In this section, we first give a description of the perturbed compressible electrohydrodynamics and then prove its unique existence of solutions.
For given \(\varepsilon\in(0,1]\), we consider the following initial boundary value problem:
$$\begin{aligned} \textstyle\begin{cases} a_{1}\frac{\partial v_{\epsilon}}{\partial t} +a_{1}\mathcal{B}(v _{\epsilon},v_{\epsilon})+\frac{1}{2}a_{1}(\operatorname{div}v_{\epsilon})v _{\epsilon}+a_{2}\mathcal{A}_{0}v_{\epsilon}+\mathcal{N}(q_{\epsilon }+\tilde{Q}_{0},\varphi_{\epsilon}+\tilde{\phi}_{0}) \\ \quad {}-\frac{1}{2}\nabla(q_{\epsilon}+\tilde{Q}_{0})(\varphi_{\epsilon}+ \tilde{\phi}_{0})+\nabla p_{\epsilon}=f,\quad \text{in } \Omega\times(0,T), \\ \frac{\partial q_{\epsilon}}{\partial t} +a_{3}\mathcal{A}_{1}q_{ \epsilon}+a_{4}q_{\epsilon}-\mathcal{M}(q_{\epsilon}+\tilde{Q}_{0},v _{\epsilon})\\ \quad {}+\frac{1}{2}(\operatorname{div}v_{\epsilon})(q_{\epsilon}+ \tilde{Q}_{0})=0,\quad \text{in } \Omega\times(0,T), \\ \mathcal{A}_{1}\varphi_{\epsilon}=q_{\epsilon},\quad \text{in } \Omega \times(0,T), \\ \epsilon\frac{\partial p_{\epsilon}}{\partial t}+\operatorname{div}v_{ \epsilon}=0,\quad \text{in } \Omega\times(0,T), \end{cases}\displaystyle \end{aligned}$$
(3.1)
supplemented with the initial boundary value conditions
$$\begin{aligned}& v_{\epsilon}=0,\quad q_{\epsilon} =0, \quad \varphi_{\epsilon}=0, \quad x\in\partial \Omega, \end{aligned}$$
(3.2)
$$\begin{aligned}& v_{\epsilon}(x,0)=v_{0},\quad q_{\epsilon}(x,0) =q_{0},\quad \varphi_{\epsilon }(x,0)=\varphi_{0}, \quad p_{\epsilon}(x,0)=p_{0},\quad x\in\Omega, \end{aligned}$$
(3.3)
where \(v_{0}\), \(q_{0}\) satisfy (2.1) and (2.2), the function \(p_{0}\) is independent of ϵ and
$$\begin{aligned} p_{0}\in L^{2}(\Omega). \end{aligned}$$
(3.4)
Remark 3.1
The terms \(\frac{1}{2}( \operatorname{div}v_{\epsilon})v_{\epsilon}\), \(\frac{1}{2}\nabla(q_{\epsilon}+\tilde{Q}_{0})(\varphi_{\epsilon}+ \tilde{\phi}_{0})\) and \(\frac{1}{2}( \operatorname{div}v_{\epsilon})(q_{ \epsilon}+\tilde{Q}_{0})\) in (3.1) are the stabilization terms corresponding to the substitution of the trilinear form b̂ for the form b and the trilinear form n̂ for the form n, where the trilinear form b̂, n̂ is given by
$$\begin{aligned}& \hat{b}(u,v,w)=\frac{1}{2} \bigl[b(u,v,w)-b(u,w,v) \bigr],\quad \forall u,v,w \in {\mathbb{H}}^{1}_{0}(\Omega), \end{aligned}$$
(3.5)
$$\begin{aligned}& \hat{n}(q,\varphi,v)=\frac{1}{2} \bigl[n(q,\varphi,v)-n(\varphi,q,v) \bigr],\quad \forall v\in{\mathbb{H}}^{1}_{0}(\Omega),\quad q, \varphi \in H^{1}_{0}( \Omega). \end{aligned}$$
(3.6)
Note that if \(\operatorname{div}u\neq0\), then \(b(u,u,u)\neq0\), \(n(q,q,v) \neq0\). But \(\hat{b}(u,v,v)=0\), \(\hat{n}(q,q,v)=0\), for \(\forall u,v \in{\mathbb{H}}^{1}_{0}(\Omega)\), \(q\in H^{1}_{0}(\Omega)\).
Let us assume that \(\{v_{\epsilon},\varphi_{\epsilon},q_{\epsilon},p _{\epsilon}\}\) is the classical solution of system (3.1), that is, \(v_{\epsilon}\in\mathcal{C}^{2}(\overline{Q})\), \(\varphi_{ \epsilon}\in\mathcal{C}^{2}(\overline{Q})\), \(q_{\epsilon}\in \mathcal{C}^{2}(\overline{Q})\), \(p_{\epsilon}\in\mathcal{C}^{1}( \overline{Q})\). Then we have \(\phi\in{\mathcal{C}}^{\infty}_{0}( \Omega)\times{\mathcal{C}}^{\infty}_{0}(\Omega)\) and \(\varrho \in{\mathcal{C}}^{\infty}_{0}(\Omega)\). Multiplying equation (3.1)1 by ϕ, (3.1)2-(3.1)4 by ϱ, we obtain
$$\begin{aligned}& a_{1}\frac{d}{d t}(v_{\epsilon},\phi)+a_{2} \bigl((v_{\epsilon},\phi) \bigr)+a _{1}\hat{b}(v_{\epsilon },v_{\epsilon}, \phi)+\hat{n}(q_{\epsilon}+ \tilde{Q}_{0},\varphi_{\epsilon}+ \tilde{\phi}_{0},\phi)+(\nabla p _{\epsilon},\phi)=(f,\phi), \\& \frac{d}{d t}(q_{\epsilon},\varrho)+a_{3} \bigl((q_{\epsilon},\varrho) \bigr)+a_{4}(q_{\epsilon},\varrho)- \hat{n}(q_{\epsilon}+\tilde{Q}_{0}, \varrho,v_{\epsilon})=0, \\& \bigl((\varphi_{\epsilon},\varrho) \bigr)=(q_{\epsilon},\varrho), \\& \epsilon\frac{d}{d t} (p_{\epsilon},\varrho) +(\operatorname{div}v_{ \epsilon}, \varrho)=0. \end{aligned}$$
Let b̂, n̂ be defined by (3.5), (3.6) and set \(\hat{B}(u)=\hat{B}(u,u)\), \(\hat{N}(q)=\hat{N}(q,q)\) via
$$\begin{aligned}& \bigl\langle \hat{B}(u),v \bigr\rangle =\hat{b}(u,u,v),\quad \forall u,v\in{ \mathbb{H}} ^{1}_{0}(\Omega), \end{aligned}$$
(3.7)
$$\begin{aligned}& \bigl\langle \hat{N}(q),v \bigr\rangle =\hat{n}(q,q,v), \quad \forall v\in{ \mathbb{H}} ^{1}_{0}(\Omega), q\in H^{1}_{0}( \Omega). \end{aligned}$$
(3.8)
Then like \(B(u)\), \(N(q)\), the operators \(\hat{B}(u)\), \(\hat{N}(q)\) are continuous on \({\mathbb{H}}^{1}_{0}(\Omega)\), \(H^{1}_{0}(\Omega)\), respectively.
Lemma 3.1
If
\(v\in L^{2}(0,T;{\mathbb{H}}^{1}_{0}(\Omega))\cap L^{\infty}(0,T; \mathbb{H})\), \(q\in L^{2}(0,T;H^{1}_{0}(\Omega))\cap L^{\infty}(0,T;L ^{2}(\Omega))\), \(\varphi\in L^{2}(0,T;H^{1}_{0}(\Omega))\cap L^{ \infty}(0,T;L^{2}(\Omega))\)
then the function
\(t\rightarrow\hat{B}(v(t))\)
belongs to
\(L^{4/3}(0,T, \mathbb{H}^{-1}( \Omega))\), \(t\rightarrow\hat{N}(q(t),\varphi(t))\)
belongs to
\(L^{4/3}(0,T,\mathbb{H}^{-1}(\Omega))\), \(t\rightarrow \hat{M}(q(t),v(t))\)
belongs to
\(L^{4/3}(0,T,H^{-1}(\Omega))\).
Proof
For almost all \(t\in(0,T)\), \(\hat{B}(v(t))\), \(\hat{N}(q(t))\) are the elements of \(\mathbb{H}^{-1}(\Omega)\), \(\hat{M}(q(t),v(t))\) is an element of \(H^{-1}(\Omega)\), and the measurability of the functions \(t\rightarrow\hat{B}(v(t))\), \(t\rightarrow\hat{N}(q(t))\), \(t\rightarrow\hat{M}(q(t),v(t))\) is easy to check. Now for \(\forall\phi\in{\mathbb{H}}^{1}_{0}(\Omega)\), \(\forall\psi \in H^{1}_{0}(\Omega)\) by the Hölder inequality and
$${ \Vert u\Vert _{\mathbb{L}^{4}}\leq C\vert u\vert ^{1/2}\Vert u \Vert ^{1/2},\quad \forall u\in\mathbb{R}^{2}}, $$
we have
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \bigl\langle \hat{B} \bigl(v(t) \bigr),\phi \bigr\rangle \bigr\vert &= \biggl\vert \frac{1}{2} \bigl[b \bigl(v(t),v(t),\phi \bigr)-b \bigl(v(t),\phi,v(t) \bigr) \bigr] \biggr\vert \\ &\leq\frac{1}{2}\sum_{i,j=1}^{2} \biggl( \int_{\Omega} \biggl\vert v_{i}\frac{\partial v_{j}}{\partial x_{i}} \phi_{j} \biggr\vert \,dx+ \int_{\Omega} \biggl\vert v_{i}\frac{\partial\phi_{j}}{\partial x_{i}}v_{j} \biggr\vert \,dx \biggr) \\ &\leq C \bigl(\Vert v\Vert _{\mathbb{L}^{4}(\Omega)}\Vert v\Vert \Vert \phi \Vert _{\mathbb{L} ^{4}(\Omega)}+\Vert v\Vert ^{2}_{\mathbb{L}^{4}(\Omega)}\Vert \phi \Vert \bigr) \\ &\leq C \bigl(\vert v\vert ^{\frac{1}{2}}\Vert v\Vert ^{\frac{3}{2}}+ \vert v\vert \Vert v\Vert \bigr)\Vert \phi \Vert , \end{aligned} \end{aligned}$$
(3.9)
