3.1 Stream function
First we introduce a stream function ψ which solves the Stokes system (see [8])
$$ \textstyle\begin{cases} -\triangle u+\nabla q= 0, & \mbox{in } \Omega,\\ \operatorname {div}u=0, &\mbox{in }\Omega,\\ u=\varphi &\mbox{a.e. on }\partial\Omega \mbox{ in the sense of nontangential convergence}, \end{cases} $$
(3.1)
and ψ satisfies
$$\begin{aligned}& \sup_{x\in\Omega} \bigl\vert \psi(x) \bigr\vert + \sup_{x\in\Omega } \bigl\vert \nabla \psi(x) \bigr\vert \operatorname {dist}(x, \partial \Omega)\leq C_{4}\Vert \varphi \Vert _{L^{\infty }(\partial \Omega)}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \bigl\Vert \vert \nabla\psi \vert \operatorname {dist}(\cdot,\partial \Omega)^{1-\frac{1}{p}} \bigr\Vert _{L^{p}(\Omega)}\leq C_{5}\Vert \varphi \Vert _{L^{p}(\partial\Omega)},\quad 2\leq p\leq\infty. \end{aligned}$$
(3.3)
It follows that
$$\begin{aligned} &\Vert \psi \Vert _{L^{\infty}(\Omega)}\leq C_{4}\Vert \varphi \Vert _{L^{\infty }(\partial\Omega)}. \end{aligned}$$
(3.4)
Let \(\varepsilon\in(0,c\cdot \operatorname {diam}(\Omega))\) be a constant to be determined later, and \(\eta_{\varepsilon}\in C_{0}^{\infty}(\mathbb {R}^{2})\) such that
$$ \textstyle\begin{cases} \eta_{\varepsilon}= 1, &\mbox{in } \{x\in R^{2}\vert \operatorname {dist}(x,\partial \Omega )\leq C'_{1}\varepsilon\},\\ \eta_{\varepsilon}= 0, &\mbox{in } \{x\in R^{2}\vert \operatorname {dist}(x,\partial \Omega )\geq C'_{2}\varepsilon\},\\ 0\leq\eta_{\varepsilon}\leq1, & \mbox{otherwise}, \end{cases} $$
(3.5)
and
$$\begin{aligned} & \bigl\vert \nabla^{\alpha}\eta_{\varepsilon} \bigr\vert \leq C'_{\alpha }/\varepsilon^{\vert \alpha \vert }, \end{aligned}$$
(3.6)
where \(\eta_{\varepsilon}\) is in the form \(h(\frac{\rho (x)}{\varepsilon })\), h is a standard bump function, and \(\rho\in C^{\infty}\) is a regularized distance bump function to ∂Ω.
We also know
$$\begin{aligned}& \operatorname {div}\psi=0,\qquad x\in\Omega;\qquad \psi=u,\qquad x\in \bigl\{ x \in \Omega ; \operatorname {dist}(x,\partial \Omega)< C'_{1}\varepsilon \bigr\} , \end{aligned}$$
(3.7)
$$\begin{aligned}& \psi=\varphi, \quad \mbox{on }\partial\Omega \mbox{ in the sense of nontangential convergence}, \end{aligned}$$
(3.8)
and
$$\begin{aligned} &\operatorname {Supp}\psi\subset \bigl\{ x\in\bar{\Omega}; \operatorname {dist}(x,\partial \Omega)< C'_{2} \varepsilon \bigr\} . \end{aligned}$$
(3.9)
Lemma 3.1
Assume
ψ
satisfies (3.7)-(3.9), then we have
$$\begin{aligned} &\Delta\psi=\nabla(q\eta_{\varepsilon})+F, \end{aligned}$$
(3.10)
where
$$\begin{aligned}& \Vert F\Vert _{L^{2}(\Omega)}\leq C/\varepsilon^{\frac {3}{2}} \Vert \varphi \Vert _{L^{2}(\partial\Omega)}, \qquad \nabla q=\triangle u, \end{aligned}$$
(3.11)
$$\begin{aligned}& F=0,\quad \textit{if }x\in \bigl\{ x|\operatorname {dist}(x,\partial\Omega)< C'_{1} \varepsilon\textit{ or }\operatorname {dist}(x,\partial\Omega)> C'_{2}\varepsilon \bigr\} . \end{aligned}$$
(3.12)
3.2 Assumptions and abstract equation
For any \(t\in(\tau,T)\), we define \(u:(\tau-h,T)\rightarrow (L^{2}(\Omega))^{2}\), and \(u_{t}\) is a function defined on \((-h,0)\) satisfying \(u_{t}=u(t+s)\), \(s\in(-h,0)\). Let
$$C_{H}= C^{0} \bigl([-h,0];H \bigr), \qquad C_{D(A^{3/4})}=C^{0} \bigl([-h,0];D \bigl(A^{3/4} \bigr) \bigr), \qquad C_{V}=C^{0} \bigl([-h,0];V \bigr) $$
be three Banach spaces with the norms
$$\begin{aligned}& \Vert u\Vert _{C_{H}}= \sup_{\theta\in[-h,0]} \bigl\vert u(t+\theta) \bigr\vert , \qquad \Vert u\Vert _{C_{V}}= \sup _{\theta\in[-h,0]} \bigl\Vert u(t+\theta) \bigr\Vert , \\& \Vert u\Vert _{C_{D(A^{3/4})}}= \sup_{\theta\in[-h,0]} \bigl\Vert u(t+\theta) \bigr\Vert _{3/4}, \end{aligned}$$
respectively, and
$$\begin{aligned}& L_{H}^{2}=L^{2}(-h,0;H),\qquad L_{V}^{2}=L^{2}(-h,0;V), \\& L_{H}^{\infty }=L^{\infty}(-h,0;H),\qquad L_{V}^{\infty}=L^{\infty}(-h,0;V). \end{aligned}$$
The problem (1.1) can be written as the abstract form
$$ \textstyle\begin{cases} \frac{du}{dt}+\nu Au+\alpha^{2} Au_{t}+B(u)=f_{\rho}(u)+g(t,u_{t}), \\ u(\tau)=u_{\tau},\qquad u(t)=\phi(t-\tau),\quad t \in(\tau-h,\tau), \end{cases} $$
(3.13)
where \(f_{\rho}(u)=f(t-\rho(t),u(t-\rho(t)))\), \(g(t,u_{t})=\int _{-h}^{0}G(t,s,u(t+s))\,ds\), and it satisfies
-
(a)
\(\forall \xi\in C_{H}\), \(t\in\mathbb{R}\mapsto g(t,\xi) \in(L^{2}(\Omega))^{2} \) is measurable and \(g(t,0)=0\), \(\forall t\in\mathbb{R}\);
-
(b)
there exists \(L_{g}>0\) such that for all \(t\in\mathbb{R}\), \(\xi ,\eta \in C_{H}\),
