\|DW\|^2_L^2(Ω),\end{split} $$ then we have $$ \begin{split}\|(λ-G)(ξ^1,W)^T\|^2_L^2(Σ_0)× L^{2 (Ω)}≥λ^2\|(ξ^1,W)^T\|^2_L^2(Σ_0)× L^{2 (Ω)},\end{split} $$ which finishes the proof of Lemma 3.1. ∎ Let’s now prove Proposition 3.1. * Proof of Proposition3.1* Thanks to Lemma 3.1, we know that $G$ is a dissipative operator, and the right half plane belongs to the resolvent set, and $G$ has a closed extension, still denoted by $G$ , which generates a contraction semigroup $e^tG$ . Hence, similar to Lemma 2.2.1 in Chapter V.2 of [19], we obtain there is a weak solution $(ξ^1,W)$ of (3.43) in $L^2_loc((0,T);L^2(Σ_0))×(L^∞_loc((0,T);L_σ^ 2(Ω))∩ L^2_loc((0,T);{{}_0{H}}_σ^1(Ω)))$ . Therefore, in order to complete the proof of Proposition 3.1, we need to give necessary a priori energy estimate (3.41). In fact, by using the energy method, we can deduce from (3.43) that $$ \begin{split}&\frac{1}{2}\frac{d}{dt}≤ft(\|Λ_h^s W\|^2_L^2(Ω)+g\|ξ^1\|^2_H^s(Σ_0)\right)+ν\| Λ_h^sD(W)\|^2_L^2(Ω)\\ &=(Λ_h^s-1\widetilde{f}_1,Λ_h^s+1W)_L^2(Ω)+( Λ_h^s\widetilde{f}_0,gΛ_h^sξ^1)_L^2(Σ_0). \end{split} $$ Thanks to H $\ddot{o}$ lder’s inequality and Lemma 2.4, we have $$ \begin{split}&\frac{d}{dt}≤ft(\|Λ_h^sW\|^2_L^ {2(Ω)}+g\|ξ^1\|^2_H^s(Σ_0)\right)+\frac{3ν}{2}\| Λ_h^sD(W)\|^2_L^2(Ω)\\ &≤\frac{2}{ν}\|Λ_h^s-1\widetilde{f}_1\|^2_L^2(Ω)+g \|\widetilde{f}_0\|_H^s(Σ_0)\|ξ^1\|_H^s(Σ_0).\end{split} $$ On the other hand, in order to avoid the loss of regularity on the boundary, similar to the proof of Theorem 1.3 in [13], we introduce new unknowns as follows: $$ \begin{cases}G_Q:=Q-gH(ξ^1)+2ν∇_h· W^h ,\\ G_W:=≤ft(\begin{array}[]{c}0\\ G_W^2\\ G_W^3\end{array}\right):=≤ft(\begin{array}[]{c}0\\ ∂_1W^2+∂_2W^1\\ ∂_1W^3+∂_3W^1\end{array}\right).\end{cases} $$ Then, one can deduce that $$ \|Λ_h^s-1W_t\|^2_L^2(Ω)+I+II=(Λ_h^s-1 \widetilde{f}_1,Λ_h^s-1W_t)_L^2(Ω). $$ where $I:=∫_ΩΛ_h^s-1W_t·Λ_h^s-1∇ Qdx$ , and $II:=ν∫_ΩΛ_h^s-1W_t·(∇·D(Λ_ {h}^s-1W))dx$ . Using (3.45) 1, the integral $I$ can be split into three parts $$ \begin{split}I=&∫_ΩΛ_h^s-1W_t·Λ_h^s-1 ∇G_Qdx+∫_ΩΛ_h^s-1W_t· g Λ_h^s-1∇H(ξ^1)dx\\ &-2ν∫_ΩΛ_h^s-1W_t·Λ_h^s-1∇(∇_ h· W^h)dx.\end{split} $$ Due to $∇· W=0$ , one can show that $$ \begin{split}I=&\frac{d}{dt}∫_ΩΛ_h^s-1W · gΛ_h^s-1∇H(ξ^1)dx-∫_Ω Λ_h^s-1W· gΛ_h^s-1∇H(W^1+\tilde{η} )dx\\ &+ν\frac{d}{dt}\|Λ_h^s-1∂_1W^1\|^2_ {L^2(Ω)}-2ν∫_ΩΛ_h^s-1W^1_t·Λ_h^s -1∂_1(∇_h· W^h)dx.\end{split} $$ For $II$ , thanks to the new unknowns $G_W$ defined by (3.45) 2 and $∇· W=0$ , we obtain that $$ \begin{split}II=&-ν∫_ΩΛ_h^s-1W_t·Λ_h^s-1 ∂_1^2Wdx+\frac{ν}{2}\frac{d}{dt}\| Λ_h^s-1∇_hW\|^2_L^2(Ω)\\ =&-ν∫_ΩΛ_h^s-1W_t·Λ_h^s-1∂_1 G_Wdx+ν∫_ΩΛ_h^s-1W^1_t· Λ_h^s-1∂_1(∇_h· W^h)dx\\ &+\frac{ν}{2}\frac{d}{dt}≤ft(\|Λ_h^s-1∂ _1W^1\|^2_L^2(Ω)+\|Λ_h^s-1∇_hW\|^2_L^2( Ω)\right).\end{split} $$ In view of the boundary conditions $W|_Σ_{b}=0$ and $G_W|_Σ_0=0$ , one can deduce that $$ \begin{split}&-ν∫_ΩΛ_h^s-1W_t·Λ_h^s-1 ∂_1G_Wdx=\frac{ν}{2}\frac{d}{ dt}\|Λ_h^s-1∂_1W^h\|^2_L^2{Ω}\\ & +ν\frac{d}{dt}∫_ΩΛ_h ^s-1∂_1W^2·Λ_h^s-1∂_2W^1dx+ν ∫_ΩΛ_h^s-1∂_2∂_1W^2·Λ_h^s -1W_t^1dx\\ & +ν\frac{d}{dt}∫_ΩΛ_h ^s-1∂_1W^3·Λ_h^s-1∂_3W^1dx+ν ∫_ΩΛ_h^s-1∂_3∂_1W^3·Λ_h^s -1W_t^1dx.\end{split} $$ Define $$ \begin{split}\tilde{E}_s-1:=&\|Λ_h^s-1∇ W\|^2_L^ {2(Ω)}+2\|Λ_h^s-1∂_1W^1\|^2_L^2(Ω)+\frac {2}{ν}∫_ΩΛ_h^s-1W· gΛ_h^s-1∇ H(ξ^1)dx\\ &+2∫_ΩΛ_h^s-1∂_1W^h·Λ_h^s-1∇ _hW^1dx,\end{split} $$ combining $I$ , $II$ with (3.46) leads to $$ \begin{split}\frac{ν}{2}\frac{d}{dt}\tilde{E}_s -1+\|Λ_h^s-1W_t\|^2_L^2(Ω)=&(Λ_h^s-1 \widetilde{f}_1,Λ_h^s-1W_t)_L^2(Ω)+(gΛ_h^s-1 ∇H(W^1+\widetilde{f}_0),Λ_h^s-1W)_L^2(Ω) .