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \bigl\langle \hat{N} \bigl(q(t), \varphi (t) \bigr), \phi \bigr\rangle \bigr\vert &= \biggl\vert \frac{1}{2} \bigl[n \bigl(q(t), \varphi(t),\phi \bigr)-n \bigl( \varphi(t),q(t),\phi(t) \bigr) \bigr] \biggr\vert \\ &\leq\frac{1}{2}\sum_{i=1}^{2} \biggl( \int_{\Omega} \biggl\vert q\frac{\partial\varphi}{\partial x}\phi_{i} \biggr\vert \,dx+ \int_{\Omega} \biggl\vert \varphi\frac{\partial q}{\partial x} \phi_{i} \biggr\vert \,dx \biggr) \\ &\leq C \bigl(\Vert q\Vert _{\mathbb{L}^{4}(\Omega)}\Vert \varphi \Vert _{\mathbb{L}^{4}( \Omega)}\vert \phi \vert +\Vert \varphi \Vert _{\mathbb {L}^{4}(\Omega)} \Vert q \Vert \Vert \phi \Vert _{\mathbb{L}^{4}(\Omega)} \bigr) \\ &\leq C \bigl(\vert q\vert ^{\frac{1}{2}}\Vert q\Vert ^{\frac{1}{2}} \vert \varphi \vert ^{\frac{1}{2}} \Vert \varphi \Vert ^{\frac{1}{2}}+ \vert \varphi \vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{ \frac{1}{2}} \Vert q \Vert \bigr)\Vert \phi \Vert , \end{aligned} \end{aligned}$$
(3.10)
and
$$\begin{aligned} & \bigl\vert \bigl\langle \hat{M} \bigl(q(t),v(t) \bigr),\psi \bigr\rangle \bigr\vert \\ &\quad = \biggl\vert \frac{1}{2} \bigl[n \bigl(q(t),\psi (t),v(t) \bigr)-n \bigl(\psi(t),q(t),v(t) \bigr) \bigr] \biggr\vert \\ &\quad \leq\frac{1}{2}\sum_{i=1}^{2} \biggl( \int_{\Omega} \biggl\vert q\frac{\partial\varphi}{\partial x}\phi_{i} \biggr\vert \,dx+ \int_{\Omega} \biggl\vert \varphi\frac{\partial q}{\partial x} \phi_{i} \biggr\vert \,dx \biggr) \\ &\quad \leq C \bigl(\Vert q\Vert _{\mathbb{L}^{4}(\Omega)}\Vert v \Vert _{\mathbb{L}^{4}(\Omega )} \Vert \psi \Vert +\Vert \psi \Vert _{\mathbb{L}^{4}(\Omega )}\Vert q\Vert \Vert v \Vert _{ \mathbb{L}^{4}(\Omega)} \bigr) \\ &\quad \leq C \bigl(\vert q\vert ^{\frac{1}{2}}\Vert q \Vert ^{\frac{1}{2}} \vert v\vert ^{\frac{1}{2}}\Vert v \Vert ^{\frac{1}{2}}+\vert v\vert ^{\frac{1}{2}}\Vert v \Vert ^{\frac{1}{2}}\Vert q\Vert \bigr)\Vert \psi \Vert , \end{aligned}$$
(3.11)
(3.9), (3.10), and (3.11) imply that
$$\begin{aligned}& \bigl\Vert \hat{B} \bigl(v(t) \bigr) \bigr\Vert _{\mathbb {H}^{-1}(\Omega)} \leq C \bigl(\vert v\vert ^{\frac{1}{2}} \Vert v\Vert ^{\frac{3}{2}}+ \vert v\vert \Vert v\Vert \bigr), \end{aligned}$$
(3.12)
$$\begin{aligned}& \bigl\Vert \hat{N} \bigl(q(t),\varphi(t) \bigr) \bigr\Vert _{\mathbb{H}^{-1}(\Omega)}\leq C \bigl(\vert q\vert ^{ \frac{1}{2}}\Vert q\Vert ^{\frac{1}{2}}\vert \varphi \vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{\frac{1}{2}}+\vert \varphi \vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{\frac{1}{2}} \Vert q \Vert \bigr), \end{aligned}$$
(3.13)
$$\begin{aligned}& \bigl\Vert \hat{M} \bigl(q(t),v(t) \bigr) \bigr\Vert _{H^{-1}(\Omega)}\leq C \bigl(\vert q\vert ^{\frac{1}{2}}\Vert q\Vert ^{\frac{1}{2}}\vert v\vert ^{\frac{1}{2}}\Vert v\Vert ^{\frac{1}{2}}+ \vert v\vert ^{\frac{1}{2}} \Vert v\Vert ^{\frac{1}{2}}\Vert q\Vert \bigr). \end{aligned}$$
(3.14)
Therefore, the lemma follows from (3.12)-(3.14). □
Now if \(v_{\epsilon}\in L^{2}(0,T;{\mathbb{H}}^{1}_{0}(\Omega)) \cap L^{\infty}(0,T;\mathbb{H})\), \(q_{\epsilon}\in L^{2}(0,T;H^{1} _{0}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))\), \(\varphi_{\epsilon }\in L^{2}(0,T;H^{1}_{0}(\Omega))\cap L^{\infty}(0,T;L^{2}(\Omega))\), and \(p_{\epsilon}\in L^{2}(0,T;L^{2}(\Omega))\) satisfy (3.1)-(3.2) in the distribution sense, then
$$\begin{aligned}& a_{1}\frac{d}{dt}(v_{\epsilon},\phi)= \bigl\langle f-a_{1}\hat{B}(v_{ \epsilon},v_{\epsilon})-a_{2}{ \mathcal{A}_{0}}v_{\epsilon } -\hat{N}(q_{\epsilon}+ \tilde{Q}_{0},\varphi_{\epsilon}+ \tilde{\phi}_{0})- \nabla p_{\epsilon},\phi \bigr\rangle , \end{aligned}$$
(3.15)
$$\begin{aligned}& \frac{d}{dt}(q_{\epsilon},\psi)=- \bigl\langle a_{3}{ \mathcal{A}_{1}}q_{\epsilon}+a_{4}q_{\epsilon} - \hat{M}(q_{\epsilon }+\tilde{Q}_{0},v_{\epsilon}),\psi \bigr\rangle . \end{aligned}$$
(3.16)
Since \(p_{\epsilon}\in L^{2}(0,T;L^{2}(\Omega))\), we see from Lemma 3.1 that
$$\begin{aligned}& \begin{aligned}[b] v^{\prime}_{\epsilon} &= \frac{1}{a_{1}} \bigl(f-a_{1} \hat{B}(v_{\epsilon },v_{\epsilon})-a_{2}{ \mathcal{A}_{0}}v_{\epsilon}- \hat{N}(q_{\epsilon}+ \tilde{Q}_{0},\varphi_{\epsilon}+\tilde{ \phi} _{0})- \nabla p_{\epsilon} \bigr) \\ &\in L^{4/3} \bigl(0,T;\mathbb{H}^{-1} \bigr), \end{aligned} \end{aligned}$$
(3.17)
$$\begin{aligned}& q^{\prime}_{\epsilon} =- \bigl(a_{3}{ \mathcal{A}_{1}}q _{\epsilon}+a_{4}q_{\epsilon}- \hat{M}(q_{\epsilon}+ \tilde{Q}_{0},v _{\epsilon}) \bigr)\in L^{4/3} \bigl(0,T;H^{-1} \bigr); \end{aligned}$$
(3.18)
similarly,
$$ p^{\prime}_{\epsilon}\in L^{2} \bigl(0,T;L^{2}(\Omega) \bigr). $$
(3.19)
The above analysis leads to the following weak formulation of the problem described by (3.1).
Problem 3.1
Let \(\epsilon\in(0,1]\) be fixed. For any given \(v_{0}\), \(q_{0}\) and \(p_{0}\) satisfying (2.1), (2.2), and (3.4), find \(\{v_{\epsilon},q_{\epsilon},\varphi_{\epsilon},p _{\epsilon}\}\) such that
$$\begin{aligned} &v_{\epsilon}\in L^{2} \bigl(0,T;{\mathbb{H}}^{1}_{0}( \Omega) \bigr)\cap L^{ \infty}(0,T;\mathbb{H}),\quad v^{\prime}_{\epsilon} \in L^{4/3} \bigl(0,T;{ \mathbb{H}}^{-1}(\Omega) \bigr), \end{aligned}$$
(3.20)
$$\begin{aligned} &q_{\epsilon}\in L^{2} \bigl(0,T;H^{1}_{0}( \Omega) \bigr)\cap L^{\infty} \bigl(0,T;L ^{2}(\Omega) \bigr), \quad q^{\prime}_{\epsilon}\in L^{4/3} \bigl(0,T;H^{-1}( \Omega) \bigr), \end{aligned}$$
(3.21)
$$\begin{aligned} &\varphi_{\epsilon}\in L^{2} \bigl(0,T;H^{1}_{0}( \Omega) \bigr); \end{aligned}$$
(3.22)
$$\begin{aligned} &p_{\epsilon}\in L^{2} \bigl(0,T;L^{2}(\Omega) \bigr), \quad p^{\prime}_{\epsilon}\in L^{2} \bigl(0,T;L^{2}( \Omega) \bigr), \end{aligned}$$
(3.23)
$$\begin{aligned} & \begin{aligned}[b] &a_{1}v'_{\epsilon}+a_{1}{ \mathcal{B}}(v_{\epsilon},v _{\epsilon})+\frac{1}{2}a_{1}( \operatorname{div}v_{\epsilon})v_{\epsilon}+a _{2}{ \mathcal{A}_{0}}v_{\epsilon}+ \mathcal{N}(q_{\epsilon }+ \tilde{Q}_{0},\varphi_{\epsilon}+ \tilde{\phi}_{0}) \\ &\quad {}- \frac{1}{2} \nabla(q_{\epsilon}+ \tilde{Q}_{0}) ( \varphi_{\epsilon}+ \tilde{\phi}_{0})+ \nabla p_{\epsilon}=f, \end{aligned} \end{aligned}$$
(3.24)
$$\begin{aligned} &q'_{\epsilon}+a_{3}{ \mathcal{A}_{1}}q_{\epsilon}+a_{4}q _{\epsilon}-{ \mathcal{M}}(q_{\epsilon}+\tilde{Q}_{0},v _{\epsilon})+\frac{1}{2}(\operatorname{div}v_{\epsilon}) (q_{\epsilon}+ \tilde{Q}_{0})=0, \end{aligned}$$
(3.25)
$$\begin{aligned} &{ \mathcal{A}_{1}}\varphi_{\epsilon}=q_{\epsilon}, \end{aligned}$$
(3.26)
$$\begin{aligned} &\epsilon p'_{\epsilon}+\operatorname{div}v_{\epsilon}=0, \end{aligned}$$
(3.27)
$$\begin{aligned} &v_{\epsilon}(x,0)=v_{0},\quad q_{\epsilon}(x,0)=q_{0}, \quad \varphi_{\epsilon }(x,0)=\varphi_{0},\quad p_{\epsilon}(x,0)=p_{0},\quad x \in\Omega. \end{aligned}$$
(3.28)
Theorem 3.1
Let
\(\epsilon\in(0,1]\)
be fixed. For any given
\(v_{0}\), \(q_{0}\), and
\(p_{0}\)
satisfying (2.1), (2.2), and (3.4). Then there exists a unique solution
\(\{v_{\epsilon},q_{\epsilon}, \varphi_{\epsilon},p_{\epsilon}\}\)
of Problem
3.1.
Proof
(i) In order to apply the Galerkin procedure, we consider a basis of \(\mathbb{H}^{1}_{0}(\Omega)\) constituted of elements \(\omega_{i}\) of \(\mathcal{D}(\Omega)\), a basis of \(H^{1}_{0}(\Omega)\) constituted of elements \(w_{i}\) of \(\mathcal{C}^{\infty}_{0}(\Omega)\), and a basis of \(L^{2}(\Omega)\) constituted of elements \(\gamma_{i}\) of \(\mathcal{C}^{\infty}_{0}(\Omega)\).