$$\bigl\vert g(t,\xi)-g(t,\eta) \bigr\vert \leq L_{g}\Vert \xi- \eta \Vert _{C_{H}}; $$
-
(c)
\(\exists m_{0}\geq0\), \(C_{g}>0: \forall m\in[0,m_{0}]\), \(\tau\leq t\), \(u,v\in C^{0}([\tau-h,t];H)\),
$$\int_{\tau}^{t}e^{ms} \bigl\vert g(s,u_{s})-g(s,v_{s}) \bigr\vert ^{2}\,ds\leq C_{g}^{2} \int_{\tau -h}^{t}e^{ms} \bigl\vert u(s)-v(s) \bigr\vert ^{2}\,ds; $$
-
(d)
\(\rho: [0,\infty)\rightarrow[0,h]\), \(\vert \frac{d\rho}{dt} \vert \leq M <1\);
-
(e)
\(f(t,u)\) satisfies the Lipschitz condition with respect to u: \(\exists L(\beta)>0\) such that
$$\bigl\vert f(t,u)-f(t,v) \bigr\vert \leq L(\beta)\vert u-v\vert ; $$
-
(f)
\(\exists a>0, b>0\) such that \(\vert f(t,u)\vert ^{2}\leq a\vert u\vert ^{2}+b\);
-
(g)
\(\nu>\frac{3C_{g}}{\lambda_{1}}\);
-
(h)
under the conditions (a)-(g), \(\exists K_{1}>0\), and let
$$\nu>\frac{6C_{1}^{4}}{\nu\lambda_{1}}K_{1}^{2}+\frac{6C_{2}^{2}C_{4}^{2}}{\nu \lambda_{1}}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega)}^{2}+\frac{6C_{2}^{2}C_{3}C_{4}^{2}}{\nu \lambda_{1}}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega)}^{2}+\frac {2C_{2}C_{g}}{3\nu \lambda_{1}}+\frac{2C_{g}}{\lambda_{1}^{\frac{3}{2}}}. $$
Let \(v=u-\psi\), (1.1) can be reduced to the following system:
$$ \textstyle\begin{cases} \frac{\partial v}{\partial t}-\nu\Delta v-\alpha^{2}\Delta v_{t}+(v \cdot\nabla)v+(v \cdot\nabla)\psi\\ \qquad {}+(\psi\cdot\nabla)v +\nabla(p-\nu q\eta_{\varepsilon})\\ \quad = \bar{f}-(\psi\cdot\nabla)\psi+g(t,v_{t}+\psi), &(x,t)\in\Omega _{\tau },\\ \operatorname {div}v=0, &(x,t)\in\Omega_{\tau},\\ v=0, &(x,t)\in\partial\Omega_{\tau},\\ v(\tau,x)=v_{\tau}(x), &x\in\Omega,\\ v(t,x)=\phi(t-\tau,x)-\psi(x)=\eta(t-\tau,x), &(x,t)\in\Omega _{\tau h}, \end{cases} $$
(3.14)
where \(\bar{f}=f_{\rho}(v+\psi)+\nu F\), \(g(t,v_{t}+\psi)=\int _{-h}^{0}G(s,v(t+s)+\psi)\,ds\), \(\phi\in L^{2}_{V}\cap L^{\infty}_{H}\).
Let \(v_{0}\in H\), \(\eta\in L_{H}^{2}\), we consider the equivalent abstract system of (3.14)
$$ \textstyle\begin{cases} \frac{dv}{dt}+\nu Av+\alpha^{2}Av_{t}+B(v)+R(v)=P\bar{f}-B(\psi)+g(t,v_{t}+\psi), \\ v(\tau)=v_{\tau},\\ v(t)=\eta(t-\tau), \end{cases} $$
(3.15)
where \(R(v)=B(v,\psi)+B(\psi,v)\), which is also a linear continuous operator from V into \(V'\) and maps W into H (see [21]).
Definition 3.1
Let \(u_{\tau},f\in H\), \(\varphi\in L^{\infty }(\partial \Omega)\) and \(\varphi\cdot n=0\) on ∂Ω, u is called a weak solution of the problem (1.1) provided
-
(i)
\(u\in C([\tau-h,T];V)\), \(u(\cdot,\tau)=u_{\tau}\), and \(du/dt\in L^{2}([\tau,T];V')\);
-
(ii)
\(\forall v\in C^{\infty}_{0}(\Omega)\) with \(\operatorname {div}v=0\), we get
$$\begin{aligned} &\frac{d}{dt}\langle u,v\rangle-\nu\langle u,\Delta v\rangle - \alpha^{2} \frac {d}{dt}\langle u,\Delta v\rangle - \int_{\Omega}\sum^{2}_{i,j=1}u^{i}u^{j} \frac {\partial v^{i}}{x_{j}}\,dx \\ &\quad =\langle f,v\rangle +\biggl\langle \int _{-h}^{0}G \bigl(s,u(t+s) \bigr)\,ds,v\biggr\rangle ; \end{aligned} $$
-
(iii)
\(\exists \psi\in C^{2}(\Omega)\cap L^{\infty}(\Omega)\), \(q\in C^{1}(\Omega)\) and \(g\in L^{2}(\Omega)\) such that
$$\textstyle\begin{cases} \triangle\psi=\nabla q+g, &\mbox{in }\Omega,\\ \operatorname {div}\psi=0, &\mbox{in } \Omega,\\ \psi=\varphi &\mbox{on }\partial\Omega, \end{cases} $$
where we assume that ψ obtain its boundary values in sense of non-tangential convergence and \(u-\psi\in L^{2}([\tau,T];V)\).
3.3 Existence of solutions and uniqueness
We shall give the main result in this section.
Theorem 3.1
Let
\(v_{\tau}\in V\), \(\eta\in L_{H}^{2}\), and the assumptions (a)-(h) hold, then there exists a unique global weak solution of (3.15) which satisfies
$$v(t)\in L^{\infty}(\tau,T;V)\cap L^{2}(\tau,T;V), $$
and
\(\frac{dv}{dt}\)
is uniformly bounded in
\(L^{2}(\tau,T;V')\).
Proof
We first use the standard Faedo-Galerkin method to establish the existence of a solution to (3.15).
Fix \(n\geq1\), we define an approximate solution \(v_{n}\) to (3.15) as \(v_{n}(t)={\sum_{j=0}^{n}}a_{nj}(t)w_{j}\), which satisfies
$$ \textstyle\begin{cases} \frac{dv_{n}}{dt}+\nu Av_{n}+\alpha^{2}Av_{nt}+B(v_{n})+R(v_{n})=P_{n}\bar{f}-B(\psi )+g(t,v_{nt}+\psi) ,\\ v_{n}(\tau)=v_{n\tau},\\ v_{n}(t)=\eta_{n}(t-\tau),\quad t \in(\tau-h,\tau). \end{cases} $$
(3.16)
We also denote \(f_{n}=f(t,v_{n}(t)+\psi)\), \(f_{n\rho}=f(t-\rho (t),v_{n}(t-\rho (t))+\psi)\), and \(g_{n}=g(t,v_{n}(t)+\psi)\).