\end{split} $$ Then we have $$ \begin{split}&ν\frac{d}{dt}\tilde{E}_s-1+\| Λ_h^s-1W_t\|^2_L^2(Ω)\\ &≤\|Λ_h^s-1\widetilde{f}_1\|^2_L^2(Ω)+2g≤ft(\| Λ_h^s-1∇ W\|_L^2(Ω)+\|\widetilde{f}_0\|_H^s-\frac{ 1{2}(Σ_0)}\right)\|Λ_h^s-1W\|_L^2(Ω),\end{split} $$ due to Hölder’s inequality and Lemma 2.4. Now, let’s consider the estimates of $\|Λ_h^s-1∂_1^2W\|_L^2(L^{2(Ω))}$ . Since there are different boundary conditions on the upper boundary $Σ_0$ and the bottom boundary $Σ_b$ , we need to deal with these two different situations separately. Define $W_b:=Wχ_b$ , $Q_b:=Qχ_b$ where $χ_b$ defined by (3.9), then according to (3.40), $(W_b,Q_b)$ solves $$ \begin{cases}-ν∇·D(W_b)+∇ Q_b=-χ_bW_t+ \widetilde{f}_b,\\ ∇· W_b=χ^\prime_bW^1_b,\\ (Q_bI-νD(W_b))n_0|_Σ_0=0,\\ W_b|_Σ_{b}=0,\end{cases} $$ where $$ \widetilde{f}_b=\widetilde{f}_1χ_b-ν[∇·D;χ_b ]W+[∇;χ_b]Q. $$ According to Lemma (2.9), we obtain the estimates near the bottom boundary as follow, $$ \begin{split}\|Λ_h^s-1∇ W\|^2_H^1(Ω_b)+\|Λ_ h^s-1∇ Q\|^2_L^2(Ω_b)≤&\|Λ_h^s-1\widetilde{f }_1\|^2_L^2(Ω_b)+\|Λ_h^s-1W_t\|^2_L^2(Ω_ b)\\ &+\|Λ_h^s-1W\|_H^1(Ω)+\|Λ_h^s-1Q\|^2_L^2( Ω_f).\end{split} $$ Next, we need to consider the estimates near the upper boundary. Consider the equation of $∂_1^2W^h=(∂_1^2W^2,∂_1^2W^3)^T$ $$ -ν∂_1^2W^h+∇_hQ=-W^h_t+νΔ_hW^h+\widetilde {f}_1^h. $$ By using the energy method, one can deduce that $$ \begin{split}ν\|Λ_h^s-1∂_1^2W^h\|^2_L^2(Ω_ f)+IV=∫_ΩΛ_h^s-1(W^h_t-νΔ_hW^h-\widetilde {f}_1^h)·Λ_h^s-1∂_1^2W^hχ_fdx, \end{split} $$ where $$ \begin{split}IV=&-∫_ΩΛ_h^s-1∇_hQ·Λ_h^s -1∂_1^2W^hχ_fdx\\ =&-∫_ΩΛ_h^s-1∂_1Q·Λ_h^s-1∂_ 1(∇_h· W^h)χ_fdx-∫_ΩΛ_h^s-1 ∂_1Q·Λ_h^s-1∂_1(∇_h· W^h)χ^ \prime_fdx\\ &-∫_Σ_0Λ_h^s-1∇_hQ·Λ_h^s-1∂_ {1}W^hdx\end{split} $$ Thanks to the boundary conditions of $∇_hW^1$ and $Q$ on $Σ_0$ , we obtain $$ \begin{split}-∫_Σ_0Λ_h^s-1∇_hQ·&Λ_h^ s-1∂_1W^hdx=∫_Σ_0Λ_h^s-1∇_h( gξ^1-2ν∇_h· W^h)·Λ_h^s-1∇_hW^1 dx\\ =&\frac{g}{2}\frac{d}{dt}\|Λ_h^s-1∇_hξ^1 \|^2_L^2(Σ_0)-∫_Σ_0Λ_h^s-1∇_hξ^1 ·Λ_h^s-1∇_h\tilde{η}dx\\ &+2ν∫_Σ_0Λ_h^s-1(∇_h· W^h)·Λ_h ^s-1Δ_hW^1dx.\end{split} $$ Hence, we have $$ \begin{split}&ν\|Λ_h^s-1∂_1^2W^h\|^2_L^2(Ω_ {f)}+\frac{g}{2}\frac{d}{dt}\|Λ_h^s-1∇_h ξ^1\|^2_L^2(Σ_0)\\ &=∫_ΩΛ_h^s-1(W^h_t-νΔ_hW^h-\widetilde{f}_1 ^h)·Λ_h^s-1∂_1^2W^hχ_fdx\\ & +∫_ΩΛ_h^s-1∂_1Q·Λ_h^s-1 ∂_1(∇_h· W^h)χ_fdx+∫_ΩΛ_h ^s-1∂_1Q·Λ_h^s-1∂_1(∇_h· W^h) χ^\prime_fdx\\ & +∫_Σ_0gΛ_h^s-1∇_hξ^1·Λ_h^s -1∇_h\widetilde{f}_0dx-2ν∫_Σ_0Λ_h^s- \frac{1{2}}(∇_h· W^h)·Λ_h^s-\frac{3{2}}Δ_hW ^1dx.\end{split} $$ Since $∂_1Q=\tilde{f}^1+Δ_hW^1-∂_1(∇_h· W^h )-W_t$ , one has $$ \begin{split}&\frac{ν}{2}\|Λ_h^s-1∂_1^2W^h\|^2_L^ 2(Ω_f)+\frac{g}{2}\frac{d}{dt}\|Λ_h^s-1 ∇_hξ^1\|^2_L^2(Σ_0)\\ &≤ C≤ft(\|Λ_h^s-1W^h_t\|^2_L^2(Ω_f)+\|Λ_ {h}^s∇ W^h\|_L^2(Ω)+\|Λ_h^s-1\widetilde{f}_1\|^ 2_L^2(Ω_f)+\|∇_hξ^1\|_H^s-1\|∇_h\widetilde{f }_0\|_H^s-1\right).\end{split} $$ Moreover, we have the estimates of $∇ Q$ as follow, $$ \begin{split}&\|Λ_h^s-1∂_1Q\|^2_L^2(Ω)≤\| Λ_h^s-1(W^h_t,\widetilde{f}_1)\|^2_L^2(Ω)+\|Λ _h^s∇ W^h\|^2_L^2(Ω),\\ &\|Λ_h^s-1(∇_hQ-ν∂_1^2W^h)\|^2_L^2(Ω _f)≤\|Λ_h^s-1(W^h_t,\widetilde{f}_1)\|^2_L^2(Ω _f)+\|Λ_h^s+1W^h\|^2_L^2(Ω_f),\\ &\|Λ_h^s-1(∇_hQ-ν∂_1^2W^h)\|^2_L^2(Ω )≤\|Λ_h^s-1(W^h_t,\widetilde{f}_1)\|^2_L^2(Ω)+ \|Λ_h^s+1W^h\|^2_L^2(Ω).\end{split} $$ Thanks to Lemma 2.5, we may get $$ \begin{split}\|Λ_h^s-1Q\|^2_L^2(Ω)&≤sssim\|Λ_h^ {s-1}Q\|^2_L^2(Σ_0)+\|Λ_h^s-1∂_1Q\|^2_L^2 (Ω)\\ &≤\|ξ^1\|^2_H^s(Σ_0)+\|Λ_h^s-1(W^h_t, \widetilde{f}_1)\|^2_L^2(Ω)+\|Λ_h^s-1∇_hW^h\|^ {2}_H^1(Ω).\end{split} $$ Then combining (3.49), (3.51), (3.52) with (3.50), we may get $$ \begin{split}&c_0\|Λ_h^s-1(∂_1^2W^h,Q,∇ Q)\|^2 _L^2(Ω)+\frac{d}{dt}\|Λ_h^s-1∇_h ξ^1\|^2_L^2(Σ_0)\\ &≤ C_0≤ft(\|Λ_h^s-1W^h_t\|^2_L^2(Ω)+\|Λ_ {h}^s∇ W^h\|_L^2(Ω)+\|Λ_h^s-1\widetilde{f}_1\|^ 2_L^2(Ω)+\|ξ^1\|_H^s+\|\widetilde{f}_0\|^2_H^s\right ),\end{split} $$ for some suitably positive constant $c_0$ and $C_0$ . Finally, we will show the total energy estimates. For any given small positive constant $κ$ , we set $$ \begin{split}&\bar{E}_s-1:=\|Λ_h^sW\|^2_L^2(Ω) +\|ξ^1\|_H^s(Σ_0)+κν\tilde{E}_s-1+κ^2 \|∇_hξ^1\|_H^s-1(Σ_0),\\ &\bar{D}_s-1:=\frac{3ν}{2}\|Λ_h^sD(W)\|^2_ L^2(Ω)+κ\|Λ_h^s-1W_t\|^2_L^2(Ω)+κ^2 c_0\|Λ_h^s-1(∂_1^2W^h,Q,∇ Q)\|^2_L^2(Ω ),\end{split} $$ From this, we find that there is a small enough $κ>0$ such that the following inequalities hold $$ \begin{split}&\bar{E}_s-1≥ c_1(\|Λ_h^s-1W\|^2_H^ {1(Ω)}+\|ξ^1\|_H^s(Σ_0)),\\ &\bar{D}_s-1≥ c_1(\|Λ_h^s-1∇ W\|^2_H^1( Ω)+\|Λ_h^s-1W_t\|^2_L^2(Ω)+\|Λ_h^s-1Q\| ^2_H^1(Ω)),\end{split} $$ for some suitably small positive constant $c_1$ . Combining (3.54) with (3.44),(3.48),(3.53), we obtain $$ \begin{split}&c_1\frac{d}{dt}≤ft(\|Λ_h^s-1W\|^ {2}_H^1(Ω)+\|ξ^1\|^2_H^s(Σ_0)\right)+c_1≤ft(\| Λ_h^s-1(∇ W,Q)\|^2_H^1(Ω)+\|Λ_h^s-1W_t\| ^2_L^2(Ω)\right)\\ &≤\frac{2}{ν}\|Λ_h^s-1\widetilde{f}_1\|^2_L^2(Ω)+ \|\widetilde{f}_0\|_H^s(Σ_0)\|ξ^1\|_H^s(Σ_0)\\ & +κ≤ft(\|Λ_h^s-1\widetilde{f}_1\|^2_L^2(Ω)+ 4C_korn\|Λ_h^s-1∇ W\|^2_L^2(Ω)+\|\tilde{η}\|^2 _H^s-\frac{1{2}(Σ_0)}\right)\\ & +κ^2C_0≤ft(\|Λ_h^s-1W^h_t\|^2_L^2(Ω) +\|Λ_h^s∇ W^h\|_L^2(Ω)^2+\|Λ_h^s-1 \widetilde{f}_1\|^2_L^2(Ω)+\|ξ^1\|_H^s^2+\|\widetilde{f }_0\|^2_H^s\right).\end{split} $$ Let $κ≤\min\{1,\frac{c_0}{16C_korn},√{\frac{c_0}{4C_0}}\}$ , then we have $$ \begin{split}&c_1\frac{d}{dt}≤ft(\|Λ_h^s-1W\|^ {2}_H^1(Ω)+\|ξ^1\|^2_H^s(Σ_0)\right)+\frac{c_1}{2} ≤ft(\|Λ_h^s-1(∇ W,Q)\|^2_H^1(Ω)+\|Λ_h^s-1 W_t\|^2_L^2(Ω)\right)\\ &≤ C\|Λ_h^s-1\widetilde{f}_1\|^2_L^2(Ω)+\|\widetilde {f}_0\|_H^s(Σ_0)\|ξ^1\|_H^s(Σ_0)+2κ\| \widetilde{f}_0\|^2_H^s(Σ_0)+κ^2c^\prime_0\|ξ^1 \|^2_H^s(Σ_0).\end{split} $$ Due to $∂_tξ^1-W^1=\widetilde{f}_0$ on $[0,T]×Σ_0$ , one can see that $∀ t∈[0,T]$ $$ \begin{split}\sup_τ∈[0,t]\|ξ^1(τ)\|^2_H^s(Σ_0)≤ t ∫_0^t≤ft(2C_korn\|Λ_h^s-1∇ W\|^2_L^2(Ω)+ \|\widetilde{f}_0\|^2_H^s(Σ_0)\right)dτ+\|ξ^1_ 0\|_H^s(Σ_0)^2,\end{split} $$ which along with (3.55) and Young’s inequality implies $$ \begin{split}c_1\frac{d}{dt}&≤ft(\|Λ_h^s-1W\|^ {2}_H^1(Ω)+\|ξ^1\|^2_H^s(Σ_0)\right)+\frac{c_1}{2} ≤ft(\|Λ_h^s-1(∇ W,Q)\|^2_H^1(Ω)+\|Λ_h^s-1 W_t\|^2_L^2(Ω)\right)\\ ≤&C≤ft(\|Λ_h^s-1\widetilde{f}_1\|^2_L^2(Ω)+\| \widetilde{f}_0\|^2_H^s(Σ_0)\right)+(\frac{c_1}{8}+κ^2 C_0)t∫_0^t\|Λ_h^s-1∇ W\|^2_L^2(Ω)d τ\\ &+(1+κ^2C_0)≤ft(t∫_0^t\|\widetilde{f}_0\|^2_H^s( Σ_0)dτ+\|ξ^1_0\|_H^s(Σ_0)^2\right)\end{split} $$ Taking $T=1$ and $0<κ≤√{\frac{c_1}{8C_0}}$ , we have $$ \begin{split}c_1\frac{d}{dt}&≤ft(\|Λ_h^s-1W\|^ {2}_H^1(Ω)+\|ξ^1\|^2_H^s(Σ_0)\right)+\frac{c_1}{2} ≤ft(\|Λ_h^s-1(∇ W,Q)\|^2_H^1(Ω)+\|Λ_h^s-1 W_t\|^2_L^2(Ω)\right)\\ ≤&C≤ft(\|Λ_h^s-1\widetilde{f}_1\|^2_L^2+\|\widetilde{f} _0\|^2_H^s+∫_0^t\|\widetilde{f}_0\|^2_H^sdτ +\|ξ^1_0\|_H^s^2\right)+\frac{c_1}{4}∫_0^t\|Λ_h^ s-1∇ W\|^2_L^2dτ,\end{split} $$ which leads to (3.41). We thus complete the proof of Theorem 3.1. ∎ ## 4 Proof of Theorem 1.1 Following the idea from [13], we rewrite the system (1.10) as the following linearized form $$ \begin{cases}∂_tξ-v=0& in Ω,\\ ∂_tv-ν∇·D(v)+∇ q=F_1(∇ξ,v,q)& in Ω,\\ ∇· v=F_2(∇ξ,v)& in Ω,\\ qn_0-gξ^1n_0-νD(v)n_0=F_3(∇ξ,v)& on Σ_0,\\ v=0& on Σ_b,\\ (ξ,v)|_t=0=(ξ_0,v_0)& in Ω,\end{cases} $$ where $$ \begin{split}&F_1(A(∇ξ),v,q):=-ν(∇·D(v) -∇_A·D_A(v))+∇_I- Aq,\\ &F_2(∇ξ,v):={B}^1(∇ξ):∇ v, F_3(∇ξ,v):={B} ^2(∇ξ):∇_hv.\end{split} $$ and $A=A(∇ξ)=(∇ξ+I)^-1$ , ${B}^1=I-a_11^-1JA$ and $({B}^2:∇_hv)^j={B}^2,j_α,i∂_αv^i$ $(i,j=1,2,3,α=2,3)$ . Here, ${B}^2={B}^2(∇ ξ)$ is a $3× 2× 3$ tensor defined in Appendix (Sect. 5). Notice that the matrix $A-I$ depends only on $∇ξ$ from the explicit form $A=(∇ξ+I)^-1$ , then we can get the following estimates about ${B}^1$ and ${B}^2$ . **Lemma 4.1 (Lemma 4.7 in[13])** *Let $s>2$ ,and assume $∇ξ$ and $∇\widetilde{ξ}$ satisfy $$ \sup_0≤ t≤ T\big{(}\|Λ_h^s-1∇ξ\|_H^1(Ω)+\| Λ_h^s-1∇\widetilde{ξ}\|_H^1(Ω)\big{)}≤ε_0, $$ for some suitably small positive constant $ε_0∈(0,1]$ , then there exists a positive constant $C$ such that $$ \begin{split}&\|Λ_h^s-1(A-I,{B}^1,{B}^2)\|_H ^1(Ω)≤ C\|Λ_h^s-1∇ξ\|_H^1(Ω),\\ &\|Λ_h^s-1{B}^1_t\|_L^2(Ω)≤ C(1+\|Λ_h^s-1 ∇ξ\|_H^1(Ω))\|Λ_h^s-1∇ξ_t\|_L^2(Ω) ,\\ &\|Λ_h^s-1{B}^2_t\|_L^2(Ω)≤ C\|Λ_h^s-1 ∇ξ\|_H^1(Ω)\|Λ_h^s-1∇ξ_t\|_L^2(Ω), \end{split} $$ and $$ \begin{split}&\|Λ_h^s-1(A-\widetilde{A},B- \widetilde{B},{B}^2-\widetilde{B}^2)\|_H^1(Ω)≤ C\|Λ_h^ {s-1}(∇ξ-∇\widetilde{ξ})\|_H^1(Ω),\\ &\|Λ_h^s-1(B^1_t-\widetilde{B}^1_t,{B}^2_t-\widetilde{B} ^2_t)\|_L^2(Ω)≤ C\big{(}\|Λ_h^s-1(∇ξ_t- ∇\widetilde{ξ}_t)\|_L^2(Ω)\\ & +\|Λ_h^s-1(∇ξ_t,∇ \widetilde{ξ}_t)\|_L^2(Ω)\|Λ_h^s-1(∇ξ-∇ \widetilde{ξ})\|_H^1(Ω)).