For each m, we define an approximate solution \(v_{\epsilon m}\), \(q _{\epsilon m}\), \(\varphi_{\epsilon m}\), \(p_{\epsilon m}\) of Problem 3.1 by
$$\begin{aligned} \begin{aligned}[c] &v_{\epsilon m}(t)=\sum _{i=1}^{m} g_{im}(t)\omega_{i},\quad\varphi_{\epsilon m}(t) =\sum_{i=1}^{m} h_{im}(t)w_{i}, \\ &p_{\epsilon m}(t)=\sum_{j=1}^{m} \xi_{jm}(t)\gamma_{j},\quad q_{\epsilon m}(t)={ \mathcal{A}_{1}}\varphi_{\epsilon m}(t), \end{aligned} \end{aligned}$$
(3.29)
and
$$\begin{aligned} &a_{1} \bigl(v'_{\epsilon m},\omega_{k} \bigr)+a_{2} \bigl((v_{\epsilon m},\omega_{k}) \bigr) +a _{1}\hat{b}(v_{\epsilon m},v_{\epsilon m},\omega_{k})+ \hat{n}(q_{ \epsilon m}+\tilde{Q}_{0},\varphi_{\epsilon m}+\tilde{ \phi}_{0}, \omega_{k}) \\ &\quad {}+(\nabla p_{\epsilon m}, \omega_{k})=(f, \omega_{k}), \end{aligned}$$
(3.30)
$$\begin{aligned} & \bigl(q'_{\epsilon m},w_{k} \bigr)+a_{3} \bigl((q_{\epsilon m},w_{k}) \bigr)+a_{4}(q_{ \epsilon m},w_{k})- \hat{n}(q_{\epsilon m}+\tilde{Q}_{0},w_{k},v_{ \epsilon m})=0, \end{aligned}$$
(3.31)
$$\begin{aligned} & \bigl((\varphi_{\epsilon m},w_{k}) \bigr)=(q_{\epsilon m},w_{k}), \end{aligned}$$
(3.32)
$$\begin{aligned} &\epsilon \bigl(p'_{\epsilon m},\gamma_{l} \bigr) +( \operatorname{div}v_{\epsilon m}, \gamma_{l})=0, \quad k,l=1,\ldots,m. \end{aligned}$$
(3.33)
Moreover, this differential system is required to satisfy the initial conditions
$$\begin{aligned} v_{\epsilon m}(0)=v_{0m},\quad q_{\epsilon m}(0)=q_{0m}, \quad \varphi_{\epsilon m}(0)=\varphi_{0m},\quad p_{\epsilon m}(0)=p_{0m}. \end{aligned}$$
(3.34)
Here \(v_{0m}\) (or \(q_{0m}\) or \(\varphi_{0m}\) or \(p_{0m}\)) is the orthogonal projection of \(v_{0}\) (or \(q_{0}\) or \(\varphi_{0}\) or \(p_{0}\)) onto the space spanned by \(\omega_{1},\ldots,\omega_{m}\) (or \(w_{1},\ldots,w_{m}\) or \(w_{1},\ldots,w_{m}\) or \(\gamma_{1},\ldots ,\gamma_{m}\)) in \(\mathbb{H}^{1}_{0}\) (resp. \(H^{1}_{0}\) or \(H^{1}_{0}\) or \(L^{2}(\Omega)\)).
Equations (3.30)-(3.33) form a nonlinear differential system for the functions \(g_{1m},\ldots, g_{mm},h_{1m}, \ldots,h_{mm},\xi_{1m},\ldots,\xi_{mm}\). By the standard theory of ODE, we have the existence of a solution defined at least on some interval \([0,t _{m})\), \(0< t_{m}\leq T\). And the following a priori estimates show that in fact \(t_{m}=T\).
(ii) If we multiply (3.30) by \(g_{km}(t)\), multiply (3.31)-(3.32) by \(h_{km}(t)\), multiply (3.33) by \(\xi_{lm}(t)\), we have
$$\begin{aligned} &a_{1} \bigl(v'_{\epsilon m},v_{\epsilon m} \bigr)+a_{2} \bigl((v_{\epsilon m},v_{\epsilon m}) \bigr) +a_{1}\hat{b}(v_{\epsilon m},v_{\epsilon m},v_{\epsilon m})+ \hat{n}(q_{\epsilon m}+\tilde{Q}_{0},\varphi_{\epsilon m}+ \tilde{ \phi}_{0},v_{\epsilon m}) \\ &\quad {}+(\nabla p_{\epsilon m},v_{\epsilon m})=(f,v _{\epsilon m}), \end{aligned}$$
(3.35)
$$\begin{aligned} & \bigl(q'_{\epsilon m},\varphi_{\epsilon m} \bigr)+a_{3} \bigl((q_{\epsilon m}, \varphi_{\epsilon m}) \bigr)+a_{4}(q_{\epsilon m}, \varphi_{\epsilon m})- \hat{n}(q_{\epsilon m}+\tilde{Q}_{0}, \varphi_{\epsilon m},v_{\epsilon m})=0, \end{aligned}$$
(3.36)
$$\begin{aligned} & \bigl((\varphi_{\epsilon m},\varphi_{\epsilon m}) \bigr)=(q_{\epsilon m}, \varphi_{\epsilon m}), \end{aligned}$$
(3.37)
$$\begin{aligned} &\epsilon \bigl(p'_{\epsilon m},p_{\epsilon m} \bigr) +( \operatorname{div}v_{\epsilon m},p _{\epsilon m})=0, \quad k,l=1,\ldots,m. \end{aligned}$$
(3.38)
Due to (3.5), \(\hat{b}(v_{\epsilon m},v_{\epsilon m},v_{\epsilon m})=0\), and since \(v_{\epsilon m}\vert_{\partial\Omega}=0\), we have
$$(\nabla p_{\epsilon m},v_{\epsilon m}) + (\operatorname{div}{v_{\epsilon m}},p _{\epsilon m})=0. $$
Then add all these equations for \(k,l=1,\ldots,m\), and as a result
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[a_{1} \bigl\vert v_{\epsilon m}(t) \bigr\vert ^{2}+ \bigl\Vert \varphi_{\epsilon m}(t) \bigr\Vert ^{2} +\epsilon \bigl\vert p_{\epsilon m}(t) \bigr\vert ^{2} \bigr]+a _{2}\Vert v_{\epsilon m(t)}\Vert ^{2} +a_{3} \bigl\vert q_{\epsilon m}(t) \bigr\vert ^{2}+a_{4} \bigl\Vert \varphi_{\epsilon m}(t) \bigr\Vert ^{2} \\ &\quad =-\hat{n} \bigl(q_{\epsilon m}(t),\tilde{\phi}_{0},v_{\epsilon m}(t) \bigr)- \hat{n} \bigl(\tilde{Q}_{0},\tilde{\phi}_{0},v_{\epsilon m}(t) \bigr)+ \bigl(f,v_{ \epsilon m}(t) \bigr) \\ &\quad \leq C \biggl[\delta \bigl( \bigl\vert q_{\epsilon m}(t) \bigr\vert ^{2}+ \bigl\Vert q_{\epsilon m}(t) \bigr\Vert ^{2} \bigr)+ \frac{2}{ \delta} \bigl\vert v_{\epsilon m}(t) \bigr\vert ^{2}+1 \biggr]. \end{aligned}$$
(3.39)
Taking δ sufficiently small and by integration of (3.39) from 0 to s one shows that
$$\begin{aligned} &a_{1} \bigl\vert v_{\epsilon m}(s) \bigr\vert ^{2}+ \bigl\Vert \varphi_{\epsilon m}(s) \bigr\Vert ^{2}+\epsilon \bigl\vert p_{\epsilon m}(s) \bigr\vert ^{2}+ 2a_{2} \int^{s}_{0} \bigl\Vert v_{\epsilon m}(t) \bigr\Vert ^{2}\,dt \\ &\qquad {}+2a _{3} \int^{s}_{0} \bigl\vert q_{\epsilon m}(t) \bigr\vert ^{2}\,dt+2a_{4} \int^{s}_{0} \bigl\Vert \varphi_{\epsilon m}(t) \bigr\Vert ^{2}\,dt \\ &\quad \leq \vert v_{0}\vert ^{2}+\Vert \varphi_{0} \Vert ^{2} +\epsilon \vert p_{0} \vert ^{2}+ \frac{2C}{ \delta} \int^{s}_{0} \bigl\vert v_{\epsilon m}(t) \bigr\vert ^{2}\,dt+sC \\ &\quad \leq \vert v_{0}\vert ^{2}+\Vert \varphi_{0} \Vert ^{2} +\epsilon \vert p_{0} \vert ^{2}+ \frac{2C}{ \lambda_{1}\delta} \int^{s}_{0} \bigl\Vert v_{\epsilon m}(t) \bigr\Vert ^{2}\,dt+sC. \end{aligned}$$
Hence \(t_{m}=T\), and
$$\begin{aligned} &\sup_{s\in[0,T]} \bigl\{ a_{1} \bigl\vert v_{\epsilon m}(s) \bigr\vert ^{2}+ \bigl\Vert \varphi_{\epsilon m}(s) \bigr\Vert ^{2}+\epsilon \bigl\vert p_{\epsilon m}(s) \bigr\vert ^{2} \bigr\} \leq d_{1}, \end{aligned}$$
(3.40)
$$\begin{aligned} &d_{1}=a_{1}\vert v_{0}\vert ^{2}+ \Vert \varphi_{0}\Vert ^{2}+\vert p_{0} \vert ^{2}+sC. \end{aligned}$$
(3.41)
Integrating (3.39) in t from 0 to T, we have
$$\begin{aligned} & \int_{0}^{T} \bigl\Vert v_{\epsilon m}(t) \bigr\Vert ^{2}\,dt\leq\frac{d_{1}}{2a_{2}-\frac{2C}{ \lambda_{1}\delta}}, \end{aligned}$$
(3.42)
$$\begin{aligned} & \int_{0}^{T} \bigl\vert q_{\epsilon m}(t) \bigr\vert ^{2}\,dt\leq\frac{d_{1}}{2a_{3}}, \end{aligned}$$
(3.43)
$$\begin{aligned} & \int_{0}^{T} \bigl\Vert \varphi_{\epsilon m}(t) \bigr\Vert ^{2}\,dt\leq\frac{d_{1}}{2a _{4}}. \end{aligned}$$
(3.44)
Putting \(w_{k}=q_{\epsilon m}\) in equation (3.31), we find, using \(n(q,q,v)=0\),
$$\begin{aligned} \begin{aligned}[b] \frac{1}{2}\frac{d}{dt}\vert q_{\epsilon m} \vert ^{2}+a_{3}\Vert q_{\epsilon m}\Vert ^{2}+a_{4}\vert q_{\epsilon m}\vert ^{2} &= \hat{n}(\tilde{Q}_{0},q_{\epsilon m},v _{\epsilon m}) \\ &\leq C \biggl[ \delta \bigl(\vert q_{\epsilon m}\vert ^{2}+ \Vert q_{\epsilon m}\Vert ^{2} \bigr)+\frac{2}{\delta} \vert v_{\epsilon m}\vert ^{2}+1 \biggr]. \end{aligned} \end{aligned}$$
(3.45)
Taking δ sufficiently small and by integration of (3.45) from 0 to s one shows that
$$\begin{aligned} & \bigl\vert q_{\epsilon m}(s) \bigr\vert ^{2}+2a_{3} \int^{s}_{0} \bigl\Vert q_{\epsilon m}(s) \bigr\Vert ^{2}\,dt+2a _{4} \int^{s}_{0} \bigl\vert q_{\epsilon m}(s) \bigr\vert ^{2}\,dt \\ &\quad \leq\frac{2C}{\delta} \int^{s}_{0} \bigl\vert v_{\epsilon m}(s) \bigr\vert ^{2}\,dt+\vert q_{0}\vert ^{2}+Cs \\ &\quad \leq\frac{2C}{\lambda_{1}\delta} \int^{s}_{0} \bigl\Vert v_{\epsilon m}(s) \bigr\Vert ^{2}\,dt+\vert q_{0}\vert ^{2}+Cs \triangleq d_{2}. \end{aligned}$$
(3.46)
Then we can obtain
$$\begin{aligned} \sup_{s\in[0,T]} \bigl\vert q_{\epsilon m}(s) \bigr\vert ^{2}\leq d_{2}. \end{aligned}$$
(3.47)
Integrating (3.45) in t from 0 to T, we have
$$\begin{aligned} \int_{0}^{T} \bigl\Vert q_{\epsilon m}(t) \bigr\Vert ^{2}\,dt\leq\frac{d_{2}}{2a_{3}}. \end{aligned}$$
(3.48)
(iii) In order to pass to the limit in the nonlinear term we need an estimate of the fractional derivative in time of \(v_{\epsilon m}\) and \(q_{\epsilon m}\).