Multiplying (3.16) by \(v_{n}\), we have
$$\begin{aligned} & \biggl(\frac{dv_{n}}{dt},v_{n} \biggr)+\nu(Av_{n},v_{n})+ \alpha ^{2}(Av_{nt},v_{n})+ \bigl(B(v_{n}),v_{n} \bigr)+ \bigl(R(v_{n}),v_{n} \bigr) \\ &\quad =\langle P_{n}\bar{f},v_{n}\rangle- \bigl(B(\psi ),v_{n} \bigr)+\langle g_{n},v_{n}\rangle \end{aligned}$$
(3.17)
and
$$ \begin{aligned}[b] \frac{1}{2}\frac{d}{dt} \bigl(\vert v_{n}\vert ^{2}+\alpha^{2}\Vert v_{n}\Vert ^{2} \bigr)+\nu \Vert v_{n}\Vert ^{2}\leq{}& \bigl\vert b(v_{n},v_{n},v_{n}) \bigr\vert + \bigl\vert b(\psi ,v_{n},v_{n}) \bigr\vert + \bigl\vert b(v_{n},\psi,v_{n}) \bigr\vert \\ &{}+ \bigl\vert \langle P_{n}\bar{f},v_{n}\rangle \bigr\vert + \bigl\vert \bigl(B(\psi ),v_{n} \bigr) \bigr\vert + \bigl\vert \langle g_{n},v_{n}\rangle \bigr\vert . \end{aligned} $$
(3.18)
We estimate each term on the right side of (3.18) in the following.
Using Hardy’s inequality, we obtain
$$\begin{aligned} \bigl\vert b \bigl((v_{n}),\psi,v_{n} \bigr) \bigr\vert \leq& \int_{\Omega} \vert v_{n}\vert \vert \nabla\psi \vert \vert v_{n}\vert \,dx \\ \leq&C_{4}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega)} \int _{\operatorname {dist}(x,\partial \Omega)\leq C'_{2}\varepsilon}\frac{\vert v_{n}\vert ^{2}}{\operatorname {dist}(x,\partial \Omega )}\,dx \\ \leq&C'_{2}C_{4}\varepsilon \Vert \varphi \Vert _{L^{\infty}(\partial\Omega )} \int _{\Omega}\frac{\vert v_{n}\vert ^{2}}{[\operatorname {dist}(x,\partial\Omega )]^{2}}\,dx \\ \leq&C'_{2}C_{3}C_{4}\varepsilon \Vert \varphi \Vert _{L^{\infty}(\partial\Omega )}\Vert v_{n}\Vert ^{2}, \end{aligned}$$
(3.19)
and choose suitable ε such that
$$\begin{aligned} & \bigl\vert b \bigl((v_{n}),\psi,v_{n} \bigr) \bigr\vert \leq\frac{\nu}{6}\Vert v_{n}\Vert ^{2}. \end{aligned}$$
(3.20)
By the Young inequality, the Hölder inequality, Hardy’s inequality, the Cauchy inequality, and the property of the trilinear operator, we derive
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \langle P_{n}\bar{f},v_{n}\rangle \bigr\vert \leq{}&\bigl\vert \langle \bar{f},v_{n}\rangle \bigr\vert \leq\bigl\vert \langle f_{n\rho },v_{n}\rangle \bigr\vert + \nu \bigl\vert \langle F,v_{n}\rangle \bigr\vert \\ \leq{}&\vert f_{n\rho} \vert \vert v_{n}\vert + \frac{C\nu}{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial\Omega)}\Vert v_{n} \Vert \\ \leq{}&\frac{\nu}{6}\Vert v_{n}\Vert ^{2}+ \frac{3}{2\nu\lambda_{1}}\vert f_{n\rho }\vert ^{2}+ \frac{C\nu}{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial \Omega)}\Vert v_{n} \Vert \\ \leq{}&\frac{\nu}{6}\Vert v_{n}\Vert ^{2}+ \frac{3}{2\nu\lambda _{1}} \bigl(a \bigl\vert v_{n} \bigl(t-\rho (t) \bigr)+ \psi \bigr\vert ^{2}+b \bigr)+\frac{C\nu}{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial\Omega)}\Vert v_{n}\Vert \\ \leq{}&\frac{\nu}{6}\Vert v_{n}\Vert ^{2}+ \frac{3a}{\nu\lambda_{1}} \bigl\vert v_{n} \bigl(t-\rho (t) \bigr) \bigr\vert ^{2}+\frac{3a}{\nu\lambda_{1}}\vert \psi \vert ^{2}+ \frac{3b}{2\nu\lambda _{1}} \\ &{}+\frac{C\nu}{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial \Omega)}\Vert v_{n}\Vert \\ \leq{}&\frac{\nu}{6}\Vert v_{n}\Vert ^{2}+ \frac{3a}{\nu\lambda_{1}} \bigl\vert v_{n} \bigl(t-\rho (t) \bigr) \bigr\vert ^{2}+\frac{3aC_{4}^{2}}{\nu\lambda_{1}}\Vert \varphi \Vert _{L^{\infty }(\partial \Omega)}^{2}+ \frac{3b}{2\nu\lambda_{1}} \\ &{}+\frac{C\nu}{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial \Omega)}\Vert v_{n}\Vert , \end{aligned} \end{aligned}$$
(3.21)
$$\begin{aligned}& \bigl\vert b(\psi,\psi,v_{n}) \bigr\vert \leq \int_{\Omega} \vert \psi \vert \vert \nabla\psi \vert \vert v_{n}\vert \,dx\leq C_{4}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega)} \int_{\Omega}\frac {\vert v_{n}\vert }{\operatorname {dist}(x,\partial\Omega)}\vert \psi \vert \,dx \\& \hphantom{\bigl\vert b(\psi,\psi,v_{n}) \bigr\vert }\leq C_{4}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega)} \biggl\{ \int_{\Omega }\frac {\vert v_{n}\vert ^{2}}{[\operatorname {dist}(x,\partial\Omega)]^{2}}\,dx \biggr\} ^{1/2} \biggl\{ \int _{\operatorname {dist}(x,\partial\Omega)\leq C'_{2}\varepsilon} \vert \psi \vert ^{2}\,dx \biggr\} ^{1/2} \\& \hphantom{\bigl\vert b(\psi,\psi,v_{n}) \bigr\vert }\leq C\varepsilon \Vert \varphi \Vert ^{2}_{L^{\infty}(\partial\Omega )}\vert \partial \Omega \vert ^{1/2}\Vert v_{n}\Vert \sqrt{ \varepsilon}, \end{aligned}$$