\end{split} $$* * Proof* Let $ε_0$ be suitably small, we can find that $J$ , $a_11$ and $|\overrightarrow{\rm{a}_1}|$ are near $1$ . Then, similar to the proof for Lemma 4.7 in [13], we may use the product estimates to get (4.2) and (4.3). ∎ ### 4.1 Existence part of the proof of Theorem 1.1 Setting $$ \begin{split}X_T:=\{(ξ^1,v,q)|&ξ^1∈C([0,T];H^s(Σ_ {0})),Λ_h^s-1q∈ L^2([0,T];H^1(Ω)),\\ &Λ_h^s-1v∈C([0,T];{{}_0}{H}^1(Ω))∩ L^2([0, T];H^2(Ω))\}\end{split} $$ with the norm $\|·\|_X_{T}$ defined by $$ \begin{split}\|(ξ^1,v,q)\|_X_{T}:=&\|ξ^1\|_L^∞([0,T];H^{s{( Σ_0)})}+\|Λ_h^s-1v\|_L^∞([0,T];H^{1(Ω))}+\| Λ_h^s-1v\|_L^2([0,T];H^{2(Ω))}\\ &+\|Λ_h^s-1v_t\|_L^2([0,T];L^{2(Ω))}+\|Λ_h^s-1q \|_L^2([0,T];H^{1(Ω))}.\end{split} $$ It’s obvious that $(X_T,\|·\|_X_{T})$ is a Banach space. Then, based on the fixed point method, we shall show the local existence of the system (4.1). Step 1: Construction of the initial iteration $(ξ_1,v_1,q_1)$ In order to get the first step of the iteration in the fixed point method, we will solve the linear system of $(ξ_1,v_1,q_1)$ as follows: $$ \begin{cases}∂_tξ_1-v_1=0& in Ω,\\ ∂_tv_1-ν∇·D(v_1)+∇ q_1=F_1(∇ξ _0,v_1,q_1)& in Ω,\\ ∇· v_1=F_2(∇ξ_0,v_1)& in Ω,\\ (q_1-g ξ^1_1)n_0-νD(v_1)n_0=F_3(∇ξ_0,v_1 )& on Σ_0,\\ v_1=0& on Σ_b,\\ (ξ_1,v_1)|_t=0=(ξ_0,v_0)& in Ω,\end{cases} $$ which is equivalent to $$ \begin{cases}∂_tξ_1-v_1=0& in Ω,\\ ∂_tv_1-ν∇_A_0·D_A_0 (v_1)+∇_A_0q_1=0& in Ω,\\ ∇_A_0· v_1=0& in Ω,\\ (q_1-g ξ^1_1)N_0-νD(v_1)N_0=0& on Σ_0,\\ v_1=0& on Σ_b,\\ (ξ_1,v_1)|_t=0=(ξ_0,v_0)& in Ω.\end{cases} $$ where $A_0=A(∇ξ_0)=(I+∇ξ_0)^-T$ and $N_0=J_0A_0n_0$ . Since $ξ^2$ and $ξ^3$ are decoupled with other unknowns on the boundaries $Σ_0$ and $Σ_b$ in (4.4), we will first consider the linear system $$ \begin{cases}∂_tξ_1^1-v_1^1=0& on Σ_0,\\ ∂_tv_1-ν∇_A_0·D_A_0 (v_1)+∇_A_0q_1=0& in Ω,\\ ∇_A_0· v_1=0& in Ω,\\ (q_1-g ξ^1_1)N_0-νD(v_1)N_0=0& on Σ_0,\\ v_1=0& on Σ_b,\\ ξ_1^1|_t=0=ξ_0^1& on Σ_0,\\ v_1|_t=0=v_0& in Ω.\end{cases} $$ We will use the fixed point method to solve (4.6). Indeed, given $(ξ^1_1,(0),v_1,(0),q_1,(0))∈ X_T$ satisfying $∇_J_{0A_0}· v_1,(0)|_t=0=0$ and $\|(ξ^1_1,(0),v_1,(0),q_1,(0)\|_X_{T}≤ 2C_0≤ft(\|ξ_0^1 \|_H^s{(Σ_0)}+\|Λ_h^s-1v_0\|_H^1(Ω)\right)$ as the initial iteration data, we construct $(ξ^1_1,(n),v_1,(n),q_1,(n))$ ( $∀ n∈ℕ^+$ ) satisfying the following linear system $$ \begin{cases}∂_tξ^1_1,(n)-v^1_1,(n)=0& on Σ_0 ,\\ ∂_tv_1,(n)-ν∇·D(v_1,(n))+∇ q_1,(n)=F_ 1(∇ξ_0,v_1,(n-1),q_1,(n-1))& in Ω,\\ ∇· v_1,(n)=F_2(∇ξ_0,v_1,(n-1))& in Ω,\\ (q_1,(n)-g ξ^1_1,(n))n_0-νD(v_1,(n))n_0=F_3(∇ ξ_0,v_1,(n-1))& on Σ_0,\\ v_1,(n)=0& on Σ_b,\\ ξ^1_1,(n)|_t=0=ξ^1_0& in Σ_0,\\ v_1,(n)|_t=0=v_0& in Ω.\end{cases} $$ Denote this solution mapping by $Υ_1$ . We may claim that, there exists $ε_1>0$ so small that if $\|Λ_h^s-1∇ξ_0\|_H^1(Ω)≤ε_1$ , the sequence $\{(ξ^1_1,(n),v_1,(n),q_1,(n))\}_n=1^∞$ satisfies the following uniform estimate $$ \begin{split}\|(ξ^1_1,(n),v_1,(n),q_1,(n))\|_X_{T}≤ 2C_0(\| ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_0\|_H^1(Ω) ) (∀ n∈ℕ^+).\end{split} $$ In fact, applying Theorem 3.1 to (4.7), we prove that $(ξ^1_1,(n),v_1,(n),q_1,(n))∈ X_T$ ( $∀ n∈ℕ^+$ ) and $$ \begin{split}&\|(ξ^1_1,(n),v_1,(n),q_1,(n))\|_X_{T}\\ &≤ C_0\bigg{(}\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1u_ {0}\|_H^1(Ω)\bigg{)}\\ & +C_0\bigg{(}\|Λ_h^s-1F_1\|_L^2([0,T],L^{2(Ω))}+\| Λ_h^s-1(B^1(∇ξ_0),B^2(∇ξ_0))\|_H^1(Ω) \|(ξ^1_1,(n),v_1,(n),q_1,(n))\|_X_{T}\bigg{)}.