Setting
$$\begin{aligned} &\phi_{\epsilon m}(t)=f-a_{2}{ \mathcal{A}_{0}}v_{\epsilon m}- \hat{B}(v_{\epsilon m})-\hat{N}(q_{\epsilon m}+\tilde{Q}_{0}, \varphi_{\epsilon m}+\tilde{\phi}_{0}), \\ &\psi_{\epsilon m}(t)=-a_{3}{ \mathcal{A}_{1}}q_{\epsilon m}-a _{4}q_{\epsilon m}-\hat{M}(q_{\epsilon m}+\tilde{Q}_{0},v_{\epsilon m}). \end{aligned}$$
We write equations (3.30)-(3.33) as
$$\begin{aligned} &a_{1} \bigl(v'_{\epsilon m}(t),\omega_{k} \bigr)+ \bigl(\nabla p_{\epsilon m}(t),\omega_{k} \bigr)= \bigl( \phi_{\epsilon m}(t),\omega_{k} \bigr),\quad k=1,\ldots,m, \\ & \bigl(q'_{\epsilon m}(t), w_{k} \bigr)= \bigl( \psi_{\epsilon m}(t),w_{k} \bigr),\quad k=1,\ldots,m, \\ &\epsilon \bigl(p'_{\epsilon m}(t), \gamma_{l} \bigr)+ \bigl(\operatorname{div}{v_{\epsilon m}(t)}, \gamma_{l} \bigr)=0,\quad l=1, \ldots,m, \\ &\varphi_{\epsilon m}(\cdot,t)={ \mathcal{A}_{1}}^{-1}q _{\epsilon m}(\cdot,t). \end{aligned}$$
As done several times before, we extend all functions by 0 outside the interval \([0,T]\) and consider the Fourier transform of the different equations.
The following relations then hold on \(\mathbb{R}\):
$$\begin{aligned} &a_{1}\frac{d}{dt} \bigl({\tilde{v}}_{\epsilon m}(t), \omega_{k} \bigr)+ \bigl(\nabla{\tilde{p}}_{\epsilon m}(t), \omega_{k} \bigr) \\ &\quad = \bigl({\tilde{\phi}}_{\epsilon m}(t), \omega_{k} \bigr)+(v_{0m},\omega_{k}) \delta_{(0)}- \bigl(v_{\epsilon m}(T),\omega_{k} \bigr) \delta_{(T)}, \\ &\frac{d}{dt} \bigl({\tilde{q}}_{\epsilon m}(t),w_{k} \bigr)= \bigl({\tilde{\psi}} _{\epsilon m}(t),w_{k} \bigr)+(q_{0m},w_{k}) \delta_{(0)}- \bigl(q_{\epsilon m}(T),w _{k} \bigr) \delta_{(T)}, \\ &\epsilon\frac{d}{dt} \bigl({\tilde{p}}_{\epsilon m}(t), \gamma_{l} \bigr)+ \bigl( \operatorname{div}{{\tilde{v}}_{\epsilon m}(t)}, \gamma_{l} \bigr)=\epsilon(p_{0m}, \gamma_{l}) \delta_{(0)}-\epsilon \bigl(p_{\epsilon m}(T),\gamma_{l} \bigr) \delta_{(T)}, \\ &\tilde{\varphi}_{\epsilon m}(\cdot,t)={ \mathcal{A}_{1}} ^{-1}\tilde{q}_{\epsilon m}(\cdot,t). \end{aligned}$$
After taking Fourier transforms, as a result
$$\begin{aligned} &2a_{1}i\pi\tau \bigl({\hat{v}}_{\epsilon m}(\tau), \omega_{k} \bigr)+ \bigl(\nabla{\hat{p}}_{\epsilon m}(\tau), \omega_{k} \bigr) \\ &\quad = \bigl\langle {\hat{\phi}}_{ \epsilon m}(\tau), \omega_{k} \bigr\rangle +(v_{0m},\omega_{k})- \bigl(v_{\epsilon m}(T), \omega_{k} \bigr)\exp(-2i\pi\tau T), \\ &2i\pi\tau \bigl({\hat{q}}_{\epsilon m}(\tau),w_{k} \bigr)= \bigl\langle {\hat{\psi}} _{\epsilon m}(\tau),w_{k} \bigr\rangle +(q_{0m},w_{k})- \bigl(q_{\epsilon m}(T),w _{k} \bigr)\exp(-2i\pi\tau T), \\ &2i\pi\tau\epsilon \bigl({\hat{p}}_{\epsilon m}(\tau),\gamma_{l} \bigr)+ \bigl( \operatorname{div}{{\hat{v}}_{\epsilon m}(\tau)},\gamma_{l} \bigr)= \epsilon(p _{0m},\gamma_{l})-\epsilon \bigl(p_{\epsilon m}(T), \gamma_{l} \bigr)\exp(-2i \pi\tau T). \end{aligned}$$
We multiply the first of the last three equations by \(\hat{g}_{km}( \tau)\) (\(\hat{g}_{km}=\) Fourier transform of \(\tilde{g}_{km}\)) and the second by \(\hat{h}_{km}(\tau)\) (\(\hat{h}_{km}=\) Fourier transform of \(\tilde{h}_{km}\)) and the third by \(\hat{\xi}_{lm}(\tau)\) (\(\hat{\xi}_{lm}=\) Fourier transform of \(\tilde{\xi}_{lm}\)), and then add these relations for \(k=1,\ldots,m\), \(l=1,\ldots,m\), we obtain
$$\begin{aligned} & 2i\pi\tau \bigl\{ a_{1} \bigl\vert { \hat{v}}_{\epsilon m}(\tau) \bigr\vert ^{2}+ \bigl\Vert {\hat{ \varphi}}_{\epsilon m}(\tau) \bigr\Vert ^{2}+\epsilon \bigl\vert { \hat{p}}_{\epsilon m}(\tau) \bigr\vert ^{2} \bigr\} \\ &\qquad {} + \bigl(\nabla {{ \hat{p}}_{\epsilon m}}(\tau),{ \hat{v}}_{\epsilon m}( \tau) \bigr)+ \bigl({ \operatorname{div}} {{\hat{v}}_{\epsilon m}}( \tau),{{\hat{p}}_{ \epsilon m}}(\tau) \bigr) \\ &\quad = \bigl\langle {{\hat{\phi}}_{\epsilon m}}(\tau),{{\hat {v}}_{\epsilon m}}( \tau) \bigr\rangle \\ &\qquad {}+ \bigl\langle {{\hat{\psi} }_{\epsilon m}}(\tau ),{{\hat{ \varphi}} _{\epsilon m}}(\tau) \bigr\rangle + \bigl({v_{0m}},{{ \hat{v}}_{\epsilon m}}( \tau) \bigr)+ \bigl({q_{0m}},{{\hat{\varphi }}_{\epsilon m}}(\tau) \bigr) + \epsilon \bigl({p_{0m}},{{ \hat{p}}_{\epsilon m}}(\tau) \bigr) \\ &\qquad {}- \bigl\{ \bigl({v_{\epsilon m}}(T),{{\hat{v}}_{\epsilon m}}(\tau) \bigr)+ \bigl({q_{ \epsilon m}}(T),{{\hat{\varphi}}_{\epsilon m}}(\tau) \bigr) \\ &\qquad {}+ \epsilon \bigl( {p _{\epsilon m}}(T),{{\hat{p} }_{\epsilon m}}( \tau ) \bigr) \bigr\} \exp( - 2i \pi\tau T). \end{aligned}$$
(3.49)
The term \((\nabla\hat{p}_{\epsilon m},\hat{v}_{\epsilon m})+(\operatorname{div}{{\hat{v}}_{\epsilon m}},{\hat{p} }_{\epsilon m})=0\), and we have \({{\hat{v}}_{\epsilon m}}(\tau)\vert_{\partial\Omega}=0\).
We deduce from (3.49) that
$$\begin{aligned} &2\pi \vert \tau \vert \bigl\{ a_{1}{ \bigl\vert {{ \hat{v}}_{\epsilon m}(\tau) \bigr\vert ^{2}}+ \bigl\Vert {\hat{ \varphi}}_{\epsilon m}(\tau) \bigr\Vert ^{2}+\epsilon \bigl\vert {{ \hat{p}}_{\epsilon m}(\tau) \bigr\vert ^{2}}} \bigr\} \\ &\quad \leqslant \bigl\vert \bigl\langle {{\hat{\phi} }_{\epsilon m}}( \tau),{{ \hat{v}}_{\epsilon m}}(\tau) \bigr\rangle \bigr\vert + \bigl\vert \bigl\langle {{\hat{\psi} }_{\epsilon m}}(\tau),{{\hat{\varphi}}_{\epsilon m}}( \tau) \bigr\rangle \bigr\vert + \vert {v_{0m}}\vert \bigl\vert {{ \hat{v}}_{\epsilon m}}(\tau) \bigr\vert + \vert q_{0m}\vert \bigl\vert {\hat{\varphi}}_{\epsilon m}(\tau) \bigr\vert \\ &\qquad {}+ \epsilon \vert {{p_{0m}}} \vert \bigl\vert {{{ \hat{p}}_{\epsilon m}}(\tau)} \bigr\vert + \bigl\vert {v_{\epsilon m}}(T) \bigr\vert \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\vert + \bigl\vert q_{\epsilon m}(T) \bigr\vert \bigl\vert {{\hat{\varphi }}_{\epsilon m}}(\tau) \bigr\vert +\epsilon \bigl\vert {p _{\epsilon m}}(T) \bigr\vert \bigl\vert {{\hat{p}}_{\epsilon m}}(\tau) \bigr\vert \\ &\quad \leqslant \bigl\vert { \bigl\langle {{\hat{\phi} }_{\epsilon m}}( \tau),{{ \hat{v}}_{\epsilon m}}(\tau) \bigr\rangle } \bigr\vert + \bigl\vert \bigl\langle {{\hat{\psi} }_{\epsilon m}}(\tau),{{\hat{\varphi}}_{\epsilon m}}( \tau) \bigr\rangle \bigr\vert +2\sqrt{{d _{1}}} \bigl( \bigl\vert {{ \hat{v}}_{\epsilon m}}(\tau) \bigr\vert + \bigl\vert {{\hat{\varphi }}_{\epsilon m}}(\tau) \bigr\vert + \epsilon \bigl\vert {{\hat{p} }_{\epsilon m}}(\tau) \bigr\vert \bigr) \\ &\quad {}\leqslant \bigl\vert { \bigl\langle {{\hat{\phi} }_{\epsilon m}}( \tau),{{ \hat{v}}_{\epsilon m}}(\tau) \bigr\rangle } \bigr\vert + \bigl\vert \bigl\langle {{\hat{\psi} }_{\epsilon m}}(\tau),{{\hat{\varphi}}_{\epsilon m}}( \tau) \bigr\rangle \bigr\vert +2c_{1}\sqrt{ {d_{1}}} \bigl( \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\vert + \bigl\vert {{\hat{ \varphi}}_{\epsilon m}}(\tau) \bigr\vert + \epsilon \bigl\vert {{\hat{p} }_{\epsilon m}}(\tau) \bigr\vert \bigr). \end{aligned}$$
We next estimate the term \(\vert {\langle{{\hat{\phi} }_{\epsilon m}}(\tau),{{\hat{v}}_{\epsilon m}}(\tau)\rangle} \vert \) and \(\vert {\langle{{\hat{\psi} }_{\epsilon m}}(\tau),{{\hat{\varphi }}_{\epsilon m}}(\tau)\rangle} \vert \). In fact,