(3.22)
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \langle g_{n},v_{n}\rangle \bigr\vert & \leq \vert g_{n}\vert \vert v_{n}\vert \\ & \leq \frac {\vert g_{n}\vert ^{2}}{2C_{g}}+ \frac{C_{g}}{2} \vert v_{n}\vert ^{2} \\ &\leq \frac{\vert g_{n}\vert ^{2}}{2C_{g}}+\frac{C_{g}\lambda^{-1}_{1}}{2}\Vert v_{n}\Vert ^{2}. \end{aligned} \end{aligned}$$
(3.23)
Combining (3.19)-(3.23), we conclude
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl(\vert v_{n}\vert ^{2}+\alpha ^{2}\Vert v_{n}\Vert ^{2} \bigr)+\nu \Vert v_{n}\Vert ^{2} \\ &\quad \leq\frac{\nu}{3}\Vert v_{n}\Vert ^{2}+ \frac{3a}{\nu\lambda _{1}} \bigl\vert v_{n} \bigl(t-\rho (t) \bigr) \bigr\vert ^{2}+\frac{3aC_{4}^{2}}{\nu\lambda_{1}}\Vert \varphi \Vert _{L^{\infty }(\partial \Omega)}^{2}+ \frac{3b}{2\nu\lambda_{1}}+\frac{\vert g_{n}\vert ^{2}}{2C_{g}}+ \frac {C_{g}}{2\lambda_{1}}\Vert v_{n} \Vert ^{2} \\ &\qquad {}+ \biggl(\frac{C\nu}{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial\Omega)}+C \varepsilon^{3/2}\Vert \varphi \Vert ^{2}_{L^{\infty }(\partial\Omega)} \vert \partial\Omega \vert ^{1/2} \biggr)\Vert v_{n} \Vert \\ &\quad \leq\frac{\nu}{3}\Vert v_{n}\Vert ^{2}+ \frac{3a}{\nu\lambda _{1}} \bigl\vert v_{n} \bigl(t-\rho (t) \bigr) \bigr\vert ^{2}+\frac{3aC_{4}^{2}}{\nu\lambda_{1}}\Vert \varphi \Vert _{L^{\infty }(\partial \Omega)}^{2}+ \frac{3b}{2\nu\lambda_{1}}+\frac{\vert g_{n}\vert ^{2}}{2C_{g}}+ \frac {C_{g}}{2\lambda_{1}}\Vert v_{n} \Vert ^{2} \\ &\qquad {}+\frac{\nu}{6}\Vert v_{n}\Vert ^{2}+ \frac{3}{2\nu} \biggl(\frac{C\nu }{\varepsilon ^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial\Omega)}+C\varepsilon ^{3/2}\Vert \varphi \Vert ^{2}_{L^{\infty}(\partial\Omega)}\vert \partial\Omega \vert ^{1/2} \biggr)^{2}, \end{aligned}$$
i.e.,
$$\begin{aligned} &\frac{d}{dt} \bigl(\vert v_{n}\vert ^{2}+ \alpha^{2}\Vert v_{n}\Vert ^{2} \bigr) \\ &\quad \leq \frac {6a}{\nu\lambda_{1}} \bigl\vert v_{n} \bigl(t-\rho(t) \bigr) \bigr\vert ^{2}+\frac {1}{C_{g}}\vert g_{n}\vert ^{2}+K^{2}_{0}- \biggl(\nu-\frac{C_{g}}{\lambda_{1}} \biggr)\Vert v_{n}\Vert ^{2}, \end{aligned}$$
(3.24)
where
$$K^{2}_{0}=\frac{6aC_{4}^{2}}{\nu\lambda_{1}}\Vert \varphi \Vert _{L^{\infty }(\partial \Omega)}^{2}+\frac{3b}{\nu\lambda_{1}}+\frac{3}{\nu} \biggl( \frac{C\nu }{\varepsilon^{\frac{3}{2}}}\Vert \varphi \Vert _{L^{2}(\partial\Omega )}+C\varepsilon^{3/2} \Vert \varphi \Vert ^{2}_{L^{\infty}(\partial\Omega )}\vert \partial\Omega \vert ^{1/2} \biggr)^{2}. $$
Choosing suitable \(m>0 \) such that \(\nu>\frac{3C_{g}}{\lambda _{1}}+\frac {m}{\lambda_{1}}+m\alpha^{2}+\frac{6ae^{mh}}{\nu\lambda^{2}_{1}(1-M)}\), we have
$$\begin{aligned} &\frac{d}{dt} \bigl[e^{mt} \bigl(\vert v_{n} \vert ^{2}+\alpha^{2}\Vert v_{n}\Vert ^{2} \bigr) \bigr] \\ &\quad =me^{mt} \bigl(\vert v_{n}\vert ^{2}+ \alpha^{2}\Vert v_{n}\Vert ^{2} \bigr)+e^{mt}\frac {d}{dt} \bigl(\vert v_{n}\vert ^{2}+\alpha^{2}\Vert v_{n}\Vert ^{2} \bigr) \\ &\quad \leq me^{mt} \bigl(\vert v_{n}\vert ^{2}+ \alpha^{2}\Vert v_{n}\Vert ^{2} \bigr)+e^{mt}\frac{6a}{\nu \lambda_{1}} \bigl\vert v_{n} \bigl(t- \rho(t) \bigr) \bigr\vert ^{2}+\frac {1}{C_{g}}\vert g_{n}\vert ^{2}+K^{2}_{0} \\ &\qquad {}- \biggl(\nu-\frac{C_{g}}{\lambda_{1}}\Vert v_{n}\Vert ^{2} \biggr) \\ &\quad \leq\frac{6ae^{mt}}{\nu\lambda_{1}} \bigl\vert v_{n} \bigl(t-\rho(t) \bigr) \bigr\vert ^{2}+\frac {e^{mt}}{C_{g}}\vert g_{n}\vert ^{2}+K^{2}_{0}e^{mt} \\ &\qquad {}-e^{mt} \biggl(\nu-\frac{C_{g}}{\lambda _{1}}-\frac{m}{\lambda_{1}}-m\alpha^{2} \biggr) \Vert v_{n}\Vert ^{2}. \end{aligned}$$
(3.25)
Integrating (3.25) over \([\tau,t]\), we derive