\end{split} $$ So if $\|Λ_h^s-1∇ξ_0\|_H^1(Ω)≤ε_1$ with $ε_1:=\min\{ε_0,\frac{1}{2C_0C_1}\}$ , then we get by an induction argument that ( $∀ n∈ℕ^+$ ) $$ \begin{split}&\|(ξ^1_1,(n),v_1,(n),q_1,(n)\|_X_{T}\\ &≤ C_0(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1u_0\|_ H^1(Ω))+C_0C_1\|Λ_h^s-1∇ξ_0\|_H^1(Ω)\| (ξ^1_1,(n),v_1,(n),q_1,(n))\|_X_{T}\\ &≤ C_0(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_0\|_ H^1(Ω))+2C_0^2C_1ε_1 (\|ξ_0^1\|_H^s{( Σ_0)}+\|Λ_h^s-1v_0\|_H^1(Ω))\\ &≤ 2C_0(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_0\|_ {H^1(Ω)}),\end{split} $$ which finishes the proof of (4.8). Next, we will show that the sequence $\{(ξ^1_1,(n),v_1,(n),p_1,(n))\}_n=1^∞$ is a Cauchy sequence in $X_T$ . In fact, set $δ f_(n):=f_(n)-f_(n-1)$ ( $∀ n∈ℕ$ ), then $(δξ_1,(n+1)^1,δ v_1,(n+1),δ q_1,(n+1))∈ X_T$ satisfies $$ \begin{cases}∂_tδξ^1_1,(n+1)-δ v^1_1,(n+1)=0& on Σ_0,\\ ∂_tδ v_1,(n+1)-ν∇·D(δ v_1,(n+1))+ ∇δ q_1,(n+1)=F_1(∇ξ_0,δ v_1,(n),δ q_1,(n) )& in Ω,\\ ∇·δ v_1,(n+1)=F_2(∇ξ_0,δ v_1,(n))& \text {in }Ω,\\ δ q_1,(n+1)n_0-gδξ^1_1,(n+1)-νD(δ v_1,(n+ 1))n_0=F_3(∇ξ_0,δ v_1,(n))& on Σ_0,\\ δ v_1,(n+1)=0& on Σ_b,\\ δξ_1,(n+1)^1|_t=0=0& on Σ_0,\\ δ v_1,(n+1)|_t=0=0& in Ω.\end{cases} $$ Thanks to Theorem 3.1 again, we obtain $$ \begin{split}&\|(δξ_1,(n+1)^1,δ v_1,(n+1),δ q_1,(n+1) )\|_X_{T}\\ &≤ C_0≤ft(\|Λ_h^s-1((∇·D-∇_A _0·D_A_0)δ v_1,(n),(∇-∇_ A_0)δ q_1,(n))\|_L^2(L^{2(Ω))}\right.\\ & ≤ft.+\|Λ_h^s-1(B^1(∇ξ_0),B^2(∇ξ_0)\|_ H^1(Ω)\|(0,δ v_1,(n),0)\|_X_{T}\right).\end{split} $$ Let $\|Λ_h^s-1∇ξ_0\|_H^1(Ω)≤\min\{\frac{ε_0 }{2},ε_2\}$ and let $ε_2$ be small enough such that $$ \begin{split}\|Λ_h^s-1(I-A_0)\|_L^∞(H^{ 1)}(1+\|Λ_h^s-1A_0\|_L^∞(H^{1)})≤\frac{1}{8 C_0},\end{split} $$ and $$ \|Λ_h^s-1(B^1(∇ξ_0),B^2(∇ξ_0))\|_H^1(Ω )≤\frac{1}{4C_0}, $$ which implies that $$ \begin{split}\|(δξ_1,(n+1)^1,δ v_1,(n+1),δ q_1,(n+1)) \|_X_{T}≤\frac{1}{2}\|(0,δ v_1,(n),δ q_1,(n))\|_X_{T}. \end{split} $$ From this, we know that the solution mapping $Υ_1:X_T→ X_T$ in (4.7) is a contraction mapping. Hence, the sequence $\{(ξ^1_1,(n),v_1,(n),q_1,(n))\}_n=1^∞⊂ X_T$ is a Cauchy sequence, and then there is $(ξ^1_1,v_1,q_1)∈ X_T$ , the fixed point of the mapping $Υ_1$ , so that $\{(ξ^1_1,(n),v_1,(n),q_1,(n))\}_n=1^∞$ converges to $(ξ^1_1,v_1,q_1)$ in $X_T$ . Moreover, we may check that $(ξ^1_1,v_1,q_1)$ is the solution to (4.6) in $X_T$ , and there holds $$ \|(ξ^1_1,v_1,q_1)\|_X_{T}≤ 2C_0≤ft(\|Λ_h^s-1ξ^ 1_0\|_H^1(Σ_0)+\|Λ_h^s-1v_0\|_H^1(Ω)\right). $$ With $v_1$ above in hand, let $$ T≤\frac{\|Λ_h^s-1∇ξ_0\|^2_H^1(Ω)}{8C_0^2( \|Λ_h^s-1ξ^1_0\|_H^1(Σ_0)+\|Λ_h^s-1v_0 \|_H^1(Ω))^2}, $$ and we solve $ξ_1$ in $Ω$ by the linear ODEs $$ \begin{cases}∂_tξ_1=v_1& in (0,T]×Ω,\\ ξ_1|_t=0=ξ_0& in Ω,\end{cases} $$ to get $Λ_h^s-1∇ξ_1∈C([0,T];H^1(Ω))$ , $$ \begin{split}\|Λ_h^s-1∇ξ_1\|_L^∞_T(H^{1(Ω)) }&≤ 2\|Λ_h^s-1∇ξ_0\|_H^1(Ω),\\ \|Λ_h^s-1∂_t∇ξ_1\|_L^2_T(L^{2(Ω))}&≤ T ^\frac{1{2}}\|Λ_h^s-1∇ v_1\|_L^2_T(L^{2(Ω))}≤ 2 \|Λ_h^s-1∇ξ_0\|_H^1(Ω).\end{split} $$ and then $$ \begin{split}\|Λ_h^s-1∇ξ_1\|_L^∞_T(H^{1(Ω)) }+\|Λ_h^s-1∂_t∇ξ_1\|_L^2_T(L^{2(Ω))} ≤ 4\|Λ_h^s-1∇ξ_0\|_H^1(Ω).\end{split} $$ Step 2: Construction of approximate sequence. Taking $(ξ_1,v_1,q_1)$ obtained in Step 1 as the initial value of the iteration, and letting $(ξ_n,v_n,q_n)$ be a solution of the following linear problem $(n∈ℕ^+ and n≥ 2)$ : $$ \begin{cases}∂_tξ_n-v_n=0,& in Ω,\\ ∂_tv_n-ν∇·D(v_n)+∇ q_n=F_1(∇ξ _n-1,v_n,q_n),& in Ω,\\ ∇· v_n=F_2(∇ξ_n-1,v_n),& in Ω,\\ q_nn_0-gξ^1_nn_0-νD(v_n)n_0=F_3(∇ξ_n-1,v _n),& on Σ_0,\\ v_n=0,& on Σ_b,\\ (ξ_n,v_n)|_t=0=(ξ_0,v_0),& in Ω,\end{cases} $$ which is equivalent to $$ \begin{cases}∂_tξ_n-v_n=0& in Ω,\\ ∂_tv_n-ν∇_A_n-1·D_A_n -1(v_n)+∇_A_n-1q_n=0& in Ω,\\ ∇_A_n-1· v_n=0& in Ω,\\ (q_n-g ξ^1_n)N_n-1-νD(v_n)N_n-1= 0& on Σ_0,\\ v_n=0& on Σ_b,\\ (ξ_n,v_n)|_t=0=(ξ_0,v_0)& in Ω,\end{cases} $$ where $A_n=A(∇ξ_n)=(I+∇ξ_n)^-T$ and $N_n=J_nA_nn_0$ . For fixed $n∈ℕ$ , in order to solve (4.12), we first consider the linear system $$ \begin{cases}∂_tξ^1_n-v^1_n=0& on Σ_0,\\ ∂_tv_n-ν∇·D(v_n)+∇ q_n=F_1(∇ξ _n-1,v_n,q_n)& in Ω,\\ ∇· v_n=F_2(∇ξ_n-1,v_n)& in Ω,\\ (q_n-g ξ^1_n)n_0-νD(v_n)n_0=F_3(∇ξ_n-1,v_ {n})& on Σ_0,\\ v_n=0& on Σ_b,\\ ξ^1_n|_t=0=ξ^1_0& on Σ_0,\\ v_n|_t=0=v_0& in Ω.