$$\begin{aligned} & \bigl\vert { \bigl\langle {{\hat{\phi} }_{\epsilon m}}(\tau),{{ \hat{v}}_{\epsilon m}}(\tau) \bigr\rangle } \bigr\vert \\ &\quad = \bigl\vert { \bigl\langle \hat{f}(\tau)-a_{2}{ \mathcal{A}_{0}}\hat{v} _{\epsilon m}(\tau)-\hat{B} \bigl( \hat{v}_{\epsilon m}(\tau) \bigr)-\hat{N} \bigl( \hat{q}_{\epsilon m}(\tau)+ \tilde{Q}_{0},\hat{\varphi}_{\epsilon m}( \tau)+\tilde{\phi }_{0} \bigr),{{\hat{v}}_{\epsilon m}}(\tau) \bigr\rangle } \bigr\vert \\ &\quad \leq \bigl\vert \bigl(\hat{f}(\tau),\hat{v}_{\epsilon m}(\tau) \bigr) \bigr\vert +a_{2} \bigl\Vert {\hat{v}}_{\epsilon m}(\tau) \bigr\Vert ^{2}+\hat{b} \bigl({{\hat{v}}_{\epsilon m}}(\tau), {{ \hat{v}}_{\epsilon m}}(\tau),{{\hat{v}}_{\epsilon m}}(\tau) \bigr) \\ &\qquad {}+ \hat{n} \bigl({{\hat{q}}_{\epsilon m}}(\tau)+\hat{Q}_{0},{{ \hat{\varphi}} _{\epsilon m}}(\tau)+\hat{\phi}_{0},{{ \hat{v}}_{\epsilon m}}( \tau) \bigr) \\ &\quad \leq \bigl\vert \hat{f}(\tau) \bigr\vert ^{2}+ \bigl\vert \hat{v}_{\epsilon m}(\tau) \bigr\vert ^{2}+a_{2} \bigl\Vert \hat{v}_{\epsilon m}(\tau) \bigr\Vert ^{2}+\hat{n} \bigl({{ \hat{q}}_{\epsilon m}}( \tau),{{\hat{\varphi}}_{\epsilon m}}(\tau),{{ \hat{v}}_{\epsilon m}}( \tau) \bigr) \\ &\qquad {}+\hat{n} \bigl({{\hat{q}}_{\epsilon m}}( \tau),\hat{ \phi}_{0},{{\hat{v}} _{\epsilon m}}(\tau) \bigr) \\ &\qquad {}+ \hat{n} \bigl(\hat{Q}_{0},{{\hat{\varphi}}_{\epsilon m}}( \tau ),{{\hat{v}} _{\epsilon m}}(\tau) \bigr)+\hat{n} \bigl( \hat{Q}_{0}, \hat{\phi}_{0},{{\hat{v}} _{\epsilon m}}( \tau) \bigr) \\ &\quad \leq \bigl\vert \hat{f}(\tau) \bigr\vert ^{2}+ \bigl\vert \hat{v}_{\epsilon m}(\tau) \bigr\vert ^{2}+a_{2} \bigl\Vert \hat{v}_{\epsilon m}(\tau) \bigr\Vert ^{2}+C \bigl( \bigl\Vert {\hat{q}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}}+ \bigl\Vert {\hat{\varphi}}_{\epsilon m}(\tau) \bigr\Vert ^{2}+ \bigl\Vert {\hat{v}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}} \\ &\qquad {}+ \bigl\vert {\hat{q}}_{\epsilon m}(\tau) \bigr\vert ^{2}+ \bigl\vert {\hat{v}}_{\epsilon m}(\tau) \bigr\vert ^{2}+ \bigl\Vert { \hat{\varphi}}_{\epsilon m}(\tau) \bigr\Vert ^{2} \bigr), \end{aligned}$$
(3.50)
$$\begin{aligned} & \bigl\vert { \bigl\langle {{\hat{\psi} }_{\epsilon m}}(\tau),{{\hat {\varphi }}_{\epsilon m}}(\tau) \bigr\rangle } \bigr\vert \\ &\quad = \bigl\vert { \bigl\langle -a_{3}{ \mathcal{A}_{1}} \hat{q}_{\epsilon m}( \tau)-a_{4}\hat{q}_{\epsilon m}(\tau)- \hat{M} \bigl(\hat{q}_{\epsilon m}( \tau)+\tilde{Q}_{0}, \hat{v}_{\epsilon m}(\tau) \bigr),{{\hat{\varphi}} _{\epsilon m}(\tau )} \bigr\rangle } \bigr\vert \\ &\quad \leq a_{3} \bigl\vert \hat{q}_{\epsilon m}(\tau) \bigr\vert ^{2}+a_{4} \bigl\Vert {\hat{\varphi}}_{\epsilon m}( \tau ) \bigr\Vert ^{2}+\hat{n} \bigl(\hat{q}_{\epsilon m}(\tau),{ \hat{ \varphi}} _{\epsilon m}(\tau),\hat{v}_{\epsilon m}(\tau) \bigr)+\hat{n} \bigl( \hat{Q}_{0}, {\hat{\varphi}}_{\epsilon m}(\tau), \hat{v}_{\epsilon m}(\tau) \bigr) \\ &\quad \leq a_{3} \bigl\vert \hat{q}_{\epsilon m}(\tau) \bigr\vert ^{2}+a_{4} \bigl\Vert {\hat{\varphi}}_{\epsilon m}( \tau ) \bigr\Vert ^{2} \\ &\qquad {}+C \bigl( \bigl\Vert {\hat{q}}_{\epsilon m}(\tau) \bigr\Vert ^{4} _{L^{4}}+ \bigl\Vert {\hat{\varphi }}_{\epsilon m}(\tau) \bigr\Vert ^{2}+ \bigl\Vert { \hat{v}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}}+ \bigl\vert {\hat{v}}_{\epsilon m}(\tau) \bigr\vert ^{2} \bigr). \end{aligned}$$
(3.51)
Combining (3.49) and (3.50)-(3.51), we get
$$\begin{aligned} &2\pi \vert \tau \vert \bigl\{ a_{1} \bigl\vert {{ \hat{v}}_{\epsilon m}(\tau) \bigr\vert ^{2}}+ \bigl\Vert {\hat{ \varphi}}_{\epsilon m}(\tau) \bigr\Vert ^{2}+ \bigl\vert { \hat{p}}_{\epsilon m}(\tau) \bigr\vert ^{2} \bigr\} \\ &\quad \leq \bigl\vert \hat{f}(\tau) \bigr\vert ^{2}+ \bigl\vert {{\hat{v}}_{\epsilon m}}( \tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\Vert ^{2}+ \bigl\vert { \hat{q}}_{\epsilon m}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{ \varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2} \\ &\qquad {}+ C \bigl( \bigl\Vert {\hat{q}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}}+ \bigl\Vert {\hat{v}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}}+2\sqrt{{d_{1}}} \bigl( \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\vert + \bigl\Vert {{\hat {\varphi }}_{\epsilon m}}(\tau) \bigr\Vert + \epsilon \bigl\vert {{\hat{p} }_{\epsilon m}}(\tau) \bigr\vert \bigr) \bigr). \end{aligned}$$
(3.52)
For some fixed \(\gamma\in(0,\frac{1}{4})\), we have \(\vert \tau \vert ^{2 \gamma}\leq(2\gamma+1)\frac{1+\vert \tau \vert }{1+\vert \tau \vert ^{1-2\gamma}}\), \(\forall\tau\in\mathbb{R}\). Thus
$$\begin{aligned} & \int^{+\infty}_{-\infty} \vert \tau \vert ^{2\gamma} \bigl\{ a_{1} \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{\varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2} \bigr\} \,d\tau \\ &\quad \leq(2\gamma+1) \int^{+\infty}_{-\infty}\frac{a_{1} \vert {{\hat{v}}_{\epsilon m}}(\tau)\vert ^{2}+\Vert {{\hat {\varphi}}_{\epsilon m}}(\tau)\Vert ^{2}}{1+\vert \tau \vert ^{1-2\gamma}}\,d\tau \\ &\qquad {}+(2\gamma+1) \int^{+\infty}_{- \infty}\frac{\vert \tau \vert (a_{1}\vert {{\hat{v}}_{\epsilon m}}(\tau)\vert ^{2}+\Vert {{\hat{\varphi}}_{\epsilon m}}(\tau)\Vert ^{2})}{1+\vert \tau \vert ^{1-2\gamma}}\,d\tau \\ &\quad \leq(2\gamma+1) \int^{+\infty}_{-\infty} \bigl\{ a_{1} \bigl\vert {{ \hat{v}}_{\epsilon m}}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{ \varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2} \bigr\} \,d\tau \\ &\qquad {}+(2\gamma +1) \int^{+\infty}_{-\infty}\frac{\vert \tau \vert (a _{1}\vert {{\hat{v}}_{\epsilon m}}(\tau)\vert ^{2}+ \Vert {{\hat{\varphi}}_{\epsilon m}}(\tau)\Vert ^{2})}{1+\vert \tau \vert ^{1-2\gamma}}\,d\tau. \end{aligned}$$
(3.53)
By the Parseval equality and the Poincaré inequality, we have
$$\begin{aligned} &(2\gamma+1) \int^{+\infty}_{-\infty} \bigl\{ a_{1} \bigl\vert {{ \hat{v}}_{\epsilon m}}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{ \varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2} \bigr\} \,d\tau \\ &\quad =(2 \gamma+1) \int^{+\infty}_{-\infty} \bigl\{ a_{1} \bigl\vert {{ \tilde{v}}_{\epsilon m}}(t) \bigr\vert ^{2}+ \bigl\Vert {{\tilde{ \varphi}}_{\epsilon m}}(t) \bigr\Vert ^{2} \bigr\} \,dt \\ &\quad = (2\gamma+1) \int^{T}_{0} \bigl\{ a_{1} \bigl\vert {v_{\epsilon m}}(t) \bigr\vert ^{2}+ \bigl\Vert { \varphi_{\epsilon m}}(t) \bigr\Vert ^{2} \bigr\} \,dt \\ &\quad \leq C( \lambda_{1}) (2\gamma+1) \int^{T}_{0} \bigl\{ a_{1} \bigl\Vert { v_{\epsilon m}}(t) \bigr\Vert ^{2}+ \bigl\Vert { \varphi_{\epsilon m}}(t ) \bigr\Vert ^{2} \bigr\} \,dt \\ &\quad \leq C(\lambda_{1},a_{1},a_{2},a_{4},d_{1}, \gamma) \end{aligned}$$
(3.54)
and
$$\begin{aligned} &(2\gamma+1) \int^{+\infty}_{-\infty}\frac{\vert \tau \vert (a_{1} \vert {{\hat{v}}_{\epsilon m}}(\tau)\vert ^{2}+\Vert {{\hat {\varphi}}_{\epsilon m}}(\tau)\Vert ^{2})}{1+\vert \tau \vert ^{1-2\gamma}}\,d\tau \\ &\quad \leq\frac{2\gamma+1}{2\pi} \int^{+\infty}_{-\infty} \bigl( \bigl( \bigl\vert \hat{f}( \tau) \bigr\vert ^{2}+ \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau ) \bigr\vert ^{2}+ \bigl\Vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\Vert ^{2}+ \bigl\vert {{\hat {q}}_{\epsilon m}}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{\varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2} \\ &\qquad {}+C \bigl( \bigl\Vert {\hat{q}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}}+ \bigl\Vert {\hat{v}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}} \bigr) \bigr)/ \bigl(1+\vert \tau \vert ^{1-2\gamma} \bigr)\bigr)\,d\tau \\ &\qquad {}+\frac{2\gamma+1}{2\pi} \int^{+\infty}_{-\infty}\frac{2\sqrt{d _{1}}(\vert {{\hat{v}}_{\epsilon m}}(\tau)\vert + \Vert {{\hat{\varphi}}_{\epsilon m}}(\tau)\Vert +\epsilon \vert {{\hat{p} }_{\epsilon