$$\begin{aligned} & e^{mt} \bigl(\vert v_{n}\vert ^{2}+ \alpha^{2}\Vert v_{n}\Vert ^{2} \bigr)-e^{m\tau} \bigl( \bigl\vert v_{n}(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert v_{n}(\tau) \bigr\Vert ^{2} \bigr) \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{6a}{\nu\lambda_{1}} \int_{\tau }^{t}e^{ms} \bigl\vert v_{n} \bigl(s-\rho(s) \bigr) \bigr\vert ^{2}\,ds+ \frac{1}{C_{g}} \int_{\tau }^{t}e^{ms}\vert g_{n}\vert ^{2}\,ds \\ &\qquad {}- \biggl(\nu-\frac{C_{g}}{\lambda_{1}}-\frac{m}{\lambda _{1}}-m\alpha ^{2} \biggr) \int_{\tau}^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{6ae^{mh}}{\nu\lambda _{1}(1-M)} \int _{\tau-h}^{t}e^{ms} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds+C_{g} \int_{\tau -h}^{t}e^{ms} \bigl\vert v_{n}(s)+\psi \bigr\vert ^{2}\,ds \\ &\qquad {}- \biggl(\nu-\frac{C_{g}}{\lambda_{1}}-\frac{m}{\lambda _{1}}-m\alpha ^{2} \biggr) \int_{\tau}^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{6ae^{mh}}{\nu\lambda _{1}(1-M)} \biggl( \int _{\tau-h}^{\tau}e^{ms} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds+ \int_{\tau }^{t}e^{ms} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \biggr) \\ &\qquad {}+C_{g} \int_{\tau-h}^{\tau}e^{ms} \bigl\vert v_{n}(s)+\psi \bigr\vert ^{2}\,ds \\ &\qquad {}+C_{g} \int_{\tau}^{t}e^{ms} \bigl\vert v_{n}(s)+\psi \bigr\vert ^{2}\,ds- \biggl(\nu- \frac {C_{g}}{\lambda _{1}}-\frac{m}{\lambda_{1}}-m\alpha^{2} \biggr) \int_{\tau}^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{6ae^{mh}}{\nu\lambda _{1}(1-M)} \biggl( \int _{\tau-h}^{\tau}e^{ms}\vert \phi_{n}-\psi \vert ^{2}\,ds+ \int_{\tau }^{t}e^{ms} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \biggr) \\ &\qquad {}+C_{g} \int_{\tau-h}^{\tau}e^{ms}\vert \phi_{n}\vert ^{2}\,ds \\ &\qquad {}+2C_{g} \int_{\tau}^{t}e^{ms} \bigl(\vert v_{n}\vert ^{2}+\vert \psi \vert ^{2} \bigr) \,ds- \biggl(\nu-\frac {C_{g}}{\lambda_{1}}-\frac{m}{\lambda_{1}}-m \alpha^{2} \biggr) \int_{\tau }^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{6ae^{mh}}{\nu\lambda _{1}(1-M)} \biggl(2 \int_{\tau-h}^{\tau}e^{ms} \bigl(\vert \phi_{n}\vert ^{2}+\vert \psi \vert ^{2} \bigr) \,ds+ \int _{\tau }^{t}e^{ms} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \biggr) \\ &\qquad {}+C_{g} \int_{\tau-h}^{\tau}e^{ms}\vert \phi_{n}\vert ^{2}\,ds \\ &\qquad {}+2C_{g} \int_{\tau}^{t}e^{ms} \bigl(\vert v_{n}\vert ^{2}+\vert \psi \vert ^{2} \bigr) \,ds- \biggl(\nu-\frac {C_{g}}{\lambda_{1}}-\frac{m}{\lambda_{1}}-m \alpha^{2} \biggr) \int_{\tau }^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{12ae^{mh}e^{m\tau}}{\nu \lambda _{1}(1-M)} \int_{\tau-h}^{\tau} \vert \phi_{n}\vert ^{2}\,ds+\frac{12ae^{mh}e^{m\tau }}{\nu \lambda_{1}(1-M)}\vert \psi \vert ^{2}h \\ &\qquad {}+\frac{6ae^{mh}}{\nu\lambda_{1}(1-M)} \int_{\tau }^{t}e^{ms} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds+C_{g}e^{m\tau} \int_{\tau-h}^{\tau} \vert \phi _{n}\vert ^{2}\,ds \\ &\qquad {}+2C_{g} \int_{\tau}^{t}e^{ms}\vert v_{n}\vert ^{2}\,ds+\frac {2C_{g}e^{mt}}{m}\vert \psi \vert ^{2}- \biggl(\nu-\frac{C_{g}}{\lambda_{1}}-\frac {m}{\lambda _{1}}-m \alpha^{2} \biggr) \int_{\tau}^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{12ae^{mh}e^{m\tau}}{\nu \lambda _{1}(1-M)}\vert \psi \vert ^{2}h+ \biggl(\frac{12ae^{mh}}{\nu\lambda_{1}(1-M)}+C_{g} \biggr)e^{m\tau } \int _{\tau-h}^{\tau} \vert \phi_{n}\vert ^{2}\,ds \\ &\qquad {}+\frac{2C_{g}e^{mt}}{m}\vert \psi \vert ^{2}- \biggl(\nu- \frac{3C_{g}}{\lambda _{1}}-\frac{m}{\lambda_{1}}-m\alpha^{2}-\frac{6ae^{mh}}{\nu\lambda ^{2}_{1}(1-M)} \biggr) \int_{\tau}^{t}e^{ms} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq\frac{K^{2}_{0}}{m}e^{mt}+\frac{12ae^{mh}e^{m\tau}}{\nu \lambda _{1}(1-M)}\vert \psi \vert ^{2}h+ \biggl(\frac{12ae^{mh}}{\nu\lambda_{1}(1-M)}+C_{g} \biggr)e^{m\tau } \int _{\tau-h}^{\tau} \vert \phi_{n}\vert ^{2}\,ds \\ &\qquad {}+\frac{2C_{g}e^{mt}}{m}\vert \psi \vert ^{2}, \end{aligned}$$
(3.26)
which implies
$$\begin{aligned} & \bigl\vert v_{n}(t) \bigr\vert ^{2}+ \alpha^{2} \bigl\Vert v_{n}(t) \bigr\Vert ^{2} \\ &\quad \leq \bigl\vert v_{n}(\tau) \bigr\vert ^{2}+ \alpha^{2} \bigl\Vert v_{n}(\tau) \bigr\Vert ^{2}+\frac {12ae^{mh}}{\nu\lambda_{1}(1-M)}\vert \psi \vert ^{2}h \\ &\qquad {}+ \biggl(\frac{12ae^{mh}}{\nu\lambda_{1}(1-M)}+C_{g} \biggr) \int _{-h}^{0}\vert \phi _{n}\vert ^{2}\,ds+\frac{K^{2}_{0}}{m}+\frac{2C_{g}}{m}\vert \psi \vert ^{2} \\ &\quad \leq \bigl\vert v_{n}(\tau) \bigr\vert ^{2}+ \alpha^{2} \bigl\Vert v_{n}(\tau) \bigr\Vert ^{2}+\frac {12ae^{mh}C_{4}^{2}}{\nu\lambda_{1}(1-M)}\Vert \varphi \Vert _{L^{\infty}(\partial \Omega)}^{2}h \\ &\qquad {}+ \biggl(\frac{12ae^{mh}}{\nu\lambda_{1}(1-M)}+C_{g} \biggr)\Vert \phi_{n} \Vert _{L_{H}^{2}}^{2}+\frac{K^{2}_{0}}{m}+ \frac{2C_{g}C_{4}^{2}}{m} \Vert \varphi \Vert _{L^{\infty }(\partial\Omega)}^{2}\equiv K_{1}^{2}. \end{aligned}$$