\end{cases} $$ Choose $(ξ^1_n,(0),v_n,(0),q_n,(0)):=(ξ^1_1,v_1,q_1)$ and construct the sequence $(ξ^1_n,(m),v_n,(m),q_n,(m))$ ( $∀ m∈ℕ^+$ ) of (4.13) solving $$ \begin{cases}∂_tξ^1_n,(m)-v^1_n,(m)=0& on Σ_0 ,\\ ∂_tv_n,(m)-ν∇·D(v_n,(m))+∇ q_n,(m)=F_ 1(∇ξ_n-1,v_n,(m-1),q_n,(m-1))& in Ω,\\ ∇· v_n,(m)=F_2(∇ξ_n-1,v_n,(m-1))& in Ω,\\ (q_n,(m)-g ξ^1_n,(m))n_0-νD(v_n,(m))n_0=F_3(∇ ξ_n-1,v_n,(m-1))& on Σ_0,\\ v_n,(m)=0& on Σ_b,\\ ξ^1_n,(m)|_t=0=ξ^1_0& on Σ_0,\\ v_n,(m)|_t=0=v_0& in Ω.\end{cases} $$ Thanks to Theorem 3.1, we know that the solution mapping $Υ_n:X_T→ X_T$ of the system (4.14) is well-defined, and there holds $$ \begin{split}&\|(ξ_n,(m)^1,v_n,(m),q_n,(m))\|_X_{T}\\ &≤ C_0(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_0\|_ H^1(Ω)+\|Λ_h^s-1F_1(∇ξ_n-1,v_n,(m-1),q_n,(m-1) )\|_L^2_T(L^{2(Ω))}\\ &+\|Λ_h^s-1(B^1(∇ξ_n-1),B^2(∇ξ_n-1))\|_L^ ∞(H^{1)}\|(0,v_n,(m-1),0)\|_X_{T}\\ &+\|Λ_h^s-1(B^1_t(∇ξ_n-1),B^2_t(∇ξ_n-1))\| _L^2_T(L^{2)}\|Λ_h^s-1v_n,(m-1)\|_L^∞_T(H^{1)}). \end{split} $$ For $\|Λ_h^s-1∇ξ_0\|_H^1(Ω)≤ε_2$ with $ε_2:=\min\{\frac{ε_0}{4},\frac{1}{16C_0C_2}\}$ , we may employ lemma 4.1 and an induction argument to obtain that, $∀ n≥ 2, m∈ℕ^+$ , $$ \begin{split}&\|Λ_h^s-1∇ξ_n-1\|_L^∞_T(H^{1( Ω))}+\|Λ_h^s-1∂_t∇ξ_n-1\|_L^2_T(L^{2( Ω))}≤ 4\|Λ_h^s-1∇ξ_0\|_H^1(Ω)\end{split} $$ and $$ \begin{split}&\|(ξ_n,(m)^1,v_n,(m),q_n,(m))\|_X_{T}≤ C_0(\| ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1u_0\|_H^1(Ω) )\\ & +C_0C_2\|Λ_h^s-1∇ξ_n-1\|_L^∞_T(H^{ 1(Ω))}\|(ξ^1_n,(m-1),v_1,(n),q_n,(m-1))\|_X_{T}\\ & +C_0C_2\|Λ_h^s-1∇∂_tξ_n-1\|_L^2_ {T(H^1(Ω))}\|(ξ^1_n,(m-1),v_1,(n),q_n,(m-1))\|_X_{T}\\ & ≤ C_0(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_0 \|_H^1(Ω))+16C_0^2C_2ε_2 (\|ξ_0^1\|_H^s{ (Σ_0)}+\|Λ_h^s-1v_0\|_H^1(Ω))\\ & ≤ 2C_0(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_ 0\|_H^1(Ω)).\end{split} $$ In order to show that the sequence $\{(ξ_n,(m)^1,v_n,(m),q_n,(m))\}_m=1^∞$ is a Cauchy sequence, let us consider the following system $$ \begin{cases}∂_tδξ^1_n,(m+1)-δ v^1_n,(m+1)=0& on Σ_0,\\ ∂_tδ v_n,(m+1)-ν∇·D(δ v_n,(m+1))+ ∇δ q_n,(m+1)=F_1(∇ξ_n-1,δ v_n,(m),δ q_n,(m ))& in Ω,\\ ∇·δ v_n,(m+1)=F_2(∇ξ_n-1,δ v_n,(m))& in Ω,\\ δ q_n,(m+1)n_0-gδξ^1_n,(m+1)n_0-νD(δ v_ n,(m+1))n_0=F_3(∇ξ_n-1,δ v_n,(m))& on Σ_ 0,\\ δ v_n,(m+1)=0& on Σ_b,\\ \end{cases} $$ with the initial data $δξ^1_n,(m+1)|_t=0=0$ on $Σ_0$ and $δ v_n,(m+1)|_t=0=0$ in $Ω$ . Thanks to Theorem 3.1 again, we can obtain $$ \begin{split}&\|(δξ_n,(m+1)^1,δ v_n,(m+1),δ q_n,(m+1) )\|_X_{T}≤ C_0\big{(}\|F_1(∇ξ_n-1,δ v_n,(m+1),δ q _n,(m+1))\|_L^2(L^{2(Ω))}\\ & +\|Λ_h^s-1(B^1(∇ξ_n-1),B^2(∇ξ _n-1)\|_L^∞(H^{1)}\|(0,δ v_n,(m+1),0)\|_X_{T}\\ & +\|Λ_h^s-1(B^1_t(∇ξ_n-1),B^2_t( ∇ξ_n-1)\|_L^2(L^{2)}\|(0,δ v_n,(m+1),0)\|_X_{T}\big{)}. \\ \end{split} $$ Then for $\|Λ_h^s-1∇ξ_0\|_H^1(Ω)≤ε_3$ with $ε_3:=\min\{\frac{ε_0}{4},\frac{1}{2C_0C_3}\}$ , we arrive at $$ \begin{split}\|(δξ_n,(m+1)^1,δ v_n,(m+1),δ q_n,(m+1)) \|_X_{T}≤&C_0C_3ε_3\|(δξ_n,(m)^1,δ v_n,( m),δ q_n,(m))\|_X_{T}\\ ≤&\frac{1}{2}\|(δξ_n,(m)^1,δ v_n,(m),δ q_n,(m))\|_ {X_T},\end{split} $$ which implies that $\{(ξ_n,(m)^1,v_n,(m),q_n,(m))\}_m=1^∞$ is a Cauchy sequence in $X_T$ , and denote its limit by $(ξ_n^1,v_n,q_n)$ . Similarly, we may readily verify that $(ξ_n^1,v_n,q_n)$ solves (4.13). Due to (4.10) and (4.11), we therefore get a sequence $\{(ξ_n^1,v_n,q_n)\}_m=1^∞⊂ X_T$ by an induction argument about the index $n(=2,3,4,⋯)$ and we solve $ξ_n$ in $Ω$ by the linear ODEs $$ \begin{cases}∂_tξ_n=v_n& in (0,T]×Ω,\\ ξ_n|_t=0=ξ_0& in Ω\end{cases} $$ to obtain $Λ_h^s-1∇ξ_n∈C([0,T];H^1(Ω))$ and $$ \begin{split}&\|Λ_h^s-1∇ξ_n\|_L^∞_T(H^{1(Ω) )}≤\|Λ_h^s-1∇ξ_0\|_H^1(Ω)+T^\frac{1{2}}\| Λ_h^s-1∇ v_n\|_L^2_T(H^{1(Ω))},\\ &\|Λ_h^s-1∂_t∇ξ_n\|_L^2_T(H^{1(Ω))}≤ T ^\frac{1{2}}\|Λ_h^s-1∇ v_n\|_L^∞_T(L^{2(Ω)) }.