m}}(\tau)\vert )}{1+\vert \tau \vert ^{1-2 \gamma}}\,d\tau \\ &\quad \leq\frac{(2\gamma+1)}{2\pi} \int^{+\infty}_{-\infty} \bigl(\bigl\vert \hat{f}(\tau) \bigr\vert ^{2}+ \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\Vert ^{2}+ \bigl\vert {{\hat{q}}_{\epsilon m}}(\tau) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\Vert {{\hat{\varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2}+C \bigl( \bigl\Vert {\hat{q}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}}+ \bigl\Vert {\hat{v}}_{\epsilon m}(\tau) \bigr\Vert ^{4}_{L^{4}} \bigr)\bigr)\,d\tau \\ &\qquad {}+ \frac{(2\gamma+1)\sqrt{d_{1}}}{\pi} \int^{+\infty}_{-\infty}\frac{\vert {\hat{v}}_{\epsilon m}(\tau)\vert + \Vert {\hat{\varphi}}_{\epsilon m}(\tau)\Vert +\epsilon \vert {\hat{p}}_{\epsilon m}(\tau)\vert }{1+\vert \tau \vert ^{1-2\gamma}}\,d\tau \\ &\quad \leq C \int^{T}_{0}\bigl( \bigl\vert f(t) \bigr\vert ^{2}+ \bigl\vert v_{\epsilon m}(t) \bigr\vert ^{2}+ \bigl\Vert v_{\epsilon m}(t) \bigr\Vert ^{2}+ \bigl\vert q_{\epsilon m}(t ) \bigr\vert ^{2}+ \bigl\Vert \varphi_{\epsilon m}(t) \bigr\Vert ^{2} \\ &\qquad {}+ \bigl\Vert q_{\epsilon m}(t) \bigr\Vert ^{4}_{L^{4}}+ \bigl\Vert v_{\epsilon m}(t) \bigr\Vert ^{4}_{L^{4}}\bigr)\,dt \\ &\qquad {}+ C \biggl( \int^{+\infty}_{-\infty} \frac{d\tau}{(1+\vert \tau \vert ^{1-2\gamma})^{2}} \biggr)^{\frac{1}{2}} \biggl( \int^{T} _{0} \bigl( \bigl\vert {v_{\epsilon m}}(t) \bigr\vert ^{2}+ \bigl\Vert \varphi_{\epsilon m}(t) \bigr\Vert ^{2}+\epsilon \bigl\vert {p_{\epsilon m}}(t) \bigr\vert ^{2} \bigr)\,dt \biggr)^{\frac{1}{2}} \\ &\quad \leq C(\pi,\gamma,d_{1},a_{1},a_{2},a_{4},T). \end{aligned}$$
(3.55)
Here we have also used the convergence of the infinite integral
$$\begin{aligned} \biggl( \int^{+\infty}_{-\infty} \frac{d\tau}{(1+\vert \tau \vert ^{1-2\gamma})^{2}} \biggr)^{\frac{1}{2}},\quad \mbox{for some }\gamma\in \biggl(0,\frac{1}{4} \biggr). \end{aligned}$$
We can conclude that
$$\begin{aligned} \int^{+\infty}_{-\infty} \vert \tau \vert ^{2\gamma} \bigl( \bigl\vert {{\hat{v}}_{\epsilon m}}(\tau) \bigr\vert ^{2}+ \bigl\Vert {{\hat{\varphi}}_{\epsilon m}}(\tau) \bigr\Vert ^{2} \bigr)\,d\tau\leq C,\quad \mbox{for some }\gamma\in \biggl(0,\frac{1}{4} \biggr). \end{aligned}$$
(3.56)
(iv) We want to pass to the limit as \(m\rightarrow\infty\) in (3.30)-(3.33) using the estimates (3.40), (3.42)-(3.44) and (3.55), we recall that at the present time \(\epsilon\in(0,1]\) is fixed, and we are only concerned with a passage to the limit as \(m\rightarrow\infty\). There exist a sequence \(m'\rightarrow\infty\) and some \(\{v_{\epsilon },q_{\epsilon},\varphi_{\epsilon},p_{\epsilon}\}\) such that
$$\begin{aligned} &v_{\epsilon m'}\rightharpoonup v_{\epsilon}\quad \text{in } L^{2} \bigl(0,T;{\mathbb{H}}^{1}_{0}(\Omega) \bigr) \text{ weakly}, \end{aligned}$$
(3.57)
$$\begin{aligned} &v_{\epsilon m'}\rightarrow v_{\epsilon}\quad \text{in } L^{\infty} \bigl(0,T; \mathbb{L}^{2}(\Omega) \bigr) \text{ weak-star}, \end{aligned}$$
(3.58)
$$\begin{aligned} &v_{\epsilon m'}\rightarrow v_{\epsilon} \quad \text{in } L^{2} \bigl(0,T; \mathbb{L}^{2}(\Omega) \bigr) \text{ strongly}, \end{aligned}$$
(3.59)
$$\begin{aligned} &q_{\epsilon m'}\rightharpoonup q_{\epsilon}\quad \text{in } L^{2} \bigl(0,T;H^{1}_{0}(\Omega) \bigr) \text{ weakly}, \end{aligned}$$
(3.60)
$$\begin{aligned} &q_{\epsilon m'}\rightarrow q_{\epsilon}\quad \text{in } L^{\infty} \bigl(0,T;L^{2}(\Omega) \bigr) \text{ weak-star}, \end{aligned}$$
(3.61)
$$\begin{aligned} &q_{\epsilon m'}\rightarrow q_{\epsilon}\quad \text{in } L^{2} \bigl(0,T;L^{2}(\Omega) \bigr) \text{ strongly}, \end{aligned}$$
(3.62)
$$\begin{aligned} &\varphi_{\epsilon m'}\rightharpoonup\varphi_{\epsilon} \quad \text{in } L^{\infty} \bigl(0,T;H^{1}_{0}(\Omega) \bigr) \text{ weak-star}, \end{aligned}$$
(3.63)
$$\begin{aligned} &\varphi_{\epsilon m'}\rightarrow\varphi_{\epsilon} \quad \text{in } L^{2} \bigl(0,T;H^{2}(\Omega) \bigr) \text{ strongly}, \end{aligned}$$
(3.64)
$$\begin{aligned} &p_{\epsilon m'}\rightarrow p_{\epsilon}\quad \text{in } L^{\infty} \bigl(0,T;L^{2}(\Omega) \bigr) \text{ weak-star}. \end{aligned}$$
(3.65)
Take \(\psi(t)\in\mathcal{C}^{\infty}_{c}([0,T])\) with \(\psi(T)=0\). We multiply (3.30) (resp. (3.31), (3.33)) by \(\psi(t)\), integrate over \([0,T]\), and then integrate the first term by parts:
$$\begin{aligned}& -a_{1} \int^{T}_{0} \bigl(v_{\epsilon m'}(t), \omega_{k}\psi'(t) \bigr)\,dt+a_{2} \int^{T}_{0} \bigl( \bigl(v_{\epsilon m'}(t), \omega_{k}\psi'(t) \bigr) \bigr)\,dt \\& \qquad {}+a_{1} \int^{T}_{0}\hat{b} \bigl(v_{\epsilon m'}(t),v_{\epsilon m'}(t), \omega_{k} \psi(t) \bigr)\,dt \\& \qquad {}+ \int^{T}_{0}\hat{n} \bigl(q_{\epsilon m'}+ \tilde{Q}_{0},\varphi_{\epsilon m'}+ \tilde{\phi}_{0}, \omega_{k} \psi(t) \bigr)\,dt+ \int^{T}_{0} \bigl(\nabla p_{ \epsilon m'}, \omega_{k}\psi(t) \bigr)\,dt \\& \quad = \int^{T}_{0} \bigl(f(t),\omega_{k}\psi (t) \bigr)\,dt+(v _{0m'},\omega_{k})\psi(0), \end{aligned}$$
(3.66)
$$\begin{aligned}& - \int^{T}_{0} \bigl(q_{\epsilon m'}(t),w_{k} \psi'(t) \bigr)\,dt+a_{3} \int^{T} _{0} \bigl( \bigl(q_{\epsilon m'}(t),w_{k} \psi(t) \bigr) \bigr)\,dt+a_{4} \int^{T}_{0} \bigl(q_{ \epsilon m'}(t),w_{k} \psi(t) \bigr)\,dt \\& \quad {}- \int^{T}_{0}\hat{n} \bigl(q_{\epsilon m'}+ \tilde{Q}_{0},\omega_{k}\psi(t),v _{\epsilon m'} \bigr) \,dt=(q_{0m'},w_{k})\psi(0), \end{aligned}$$
(3.67)
$$\begin{aligned}& -\epsilon \int^{T}_{0} \bigl(p_{\epsilon m'}(t), \gamma_{l}\psi'(t) \bigr)\,dt+ \int^{T}_{0} \bigl( \operatorname{div}v_{\epsilon m'}(t), \gamma_{l}\psi(t) \bigr)\,dt \\& \quad =\epsilon (p_{0m'}, \gamma_{l})\psi(0),\quad 1\leq k,l\leq m. \end{aligned}$$
(3.68)
We look at the convergence of the nonlinear terms in (3.66) and (3.67). Firstly
$$\begin{aligned} & \biggl\vert \int^{T}_{0}\hat{b} \bigl(v_{\epsilon m'}(t),v_{\epsilon m'}(t), \omega_{k}\psi(t) \bigr)\,dt- \int^{T}_{0}\hat{b} \bigl(v_{\epsilon}(t),v_{\epsilon}(t), \omega_{k}\psi(t) \bigr)\,dt \biggr\vert \\ &\quad \leq\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon m'}(t),v_{\epsilon m'}(t), \omega_{k}\psi(t) \bigr)\,dt- \int^{T}_{0}b \bigl(v_{\epsilon}(t),v_{\epsilon}(t), \omega_{k}\psi(t) \bigr)\,dt \biggr\vert \\ &\qquad {}+ \frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon m'}(t), \omega_{k}\psi(t),v _{\epsilon m'}(t) \bigr)\,dt- \int^{T}_{0}b \bigl(v_{\epsilon}(t), \omega_{k}\psi(t),v _{\epsilon}(t) \bigr)\,dt \biggr\vert \\ &\quad \leq\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon m'}(t)-v_{\epsilon}(t),v_{\epsilon m'}(t), \omega_{k}\psi(t) \bigr)\,dt \biggr\vert \\ &\qquad {}+ \frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon}(t),v_{\epsilon m'}(t)-v_{\epsilon}(t), \omega_{k}\psi(t) \bigr)\,dt \biggr\vert \\ &\qquad {}+ \frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon m'}(t)-v_{\epsilon}(t), \omega_{k}\psi(t),v_{\epsilon m'}(t) \bigr)\,dt \biggr\vert \\ &\qquad {}+ \frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon}(t), \omega_{k}\psi(t),v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr)\,dt \biggr\vert \\ &\quad \triangleq b_{1}+b_{2}+b_{3}+b_{4}, \end{aligned}$$
(3.69)
where