(3.27)
Integrating (3.24) over \([t,t+1]\), we obtain
$$\begin{aligned} & \bigl( \bigl\vert v_{n}(t+1) \bigr\vert ^{2}+ \alpha^{2} \bigl\Vert v_{n}(t+1) \bigr\Vert ^{2} \bigr)- \bigl( \bigl\vert v_{n}(t) \bigr\vert ^{2}+ \alpha^{2} \bigl\Vert v_{n}(t) \bigr\Vert ^{2} \bigr) \\ &\qquad {}+ \bigl(\nu-C_{g}\lambda_{1}^{-1} \bigr) \int_{t}^{t+1}\Vert v_{n}\Vert ^{2}\,ds \\ &\quad \leq K^{2}_{0}+\frac{1}{C_{g}} \int_{t}^{t+1}\vert g_{n}\vert ^{2}\,ds+\frac {6a}{\nu \lambda_{1}} \int_{t}^{t+1} \bigl\vert v_{n} \bigl(s- \rho(s) \bigr) \bigr\vert ^{2}\,ds \\ &\quad \leq K^{2}_{0}+C_{g} \int_{t-h}^{t+1} \bigl\vert v_{n}(s)+\psi \bigr\vert ^{2}\,ds+\frac {6a}{\nu \lambda_{1}(1-M)} \int_{t-h}^{t+1} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \\ &\quad \leq K^{2}_{0}+2C_{g} \int_{t-h}^{t+1} \bigl( \bigl\vert v_{n}(s) \bigr\vert ^{2}+\vert \psi \vert ^{2} \bigr)\,ds+ \frac {6a}{\nu\lambda_{1}(1-M)} \int_{t-h}^{t+1} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \\ &\quad \leq K^{2}_{0}+2(h+1)C_{g}\vert \psi \vert ^{2}+ \biggl(\frac{6a}{\nu\lambda _{1}(1-M)}+2C_{g} \biggr) \int_{t-h}^{t+1} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \\ &\quad \leq \biggl(\frac{6a}{\nu\lambda_{1}(1-M)}+2C_{g} \biggr) \biggl( \int _{t-h}^{t} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds+ \int_{t}^{t+1} \bigl\vert v_{n}(s) \bigr\vert ^{2}\,ds \biggr)+2(h+1)C_{g}\vert \psi \vert ^{2} +K^{2}_{0} \\ &\quad \leq \biggl(\frac{6a}{\nu\lambda_{1}(1-M)}+2C_{g} \biggr) \biggl(K_{1}^{2}h+ \int _{-h}^{0} \bigl\vert \eta _{n}(s) \bigr\vert ^{2}\,ds+\frac{1}{\lambda_{1}} \int_{t}^{t+1} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \biggr) \\ &\qquad {}+K^{2}_{0}+2(h+1)C_{g}\vert \psi \vert ^{2} \\ &\quad \leq \biggl(\frac{6a}{\nu\lambda_{1}(1-M)}+2C_{g} \biggr) \\ &\qquad {}\times \biggl(K_{1}^{2}h+2 \int _{-h}^{0} \bigl(\vert \phi _{n} \vert ^{2}+\vert \psi \vert ^{2} \bigr)\,ds+ \frac{1}{\lambda_{1}} \int_{t}^{t+1} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \biggr) \\ &\qquad {}+K^{2}_{0}+2(h+1)C_{g}\vert \psi \vert ^{2} \\ &\quad \leq \biggl(\frac{6a}{\nu\lambda_{1}(1-M)}+2C_{g} \biggr) \\ &\qquad {}\times \biggl(K_{1}^{2}h+2 \int _{-h}^{0}\vert \phi _{n}\vert ^{2}\,ds+2h\vert \psi \vert ^{2}+\frac{1}{\lambda_{1}} \int_{t}^{t+1} \bigl\Vert v_{n}(s) \bigr\Vert ^{2}\,ds \biggr) \\ &\qquad {}+K^{2}_{0}+2(h+1)C_{g}\vert \psi \vert ^{2} \end{aligned}$$
(3.28)
and
$$\begin{aligned} & \biggl(\nu-\frac{3C_{g}}{\lambda_{1}}-\frac{6a}{\nu \lambda ^{2}_{1}(1-M)} \biggr) \int_{t}^{t+1}\Vert v_{n}\Vert ^{2}\,ds \\ &\quad \leq K_{1}^{2}+ \biggl(\frac{6a}{\nu\lambda_{1}(1-M)}+2C_{g} \biggr) \biggl(K_{1}^{2}h+2 \int _{-h}^{0}\vert \phi_{n}\vert ^{2}\,ds+2h\vert \psi \vert ^{2} \biggr)+K^{2}_{0} \\ &\qquad {}+2(h+1)C_{g}\vert \psi \vert ^{2} \\ &\quad \leq K_{1}^{2}+ \biggl(\frac{6a}{\nu\lambda_{1}(1-M)}+2C_{g} \biggr) \bigl(K_{1}^{2}h+2\Vert \phi_{n}\Vert ^{2}_{L_{H}^{2}}+2hC_{4}^{2}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega )}^{2} \bigr)+K^{2}_{0} \\ &\qquad {}+2(h+1)C_{g}C_{4}^{2}\Vert \varphi \Vert _{L^{\infty}(\partial\Omega )}^{2}\equiv K_{2}^{2}. \end{aligned}$$
(3.29)
That is,
$$\begin{aligned} & \int_{t}^{t+1}\Vert v_{n}\Vert ^{2}\,ds\leq\frac{K^{2}_{2}}{\nu -\frac{3C_{g}}{\lambda_{1}}-\frac{6a}{\nu\lambda^{2}_{1}(1-M)}}\equiv I_{V}^{2}, \end{aligned}$$
(3.30)
which means \(v_{n}(t)\) is uniformly bounded in \(L^{\infty}(\tau ,T;V)\cap L^{2}(\tau,T;V)\). Using the Alaoglu compact theorem, we can find a subsequence (still written as \(v_{n}\) without confusion) such that
$$\begin{aligned} v_{n}\rightarrow^{*}v \quad \mbox{in }L^{\infty}(\tau,T;V);\qquad v_{n}\rightarrow v \quad \mbox{in }L^{2}(\tau,T;V), \end{aligned}$$
(3.31)
i.e., \(v\in L^{\infty}(\tau,T;V)\cap L^{2}(\tau,T;V)\).