\end{split} $$ Therefore, we get that $(ξ_n,v_n,q_n)$ is a solution of (4.12) for $n∈ℕ^+$ and $n≥ 2$ , and obtain the uniform estimates of $\{(ξ_n,v_n,q_n)\}_n=1^∞$ from (4.17) and (4.18), that is ( $n∈ℕ^+$ ) $$ \begin{split}&\|(ξ_n^1,v_n,q_n)\|_X_{T}≤ 2C_0\big{(}\|ξ_0 ^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v_0\|_H^1(Ω)\big{)} ,\\ &\|Λ_h^s-1∇ξ_n\|_L^∞_T(H^{1(Ω))}+\|Λ_ {h}^s-1∂_t∇ξ_n\|_L^2_T(L^{2(Ω))}≤ 4\|Λ _h^s-1∇ξ_0\|_H^1(Ω).\end{split} $$ Step 3: Convergence of the approximate sequence. With Step 1 and Step 2 in hand, we will show that the sequence $\{(ξ^1_n,v_n,q_n)\}_n=1^∞$ is a Cauchy sequence. In order to do so, we consider the system $$ \begin{cases}∂_tδξ^1_n+1-δ v^1_n+1=0& on Σ_0,\\ ∂_tδ v_n+1-ν∇·D(δ v_n+1)+∇ δ q_n+1=δ F_1(∇ξ_n,v_n+1,q_n+1)& in Ω,\\ ∇·δ v_n+1=δ F_2(∇ξ_n,v_n+1)& in Ω,\\ δ q_n+1n_0-gδξ^1_n+1n_0-νD(δ v_n+1)n_ 0=δ F_3(∇ξ_n,v_n+1)& on Σ_0,\\ δ v_n+1=0& on Σ_b,\\ δξ^1_n+1|_t=0=0& on Σ_0,\\ δ v_n+1)|_t=0=0& in Ω.\end{cases} $$ Thanks to Theorem 3.1 and (4.19), we obtain $$ \begin{split}&\|(δξ_n+1^1,δ v_n+1,δ q_n+1)\|_X_{T} ≤ C_0C_4\|Λ_h^s-1∇ξ_0\|_H^1(Ω)\|(δξ_ {n,(m+1)}^1,δ v_n,(m+1),δ q_n,(m+1))\|_X_{T}\\ & +2C_0C_4\big{(}\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^ s-1v_0\|_H^1(Ω)\big{)}\big{(}\|Λ_h^s-1∇δξ_n \|_L^∞(H^{1)}+\|Λ_h^s-1∇∂_tδξ_n\|_ L^2(L^{2)}\big{)}.\end{split} $$ Since $$ \begin{split}&\|Λ_h^s-1∇δξ_n\|_L^∞_T(H^{1)} ≤ T^\frac{1{2}}\|Λ_h^s-1∇δ v_n\|_L^2_T(H^{1)} , \|Λ_h^s-1∇∂_tδξ_n\|_L^2_T(L^{2)}≤ T ^\frac{1{2}}\|Λ_h^s-1∇δ v_n\|_L^∞_T(L^{2)}, \end{split} $$ we have $$ \begin{split}\|(δξ_n+1^1,δ v_n+1,δ q_n+1)\|_X_{T}& ≤ C_0C_4\|Λ_h^s-1∇ξ_0\|_H^1(Ω)\|(δξ_ {n,(m+1)}^1,δ v_n,(m+1),δ q_n,(m+1))\|_X_{T}\\ &+2C_0C_4\big{(}\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^s-1v _0\|_H^1(Ω)\big{)}T^\frac{1{2}}\|(δξ_n^1,δ v_n, δ q_n)\|_X_{T}.\end{split} $$ Let $\|Λ_h^s-1∇ξ_0\|_H^1(Ω)≤\frac{1}{3C_0C_4}$ and $T≤\frac{1}{8C_0C_4(\|ξ_0^1\|_H^s{(Σ_0)}+\|Λ_h^ {s-1}v_0\|_H^1(Ω))},$ then we obtain $$ \begin{split}\|(δξ_n+1^1,δ v_n+1,δ q_n+1)\|_X_{T} ≤\frac{1}{2}\|(δξ_n^1,δ v_n,δ q_n)\|_X_{T},\end{split} $$ which follows that the sequence $\{(ξ_n^1,v_n,q_n)\}_n=1^∞$ is a Cauchy sequence in $X_T$ . Hence, there exists $(ξ^1,v,q)∈ X_T$ so that $$ (ξ^1_n,v_n,q_n)→(ξ^1,v,q) in X_T. $$ With $v$ in hand, we can solve $ξ$ in $Ω$ by $$ \begin{cases}∂_tξ=v& in (0,T]×Ω,\\ ξ|_t=0=ξ_0& in Ω\end{cases} $$ to obtain $Λ_h^s-1∇ξ∈C([0,T];H^1(Ω))$ . Finally, we can readily verify that $(ξ,v,q)$ is a solution of (1.10) and satisfies (1.12). This finishes the proof of existence part of Theorem 1.1. ### 4.2 Uniqueness part of the proof of Theorem 1.1 Let $({ξ},{v},{q})$ and $(\widetilde{ξ},\widetilde{v},\widetilde{q})$ be two solutions to (4.1) with the initial data $({ξ}_0,{v}_0)$ and $(\widetilde{ξ}_0,\widetilde{v}_0)$ respectively , and satisfy $$ \begin{split}Λ_h^s-1(∇{ξ}, ∇\widetilde{ξ})&∈ C([0,T];H^1(Ω)), ({ξ}^1, \widetilde{ξ}^1)∈ C([0,T];H^s(Σ_0)),\\ Λ_h^s-1({v}, \widetilde{v})&∈C([0,T]; {{}_0}{H}^1( Ω))∩ L^2(0,T;H^2(Ω)), Λ_h^s-1({q}, \widetilde{ q})∈ L^2(0,T;H^1(Ω)).