$$\begin{aligned}& b_{1} =\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon m'}(t)-v_{\epsilon}(t),v_{\epsilon m'}(t), \omega_{k}\psi(t) \bigr)\,dt \biggr\vert \\& \hphantom{b_{1}}\leq\frac{1}{2}\sup_{t\in(0,T)} \bigl\vert \psi(t) \bigr\vert \sup_{x\in\Omega} \vert \omega_{k}\vert \int^{T}_{0} \bigl\vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\vert \bigl\vert \nabla v_{\epsilon m'}(t) \bigr\vert \,dt \\& \hphantom{b_{1}}\leq C \bigl\Vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb{L} ^{2}(\Omega))} \bigl\Vert v_{\epsilon m'}(t) \bigr\Vert _{L^{2}(0,T;{\mathbb{H}}^{1}_{0}( \Omega))} \rightarrow0 \quad \mbox{as }m'\rightarrow\infty, \end{aligned}$$
(3.70)
$$\begin{aligned}& b_{2} =\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon}(t),v_{\epsilon m'}(t)-v_{\epsilon}(t), \omega_{k}\psi(t) \bigr)\,dt \biggr\vert \\ & \hphantom{b_{2}}=\frac{1}{2} \biggl\vert \int^{T}_{0}\operatorname{div}v_{\epsilon} \bigl(v_{\epsilon m'}(t)-v _{\epsilon}(t) \bigr)\omega_{k}\psi (t) \,dt \biggr\vert \\ & \hphantom{b_{2}={}}{}+\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v _{\epsilon}(t), \omega_{k}\psi(t),v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr)\,dt \biggr\vert \\ & \hphantom{b_{2}}\leq\frac{1}{2}\sup_{t\in(0,T)} \bigl\vert \psi(t) \bigr\vert \sup_{x\in\Omega} \vert \omega_{k}\vert \int^{T}_{0} \bigl\vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\vert \bigl\vert \nabla v_{\epsilon}(t) \bigr\vert \,dt \\ & \hphantom{b_{2}={}}{}+\frac{1}{2}\sup_{t\in(0,T)} \bigl\vert \psi(t) \bigr\vert \sup_{x\in\Omega} \vert \nabla\omega_{k}\vert \int^{T}_{0} \bigl\vert v_{\epsilon}(t) \bigr\vert \bigl\vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\vert \,dt \\ & \hphantom{b_{2}}\leq C \bigl\Vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb{L} ^{2}(\Omega))} \bigl( \bigl\Vert v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb{H}^{1}_{0}( \Omega))} \\ & \hphantom{b_{2}={}}{}+ \bigl\Vert v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb {L}^{2}(\Omega))} \bigr) \rightarrow0\quad \mbox{as }m'\rightarrow0, \end{aligned}$$
(3.71)
$$\begin{aligned}& b_{3} =\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon m'}(t)-v_{\epsilon}(t), \omega_{k}\psi(t),v_{\epsilon m'}(t) \bigr)\,dt \biggr\vert \\ & \hphantom{b_{3}}\leq\frac{1}{2}\sup_{t\in(0,T)} \bigl\vert \psi(t) \bigr\vert \sup_{x\in\Omega} \vert \nabla\omega_{k}\vert \int^{T}_{0} \bigl\vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\vert \bigl\vert v_{\epsilon m'}(t) \bigr\vert \,dt \\ & \hphantom{b_{3}}\leq C \bigl\Vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb{L} ^{2}(\Omega))} \bigl\Vert v_{\epsilon m'}(t) \bigr\Vert _{L^{2}(0,T;\mathbb{L}^{2}( \Omega))} \rightarrow0\quad \mbox{as }m'\rightarrow\infty, \end{aligned}$$
(3.72)
$$\begin{aligned}& b_{4} =\frac{1}{2} \biggl\vert \int^{T}_{0}b \bigl(v_{\epsilon}(t), \omega_{k}\psi(t),v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr)\,dt \biggr\vert \\ & \hphantom{b_{4}}=\frac{1}{2}\sup_{t\in(0,T)} \bigl\vert \psi(t) \bigr\vert \sup_{x\in\Omega} \vert \nabla\omega_{k}\vert \int^{T}_{0} \bigl\vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\vert \bigl\vert v_{\epsilon}(t) \bigr\vert \,dt \\ & \hphantom{b_{4}}\leq C \bigl\Vert v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb {L}^{2}(\Omega))} \bigl\Vert v_{\epsilon m'}(t)-v_{\epsilon}(t) \bigr\Vert _{L^{2}(0,T;\mathbb {L}^{2}(\Omega ))}\rightarrow0 \quad \mbox{as } m'\rightarrow\infty. \end{aligned}$$
(3.73)
It then follows from (3.69)-(3.73) that
$$\begin{aligned} \int^{T}_{0}\hat{b} \bigl(v_{\epsilon m'}(t),v_{\epsilon m'}(t), \omega_{k} \psi(t) \bigr)\,dt\rightarrow \int^{T}_{0}\hat{b} \bigl(v_{\epsilon}(t),v_{\epsilon }(t), \omega_{k}\psi(t) \bigr)\,dt. \end{aligned}$$
(3.74)
Similarly, we can conclude that
$$\begin{aligned} & \int^{T}_{0}\hat{n} \bigl(q_{\epsilon m'}(t)+ \tilde{Q}_{0}, \varphi_{\epsilon m'}(t)+\tilde{\phi}_{0}, \omega_{k}\psi(t) \bigr)\,dt \\ &\quad \rightarrow \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}(t)+ \tilde{Q}_{0}, \varphi_{\epsilon}(t)+\tilde{\phi}_{0}, \omega_{k}\psi(t) \bigr)\,dt, \end{aligned}$$
(3.75)
$$\begin{aligned} & \int^{T}_{0}\hat{n} \bigl(q_{\epsilon m'}(t)+ \tilde{Q}_{0},\omega_{k} \psi(t),v_{\epsilon m'}(t) \bigr) \,dt \\ &\quad \rightarrow \int^{T}_{0}\hat{n} \bigl(q_{ \epsilon}(t)+ \tilde{Q}_{0},\omega_{k}\psi(t),v_{\epsilon}(t) \bigr) \,dt. \end{aligned}$$
(3.76)
Taking the limit of (3.66)-(3.68) as \(m'\rightarrow \infty\), we find that
$$\begin{aligned}& -a_{1} \int^{T}_{0} \bigl(v_{\epsilon}(t), \omega_{k}\psi'(t) \bigr)\,dt+ a_{2} \int^{T}_{0} \bigl( \bigl(v_{\epsilon}(t), \omega_{k}\psi(t) \bigr) \bigr)\,dt \\& \qquad {}+a_{1} \int^{T}_{0} \hat{b} \bigl(v_{\epsilon}(t),v_{\epsilon}(t), \omega_{k}\psi(t) \bigr)\,dt \\& \qquad {}+ \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}+ \tilde{Q}_{0},\varphi_{\epsilon}+ \tilde{\phi}_{0}, \omega_{k} \psi(t) \bigr)\,dt+ \int^{T}_{0} \bigl(\nabla p_{ \epsilon}, \omega_{k}\psi(t) \bigr)\,dt \\& \quad = \int^{T}_{0} \bigl(f(t),\omega_{k}\psi (t) \bigr)\,dt+(v _{0},\omega_{k})\psi(0), \end{aligned}$$
(3.77)
$$\begin{aligned}& - \int^{T}_{0} \bigl(q_{\epsilon}(t),w_{k} \psi'(t) \bigr)\,dt+a_{3} \int^{T}_{0} \bigl( \bigl(q _{\epsilon}(t),w_{k} \psi(t) \bigr) \bigr)\,dt+a_{4} \int^{T}_{0} \bigl(q_{\epsilon}(t),w _{k}\psi(t) \bigr)\,dt \\& \quad {}- \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}+ \tilde{Q}_{0},\omega_{k}\psi(t),v _{\epsilon} \bigr) \,dt=(q_{0},w_{k})\psi(0), \end{aligned}$$
(3.78)
$$\begin{aligned}& -\epsilon \int^{T}_{0} \bigl(p_{\epsilon}(t), \gamma_{l}\psi'(t) \bigr)\,dt+ \int^{T}_{0} \bigl( \operatorname{div}v_{\epsilon}(t), \gamma_{l}\psi(t) \bigr)\,dt \\& \quad =\epsilon (p_{0}, \gamma_{l})\psi(0),\quad 1\leq k,l\leq m. \end{aligned}$$
(3.79)
Thus, (3.77) (resp. (3.78) or (3.79)) holds for ω= (resp. w or γ) any finite linear combination of \(\omega_{k}\) (resp. \(w_{k}\) or \(\gamma_{l}\)). And by a continuity argument, (3.77) is still valid for any \(\omega\in{\mathbb{H} ^{1}_{0}}(\Omega)\), (3.78) is still valid for any \(w\in H^{1} _{0}(\Omega)\), and (3.79) is still valid for any \(\gamma \in L^{2}(\Omega)\). Hence,
$$\begin{aligned}& -a_{1} \int^{T}_{0} \bigl(v_{\epsilon}(t),\omega\psi '(t) \bigr)\,dt+ a_{2} \int^{T}_{0} \bigl( \bigl(v_{\epsilon}(t),\omega \psi(t) \bigr) \bigr)\,dt \\& \qquad {}+a_{1} \int^{T}_{0} \hat{b} \bigl(v_{\epsilon}(t),v_{\epsilon}(t), \omega\psi(t) \bigr)\,dt \\& \qquad {}+ \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}+ \tilde{Q}_{0},\varphi_{\epsilon}+ \tilde{\phi}_{0}, \omega\psi(t) \bigr)\,dt+ \int^{T}_{0} \bigl(\nabla p_{\epsilon },\omega \psi(t) \bigr)\,dt \\& \quad {}= \int^{T}_{0} \bigl(f(t),\omega\psi(t) \bigr) \,dt+(v_{0}, \omega)\psi(0), \end{aligned}$$
(3.80)
$$\begin{aligned}& - \int^{T}_{0} \bigl(q_{\epsilon}(t),w\psi '(t) \bigr)\,dt+a_{3} \int^{T}_{0} \bigl( \bigl(q _{\epsilon}(t),w\psi (t) \bigr) \bigr)\,dt+a_{4} \int^{T}_{0} \bigl(q_{\epsilon}(t),w \psi(t) \bigr)\,dt \\& \quad {}- \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}+ \tilde{Q}_{0},\omega\psi(t),v _{\epsilon} \bigr) \,dt=(q_{0},w) \psi(0), \end{aligned}$$
(3.81)
$$\begin{aligned}& -\epsilon \int^{T}_{0} \bigl(p_{\epsilon}(t),\gamma\psi '(t) \bigr)\,dt+ \int^{T} _{0} \bigl( \operatorname{div}v_{\epsilon}(t), \gamma\psi(t) \bigr)\,dt \\& \quad {}=\epsilon(p_{0}, \gamma) \psi(0),\quad 1\leq k,l\leq m. \end{aligned}$$
(3.82)
Equations (3.80)-(3.82) show that \(\{v_{\epsilon},q_{ \epsilon},\varphi_{\epsilon},p_{\epsilon}\}\) satisfy (3.24)-(3.27) in the sense of distributions.