Next, we prove that \(\frac{dv_{n}}{dt}\) is uniformly bounded in \(L^{2}(\tau,T;V')\). Since
$$\begin{aligned} \frac{dv_{n}}{dt}=-\nu Av_{n}-\alpha ^{2}Av_{nt}-B(v_{n})-R(v_{n})+P_{n} \bar{f}-B(\psi)+g_{n}, \end{aligned}$$
(3.32)
and \(v_{n}\in L^{2}(\tau,T;V)\), we derive that \(-\nu Av_{n},\alpha^{2} v_{nt},g_{n}\in L^{2}(\tau,T;V')\), and
$$\begin{aligned} \bigl\Vert B(v_{n}) \bigr\Vert _{L^{2}(\tau,T;V')}^{2} =& \int_{\tau }^{T} \Bigl(\sup_{\Vert u\Vert =1} \bigl\vert (v_{n}\cdot\nabla)v_{n},u \bigr\vert \Bigr)^{2}\,ds \\ \leq& \int_{\tau }^{T} \bigl( \bigl\vert (v_{n} \cdot \nabla)v_{n} \bigr\vert \vert u\vert \bigr)^{2} \,ds \\ \leq&C \int_{\tau}^{T} \bigl\vert (v_{n}\cdot \nabla)v_{n} \bigr\vert ^{2}\Vert u\Vert ^{2}\,ds \\ =&C \int _{\tau }^{T} \bigl\vert (v_{n}\cdot \nabla)v_{n} \bigr\vert ^{2}\,ds \\ \leq&C \int_{\tau}^{T}\vert v_{n}\vert ^{2}\vert \nabla v_{n}\vert ^{2}\,ds \\ \leq& C \int_{\tau }^{T}\vert v_{n}\vert ^{2}\Vert v_{n}\Vert ^{2}\,ds \\ \leq&C\Vert v_{n}\Vert _{L^{\infty}(\tau,T;H)}^{2}\Vert v_{n}\Vert _{L^{2}(\tau ,T;V)}^{2} \\ \leq&C\Vert v_{n}\Vert _{L^{\infty}(\tau,T;V)}^{2}\Vert v_{n}\Vert _{L^{2}(\tau ,T;V)}^{2}. \end{aligned}$$
(3.33)
Similarly, we have
$$\begin{aligned} \bigl\Vert R(v_{n}) \bigr\Vert _{L^{2}(\tau,T;V')}^{2} \leq& C\Vert \psi \Vert _{L^{\infty}(\Omega)}^{2}\Vert v_{n} \Vert _{L^{\infty}(\tau,T;H)}^{2}+ C\vert \psi \vert ^{2} \Vert v_{n}\Vert _{L^{2}(\tau,T;V)}^{2} \\ \leq&C\Vert \varphi \Vert _{L^{\infty}(\partial\Omega)}^{2} \bigl(\Vert v_{n}\Vert _{L^{\infty }(\tau,T;V)}^{2}+\Vert v_{n} \Vert _{L^{2}(\tau,T;V)}^{2} \bigr). \end{aligned}$$
(3.34)
Since \(B(\psi)\in L^{2}(\tau,T;V')\), we conclude that \(\frac {dv_{n}}{dt}\) is uniformly bounded in \(L^{2}(\tau,T;V')\). By the compact embedding theorem, we also have
$$\begin{aligned} v_{n}\rightarrow v,\quad \mbox{in }L^{2}(\tau,T;V);\qquad v_{n}( \tau )=P_{n}v_{\tau}\rightarrow v(\tau)=v_{\tau}. \end{aligned}$$
(3.35)
□
Theorem 3.2
Let
\(u_{\tau},f\in H\), \(\varphi\in L^{\infty}(\partial\Omega)\), and
\(\varphi\cdot n=0\)
on
∂Ω. Then (1.1) has a unique weak solution.
Proof
The family of stream functions \(\psi_{\varepsilon}\) was constructed in [8] which satisfied \(\psi_{\varepsilon}\in C^{\infty}(\Omega)\). In addition, the solution v of (3.16) is obtained in Theorem 3.1. Let \(u=v+\psi_{\varepsilon}\), it is easy to check that u is the weak solution of (1.1) which satisfies (i), (ii), and (iii).
Suppose that \(u_{1}\) and \(u_{2}\) are two solutions to (1.1) with stream functions \(\psi_{1}\) and \(\psi_{2}\), respectively. Let \(v\in C^{\infty}_{0}(\Omega)\), \(\operatorname {div}v=0\), from the condition (ii) we get
$$\begin{aligned} &\frac{d}{dt}\langle u_{1}-u_{2},v\rangle -\nu\langle u_{1}-u_{2}, \Delta v\rangle -\alpha^{2} \frac{d}{dt}\langle u_{1}-u_{2}, \Delta v\rangle \\ &\quad = \int_{\Omega}\sum^{2}_{i,j=1} \bigl(u_{1}^{i}u_{1}^{j}-u_{2}^{i}u_{2}^{j} \bigr)\frac{\partial v^{i}}{x_{j}}\,dx+\bigl\langle f_{\rho}(u_{1})-f_{\rho}(u_{2}),v \bigr\rangle \\ &\qquad {} +\biggl\langle \int _{-h}^{0} \bigl[G \bigl(s,u_{1}(t+s) \bigr)-G_{2} \bigl(s,u(t+s) \bigr) \bigr]\,ds,v\biggr\rangle . \end{aligned}$$
(3.36)
We claim that (3.36) holds for any \(v\in V\). In fact, from the condition (ii), we have
$$\begin{aligned} u_{1}-u_{2}=(u_{1}-\psi_{1})-(u_{2}- \psi_{2})+(\psi_{1}-\psi _{2})\in L^{2} \bigl([0,T];V \bigr), \end{aligned}$$
(3.37)
thus we can write \(\langle u_{1}-u_{2},\Delta v\rangle =-(u_{1}-u_{2},v)\) (\(l=1,2\)),
$$\begin{aligned}& \begin{aligned}[b] \biggl( \int_{\Omega }\vert u_{l}\vert ^{4}\,dx \biggr)^{1/4}\leq{}& \biggl( \int_{\Omega} \vert u_{l}-\psi_{l}\vert ^{4}\,dx \biggr)^{1/4}+ \biggl( \int_{\Omega} \vert \psi _{l}\vert ^{4} \,dx \biggr)^{1/4} \\ \leq{}&C \biggl( \int_{\Omega} \bigl\vert \nabla(u_{l}-\psi _{l}) \bigr\vert ^{2}\,dx \biggr)^{1/4}\\ &{}\times \biggl( \int_{\Omega} \vert u_{l}-\psi_{l}\vert ^{2}\,dx \biggr)^{1/4} \biggl( \int _{\Omega} \vert \psi_{l}\vert ^{4} \,dx \biggr)^{1/4}, \end{aligned} \end{aligned}$$
(3.38)
$$\begin{aligned}& \Biggl\vert \int_{\Omega}\sum^{2}_{i,j=1}u_{l}^{i}u_{l}^{j} \frac {\partial v^{i}}{x_{j}}\,dx \Biggr\vert \leq C \biggl( \int_{\Omega }\vert u_{l}\vert ^{4}\,dx \biggr)^{1/2} \biggl( \int _{\Omega} \vert \nabla v\vert ^{2}\,dx \biggr)^{1/2}, \end{aligned}$$
(3.39)
and \(u_{l}\in L^{4}(\Omega\times(\tau,T))\), thus
$$ \frac{d}{dt}(u_{1}-u_{2})\in L^{2} \bigl([ \tau,T];V' \bigr),\quad \mbox{for }v\in V, $$
(3.40)
which implies (3.36) holds for any \(v\in V\).