\end{split} $$ Set $δξ\buildrel\hbox{ def}\over{=}ξ-\widetilde{ξ}$ , $δ v=v-\widetilde{v}$ , and $δ q=q-\widetilde{q}$ . Then we have $$ \begin{cases}∂_tδξ-δ v=0,& in Ω,\\ ∂_tδ v-ν∇·D(δ v)+∇δ q=δ F _1,& in Ω,\\ ∇·δ v=δ F_2,& in Ω,\\ δ qn_0-gδξ^1n_0-νD(δ v)n_0=δ F_3,& on Σ_0,\\ δ v=0,& on Σ_b,\\ (δξ,δ v)|_t=0=(δξ_0,δ v_0),& in Ω,\end{cases} $$ where $δ F_1:=F_1(∇ξ,v,q)-F_1(∇\widetilde{ξ},\widetilde{v}, \widetilde{q})$ , $δ F_2:=B^1(∇ξ):∇ v-B^1(∇\widetilde{ξ}):∇ \widetilde{v}$ , $δ F_3:=B^2(∇ξ):∇_hv-B^2(∇\widetilde{ξ}):∇ _h\widetilde{v}$ . If $δξ_0=0$ and $δ v_0=0$ in $Ω$ , then, according to Theorem 3.1, we obtain $$ \begin{split}\|(δξ^1,δ v,δ q)\|_X_{t}≤&C_0\mathfrak{ C} t^\frac{1{2}}\|(0,δ v,δ q)\|_X_{t}\end{split} $$ for any $0≤ t≤ T$ , where the positive constant $\mathfrak{C}$ depends on $\|Λ_h^s-1(∇ξ,∇\tilde{ξ})\|_L_{T^∞(H^1( Ω))}$ , $\|(0,v,q)\|_X_{T}$ and $\|(0,\tilde{v},\tilde{q})\|_X_{T}$ . Let $0≤ t≤ t_0$ with $t_0≤\frac{1}{4\mathfrak{C}^2C^2_0}$ , there holds $$ \begin{split}\|(δξ^1,δ v,δ q)\|_X_{t}≤&\frac{1}{2}\|(0 ,δ v,δ q)\|_X_{t}.\end{split} $$ which implies that $(δξ^1,δ v,δ q)=0$ in $[0,t_0]$ , and then $∇δξ=0$ in $[0,t_0]$ from (4.1) 1. The uniqueness of such strong solutions on the time interval $[0,T]$ then follows by a bootstrap argument. At the same lines, we may get that the solution $({ξ},{v},{q})$ to (4.1) depends continuously on the initial data $({ξ}_0,{v}_0)$ in $[0,T]$ . We then complete the proof of uniqueness. ## 5 Appendix ### 5.1 The expressions of $B$ forms We denote $B^1:∇ v=B^1_i,j∂_iv^j$ $(i,j=1,2,3)$ and $(B^2:∇_hv)^j=B_α,i^2,j∂_αv^i$ $(i,j=1,2,3,α=2,3)$ . Here, $B^1:=I-a_11^-1J A$ and $B^2$ is a $3× 2× 3$ tensor defined by $$ \begin{split}&B^2,2:∇_hv\buildrel\hbox{ def}\over{=}-ν a _11^-1|\overrightarrow{\rm{a}_1}|^-4\bigg{[}-a_11(a_11^2+a_31^ {2}-|\overrightarrow{\rm{a}_1}|^4)∂_2v^1\\ & +(a_11^2+a_31^2)(a_21B_1-a_11B_2): ∇_hv+|\overrightarrow{\rm{a}_1}|^2a_21a_11^2(-∇_h· v^h+B_h:∇_hv)\\ & -a_21a_31(-a_11∂_3v^1+(a_31B_1-a_11 B_3):∇_hv)\bigg{]},\\ \end{split} $$ $$ \begin{split}&B^2,3:∇_hv\buildrel\hbox{ def}\over{=}-ν a _11^-1|\overrightarrow{\rm{a}_1}|^-4\bigg{[}-a_11(a_11^2+a_21^ {2}-|\overrightarrow{\rm{a}_1}|^4)∂_3v^1\\ & +(a_11^2+a_21^2)(a_31B_1-a_11B_3): ∇_hv+|\overrightarrow{\rm{a}_1}|^2a_31a_11^2(-∇_h· v^h+B_h:∇_hv)\\ & -a_21a_31(-a_11∂_2v^1+(a_21B_1-a_11 B_2):∇_hv)\bigg{]}.\end{split} $$ and $$ \begin{split}&B^2,1:∇_hv\buildrel\hbox{ def}\over{=}-ν a _11^-1|\overrightarrow{\rm{a}_1}|^-2J^-1\bigg{[}-2a_21| \overrightarrow{\rm{a}_1}|^2J∂_2v^1-2a_31|\overrightarrow{\rm {a}_1}|^2J∂_3v^1\\ & +2J(a_21|\overrightarrow{\rm{a}_1}|^2B^2,2+a_31| \overrightarrow{\rm{a}_1}|^2B^2,3+a_11|\overrightarrow{\rm{a}_1}|^2 B_h):∇_hv,\\ & +2a_11|\overrightarrow{\rm{a}_1}|^2((1-J)∇_h· v^h- B_4:∇_hv)-a_11(a_β 1∂_βv^1+a_j1 B_j:∇_hv)\bigg{]}.\end{split} $$ where $\overrightarrow{\rm{a}_1}:=(a_11,a_21,a_31)^T$ , $$ \begin{split}B_1:∇_hv:=≤ft(\begin{array}[]{cc}a_i2a_i1 +a_12a_11&a_i3a_i1+a_13a_11\\ a_12a_21&a_13a_21\\ a_12a_31&a_13a_31\end{array}\right):≤ft(\begin{array}[]{ccc}∂ _2v^1&∂_3v^1\\ ∂_2v^2&∂_3v^2\\ ∂_2v^3&∂_3v^3\end{array}\right),\end{split} $$ $$ \begin{split}B_2:∇_hv:=≤ft(\begin{array}[]{cc}a_22a_11 -1&a_23a_11\\ a_i2a_i1+a_22a_21&a_i3a_i1+a_23a_31\\ a_22a_31&a_23a_31\end{array}\right):≤ft(\begin{array}[]{ccc}∂ _2v^1&∂_3v^1\\ ∂_2v^2&∂_3v^2\\ ∂_2v^3&∂_3v^3\end{array}\right),\end{split} $$ $$ \begin{split}B_3:∇_hv:=≤ft(\begin{array}[]{cc}a_32a_11 &a_33a_11-1\\ a_32a_21&a_33a_21\\ a_i2a_i1+a_32a_31&a_i3a_i1+a_33a_31\end{array}\right):≤ft( \begin{array}[]{ccc}∂_2v^1&∂_3v^1\\ ∂_2v^2&∂_3v^2\\ ∂_2v^3&∂_3v^3\end{array}\right),\end{split} $$ $$ \begin{split}B_4:∇_hv:=≤ft(\begin{array}[]{cc}-a_12&-a_ {13}\\ -a_22+1&-a_23\\ -a_32&-a_33+1\end{array}\right):≤ft(\begin{array}[]{ccc}∂_2v^1 &∂_3v^1\\ ∂_2v^2&∂_3v^2\\ ∂_2v^3&∂_3v^3\end{array}\right),\end{split} $$ and $$ \begin{split}B_h:∇_hv:=-a_11^-1≤ft(\begin{array}[]{cc}a_12&a_ {13}\\ a_22-a_11&a_23\\ a_32&a_33-a_11\end{array}\right):≤ft(\begin{array}[]{ccc}∂_2v ^1&∂_3v^1\\ ∂_2v^2&∂_3v^2\\ ∂_2v^3&∂_3v^3\end{array}\right).\end{split} $$ Acknowledgments. 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