It remains to prove that \(v_{\epsilon}\), \(q_{\epsilon}\), \(\varphi_{\epsilon}\) and \(p_{\epsilon}\) satisfy (3.28). To this end, we take \(\psi(t)\in{\mathcal{C}^{\infty}_{c}}([0,T])\) with \(\psi(T)=0\) and use \(\psi(t)\) to multiply (3.24)-(3.25), (3.27), respectively, then integrate the resulting equalities over \([0,T]\) and then use integrating by parts for the first term to obtain
$$\begin{aligned}& -a_{1} \int^{T}_{0} \bigl(v_{\epsilon}(t),\omega\psi '(t) \bigr)\,dt+ a_{2} \int^{T}_{0} \bigl( \bigl(v_{\epsilon}(t),\omega \psi(t) \bigr) \bigr)\,dt \\& \qquad {}+a_{1} \int^{T}_{0} \hat{b} \bigl(v_{\epsilon}(t),v_{\epsilon}(t), \omega\psi(t) \bigr)\,dt \\& \qquad {}+ \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}+ \tilde{Q}_{0},\varphi_{\epsilon}+ \tilde{\phi}_{0}, \omega\psi(t) \bigr)\,dt+ \int^{T}_{0} \bigl(\nabla p_{\epsilon },\omega \psi(t) \bigr)\,dt \\& \quad = \int^{T}_{0} \bigl(f(t),\omega\psi(t) \bigr)\,dt+ \bigl(v_{\epsilon }(0),\omega \bigr)\psi(0), \end{aligned}$$
(3.83)
$$\begin{aligned}& - \int^{T}_{0} \bigl(q_{\epsilon}(t),w\psi '(t) \bigr)\,dt+a_{3} \int^{T}_{0} \bigl( \bigl(q _{\epsilon}(t),w\psi (t) \bigr) \bigr)\,dt+a_{4} \int^{T}_{0} \bigl(q_{\epsilon}(t),w \psi(t) \bigr)\,dt \\& \qquad {}- \int^{T}_{0}\hat{n} \bigl(q_{\epsilon}+ \tilde{Q}_{0},\omega\psi(t),v _{\epsilon} \bigr)\,dt= \bigl(q_{\epsilon}(0),w \bigr)\psi(0), \end{aligned}$$
(3.84)
$$\begin{aligned}& -\epsilon \int^{T}_{0} \bigl(p_{\epsilon}(t),\gamma\psi '(t) \bigr)\,dt+ \int^{T} _{0} \bigl( \operatorname{div}v_{\epsilon}(t), \gamma\psi(t) \bigr)\,dt \\& \quad =\epsilon \bigl(p_{ \epsilon}(0), \gamma \bigr)\psi(0),\quad 1\leq k,l \leq m. \end{aligned}$$
(3.85)
By comparing (3.80) with (3.83), (3.81) with (3.84), (3.82) with (3.85), we obtain
$$\begin{aligned} & \bigl(v_{\epsilon}(0)-u_{0},\omega \bigr)\psi(0)=0 \quad \forall \omega\in{\mathbb{H}^{1}_{0}}(\Omega), \\ & \bigl(q_{\epsilon}(0)-q_{0},w \bigr)\psi(0)=0\quad \forall w\in H^{1}_{0}(\Omega), \\ & \bigl(p_{\epsilon}(0)-p_{0},\gamma \bigr)\psi(0)=0\quad \forall \gamma\in L^{2}(\Omega). \end{aligned}$$
We can choose \(\psi(0)\neq0\) and obtain
$$\begin{aligned} & \bigl(v_{\epsilon}(0)-v_{0},\omega \bigr)=0\quad \forall\omega \in{ \mathbb{H}^{1}_{0}}(\Omega), \\ & \bigl(q_{\epsilon}(0)-q_{0},w \bigr)=0 \quad \forall w\in H^{1}_{0}(\Omega), \\ & \bigl(p_{\epsilon}(0)-p_{0},\gamma \bigr)=0\quad \forall\gamma \in L^{2}(\Omega). \end{aligned}$$
Thus, (3.28) holds.
Next, we will prove the solution is unique. For this purpose we drop the indices ϵ and denote by \(\{v_{1},q_{1},\varphi_{1},p_{1}\}\), \(\{v_{2},q_{2},\varphi_{2},p_{2}\}\) two solutions of Problem 3.1 and then set
$$\begin{aligned}& u=v_{2}-v_{1},\quad r=q_{2}-q_{1}, \quad \psi= \varphi_{2}-\varphi_{1},\quad p=p _{2}-p_{1}, \\& Q_{i}=q_{i}+\tilde{Q}_{0},\quad \phi_{i}= \varphi_{i}+ \tilde{\phi}_{0},\quad i=1,2. \end{aligned}$$
Then
$$\begin{aligned} &a_{1}u'+a_{2}{ \mathcal{A}_{0}}u+ \nabla p=-a_{1}\hat{B}(v _{2},v_{2})+a_{1} \hat{B}(v_{1},v_{1})+\hat{N}(Q_{1}, \phi_{1})-\hat{N}(Q _{2},\phi_{2}), \end{aligned}$$
(3.86)
$$\begin{aligned} &r'=-a_{3}{ \mathcal{A}_{1}}r- a_{4}r+\hat{M}(Q_{2},v_{2})- \hat{M}(Q_{1},v_{1}), \end{aligned}$$
(3.87)
$$\begin{aligned} &\epsilon p' +\operatorname{div}u=0, \end{aligned}$$
(3.88)
$$\begin{aligned} &{ \mathcal{A}_{1}}\psi=r. \end{aligned}$$
(3.89)
Taking the scalar of (3.86) with u, (3.87) with ψ and (3.88) with p, and adding these equations and using (3.89), we find
$$\begin{aligned} &\frac{d}{dt} \bigl[a_{1} \bigl\vert u(t) \bigr\vert ^{2}+ \bigl\Vert \psi(t) \bigr\Vert ^{2}+ \epsilon \bigl\vert p(t) \bigr\vert ^{2} \bigr]+2a _{2} \bigl\Vert u(t) \bigr\Vert ^{2}+2a_{3} \bigl\vert r(t) \bigr\vert ^{2}+2a_{4} \bigl\Vert \psi(t) \bigr\Vert ^{2} \\ &\quad = -2a_{1}\hat{b}(u,v_{1},u)+2\hat{n}(r,\psi ,v_{1})-2\hat{n}(r,\phi_{1},u) \\ &\quad \leq a_{1} \bigl[\Vert u\Vert \vert u\vert \Vert v_{1}\Vert +\vert u\vert ^{\frac{1}{2}}\Vert u\Vert ^{\frac{3}{2}} \Vert v_{1}\Vert ^{\frac{1}{2}}\vert v_{1}\vert ^{\frac{1}{2}} \bigr] \\ &\qquad {}+\vert r\vert ^{\frac{3}{2}} \vert v_{1} \vert ^{ \frac{1}{2}}\Vert v_{1}\Vert ^{\frac{1}{2}}\Vert \psi \Vert ^{\frac{1}{2}}+\Vert \psi \Vert ^{\frac{1}{2}}\vert \psi \vert ^{\frac{1}{2}}\vert r\vert \vert v_{1} \vert ^{\frac{1}{2}}\Vert v_{1}\Vert ^{\frac{1}{2}} \\ &\qquad {}+ \vert r\vert \vert u\vert ^{\frac{1}{2}}\Vert u\Vert ^{\frac{1}{2}}\Vert \phi_{1}\Vert ^{\frac{1}{2}} \Vert \phi_{1}\Vert ^{\frac{1}{2}}_{H^{2}}+\vert \phi_{1} \vert ^{\frac{1}{2}} \Vert \phi_{1}\Vert ^{\frac{1}{2}}\vert r \vert \vert u\vert ^{\frac{1}{2}}+\Vert u\Vert ^{\frac{1}{2}} \\ &\quad \leq a_{2}\Vert u\Vert ^{2}+C_{1}(a_{2}) \bigl(\vert u\vert ^{2}\Vert v_{1}\Vert ^{2}+\vert u\vert ^{2} \Vert v_{1}\Vert ^{2}\vert v_{1}\vert ^{2} \bigr)+a_{3} \vert r\vert ^{2} \\ &\qquad {}+C_{2} \bigl(\vert v_{1} \vert ^{2} \Vert v_{1}\Vert ^{2}\Vert \psi \Vert ^{2}+ \Vert \psi \Vert \vert \psi \vert \vert v_{1} \vert \Vert v_{1}\Vert \bigr) \\ &\qquad {}+ a_{3}\vert r\vert ^{2}+a_{2}\Vert u \Vert ^{2}+C_{3}(a_{3},a_{2}) \bigl( \vert u\vert ^{2}\Vert \phi_{1}\Vert ^{2} \Vert \phi_{1}\Vert ^{2}_{H^{2}}+\vert u\vert ^{2}\vert \phi_{1}\vert ^{2}\Vert \phi_{1}\Vert ^{2} \bigr). \end{aligned}$$
(3.90)
Then we can conclude from that equation
$$\begin{aligned} &\frac{d}{dt} \bigl[a_{1} \bigl\vert u(t) \bigr\vert ^{2}+ \bigl\Vert \psi(t) \bigr\Vert ^{2}+ \epsilon \bigl\vert p(t) \bigr\vert ^{2} \bigr] \\ &\quad \leq C_{1}(a_{2}) \bigl(\vert u \vert ^{2} \Vert v_{1}\Vert ^{2}+\vert u \vert ^{2} \Vert v_{1}\Vert ^{2}\vert v_{1}\vert ^{2} \bigr)+C _{2}(a_{3}) \bigl(\vert v_{1}\vert ^{2}\Vert v_{1}\Vert ^{2} \Vert \psi \Vert ^{2}+\Vert \psi \Vert \vert \psi \vert \vert v _{1}\vert \Vert v_{1}\Vert \bigr) \\ &\qquad {}+ C_{3}(a_{3},a_{2}) \bigl(\vert u \vert ^{2}\Vert \phi_{1}\Vert ^{2}\Vert \phi_{1}\Vert ^{2}_{H^{2}}+\vert u\vert ^{2}\vert \phi_{1}\vert ^{2}\Vert \phi_{1}\Vert ^{2} \bigr) \\ &\quad \leq C_{1}(a_{2}) \bigl(\vert u \vert ^{2} \Vert v_{1}\Vert ^{2}+\vert u \vert ^{2} \Vert v_{1}\Vert ^{2}\vert v_{1}\vert ^{2} \bigr)+C _{2}(a_{3}) \bigl(\vert v_{1}\vert ^{2}\Vert v_{1}\Vert ^{2} \Vert \psi \Vert ^{2}+\Vert \psi \Vert ^{2} \vert v_{1}\vert \Vert v_{1}\Vert \bigr) \\ &\qquad {}+ C_{3}(a_{3},a_{2}) \bigl(\vert u \vert ^{2}\Vert \phi_{1}\Vert ^{2}\Vert \phi_{1}\Vert ^{2}_{H^{2}}+\vert u\vert ^{2}\vert \phi_{1}\vert ^{2}\Vert \phi_{1}\Vert ^{2} \bigr). \end{aligned}$$
(3.91)
From (3.91), we have
$$\begin{aligned} \frac{d}{dt} \bigl[a_{1} \bigl\vert u(t) \bigr\vert ^{2}+ \bigl\Vert \psi(t) \bigr\Vert ^{2}+\epsilon \bigl\vert p(t) \bigr\vert ^{2} \bigr] \leq h(t) \bigl[a_{1} \bigl\vert u(t) \bigr\vert ^{2}+ \bigl\Vert \psi(t) \bigr\Vert ^{2} \bigr], \end{aligned}$$
(3.92)
where
$$\begin{aligned} h(t) =&\frac{C_{1}(a_{2})}{a_{1}} \bigl[ \bigl\Vert v_{1}(t) \bigr\Vert ^{2}+ \bigl\vert v_{1}(t) \bigr\vert ^{2} \bigl\Vert v_{1}(t) \bigr\Vert ^{2} \bigr] \\ &{}+C_{2}(a_{3}) \bigl[ \bigl\vert v_{1}(t) \bigr\vert ^{2} \bigl\Vert v_{1}(t) \bigr\Vert ^{2}+ \bigl\vert v_{1}(t) \bigr\vert \bigl\Vert v_{1}(t) \bigr\Vert \bigr] \\ &{}+ \frac{C_{3}(a_{3},a_{2})}{a_{1}} \bigl[ \bigl\Vert \phi_{1}(t) \bigr\Vert ^{2} \bigl\Vert \phi_{1}(t) \bigr\Vert ^{2}_{H^{2}}+ \bigl\vert \phi_{1}(t) \bigr\vert ^{2} \bigl\Vert \phi(t) \bigr\Vert ^{2} \bigr]. \end{aligned}$$
(3.93)
The function \(t\rightarrow h(t)\) belongs to \(L^{1}(0,T)\) in view of equations (3.40), (3.42) and (3.47)-(3.48). Moreover, we infer \(\psi(0)=0\) from \(r(0)=0\) and \({ \mathcal{A}_{1}}\psi=r\). We apply the Gronwall inequality to equation (3.92), we have
$$\bigl\vert u(t) \bigr\vert =0,\quad \bigl\Vert \psi(t) \bigr\Vert =0, \quad \mbox{for all }t \in[0,T]. $$
Thus, \(v_{1}=v_{2}\), \(q_{1}=q_{2}\) and \(\varphi_{1}=\varphi_{2}\). The proof is complete. □