Let \(v=u_{1}-u_{2}\), we get
$$\begin{aligned} &\frac{1}{2}\frac {d}{dt} \bigl(\vert v\vert ^{2}+ \alpha^{2}\Vert v\Vert ^{2} \bigr)+\nu \Vert v\Vert ^{2} \\ &\quad \leq \Biggl\vert \int_{\Omega}\sum^{2}_{i,j=1} \bigl(u_{1}^{i}u_{1}^{j}-u_{2}^{i}u_{2}^{j} \bigr)\frac{\partial v^{i}}{x_{j}}\,dx \Biggr\vert + \bigl\vert \bigl\langle f_{\rho}(u_{1})-f_{\rho}(u_{2}),v\bigr\rangle \bigr\vert \\ &\qquad {}+ \bigl\vert \bigl\langle g(t,u_{1t})-g(t,u_{2t}),v \bigr\rangle \bigr\vert \\ &\quad \leq \Biggl\vert \int_{\Omega}\sum^{2}_{i,j=1} \biggl(u_{1}^{i}v^{j}\frac {\partial v^{i}}{x_{j}}+u_{2}^{j} \frac{1}{2}\frac{\partial u_{2}^{j}}{x_{j}} \biggr)\,dx \Biggr\vert + \bigl\vert \bigl\langle f_{\rho}(u_{1})-f_{\rho}(u_{2}),v \bigr\rangle \bigr\vert \\ &\qquad {}+ \bigl\vert \bigl\langle g(t,u_{1t})-g(t,u_{2t}),v \bigr\rangle \bigr\vert \\ &\quad \leq \Biggl\vert \int_{\Omega}\sum^{2}_{i,j=1}u_{1}^{i}v^{j} \frac {\partial v^{i}}{x_{j}}\,dx \Biggr\vert + \bigl\vert \bigl\langle f_{\rho}(u_{1})-f_{\rho}(u_{2}),v\bigr\rangle \bigr\vert \\ &\qquad {}+ \bigl\vert \bigl\langle g(t,u_{1t})-g(t,u_{2t}),v \bigr\rangle \bigr\vert \\ &\quad \leq C \biggl( \int_{\Omega} \vert u_{1}\vert ^{4}\,dx \biggr)^{1/4} \biggl( \int _{\Omega }\vert v\vert ^{4/3}\vert \nabla v \vert ^{4/3}\,dx \biggr)^{3/4}+L(\beta) \bigl\vert v \bigl(t- \rho(t) \bigr) \bigr\vert \vert v\vert +L_{g}\Vert v \Vert _{C_{H}}\vert v\vert \\ &\quad \leq C \biggl( \int_{\Omega} \vert u_{1}\vert ^{4}\,dx \biggr)^{1/4} \biggl( \biggl( \int _{\Omega }\vert v\vert ^{4}\,dx \biggr)^{\frac{1}{3}} \biggl( \int_{\Omega} \vert \nabla v\vert ^{4/3}\,dx \biggr)^{\frac {2}{3}} \biggr)^{3/4}+C\Vert v\Vert _{C_{H}} \vert v\vert \\ &\quad \leq C\Vert u_{1}\Vert _{L^{4}}\Vert v\Vert _{L^{4}}\vert \nabla v\vert +C\Vert v\Vert _{C_{H}}\vert v\vert \\ &\quad \leq C\Vert u_{1}\Vert _{L^{4}}\Vert \nabla v\Vert ^{\frac {1}{2}}\vert v\vert ^{\frac {1}{2}}\vert \nabla v\vert +C\Vert v\Vert _{C_{H}}\vert v\vert \\ &\quad \leq\nu \Vert v\Vert ^{2}+C_{\nu} \Vert u_{1}\Vert ^{4}_{L^{4}}\vert v\vert ^{2}+C\Vert v\Vert _{C_{H}}\vert v\vert , \end{aligned}$$
(3.41)
which means
$$\begin{aligned} &\frac{d}{dt} \bigl(\Vert v\Vert ^{2} \bigr)\leq C_{\nu} \Vert u_{1}\Vert ^{4}_{L^{4}} \vert v\vert ^{2}+C\Vert v\Vert _{C_{H}}\vert v \vert . \end{aligned}$$
Since \(u_{1}\in L^{4}(\Omega\times(\tau,T))\) and \(v(\cdot,\tau )=0\), we derive \(v=0\) which means the uniqueness of solution holds. □
3.4 Continuous dependence of initial data
Consider the two solutions \(u(\cdot)\) and \(v(\cdot)\) to problem (1.1) with corresponding initial data \((u_{\tau},\phi_{1})\) and \((v_{\tau},\phi_{2})\), respectively. Let \(w=u-v\), then w satisfies the problem
$$\begin{aligned} \textstyle\begin{cases}\frac{dw}{dt}-\nu\Delta w-\alpha^{2}\Delta w_{t}+(u\cdot\nabla)u-(v\cdot\nabla )v\\ \quad =g(u_{t})-g(v_{t})+f_{\rho}(u)-f_{\rho}(v),\\ \operatorname {div}w=0, & (x,t)\in\Omega_{\tau},\\ w(t,x)|_{\partial\Omega}=0, & (x,t)\in\partial\Omega_{\tau},\\ w(\tau,x)=u_{\tau}(x)-v_{\tau}(x),& x\in\Omega,\\ u(t,x)=\phi_{1}(t-\tau,x)-\phi_{2}(t-\tau,x),& (x,t)\in\Omega_{\tau h}. \end{cases}\displaystyle \end{aligned}$$
(3.42)
Since \(B(u,u)-B(v,v)=B(w,u)+B(v,w)\), we obtain the abstract form
$$\begin{aligned} \frac{dw}{dt}+\nu Aw+\alpha ^{2}Aw_{t}+B(w,u)+B(v,w)=g(u_{t})-g(v_{t})++f_{\rho}(u)-f_{\rho}(v). \end{aligned}$$
(3.43)
Multiplying (3.43) by w, we derive
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl(\vert w\vert ^{2}+ \alpha^{2}\Vert w\Vert ^{2} \bigr)+\nu \Vert w\Vert ^{2} \\ &\quad \leq \bigl\vert b(w,u,w) \bigr\vert + \bigl\vert f_{\rho}(u)-f_{\rho }(v) \bigr\vert \vert w\vert + \bigl\vert g(u_{t})-g(v_{t}) \bigr\vert \vert w\vert \\ &\quad \leq C\vert w\vert \Vert w\Vert \Vert u\Vert +L(\beta)\vert w \vert \bigl\vert w \bigl(t-\rho (t) \bigr) \bigr\vert + \bigl\vert g(u_{t})-g(v_{t}) \bigr\vert \vert w\vert \\ &\quad \leq\frac{\nu}{4}\Vert w\Vert ^{2}+\frac{C^{2}}{\nu} \vert w\vert ^{2}\Vert u\Vert ^{2}+\frac{\nu }{8} \Vert w\Vert ^{2}+\frac{2L^{2}(\beta)}{\nu\lambda_{1}} \bigl\vert w \bigl(t- \rho(t) \bigr) \bigr\vert ^{2}+\frac{\nu }{8}\Vert w\Vert ^{2} \\ &\qquad {}+\frac{2}{\nu\lambda_{1}} \bigl\vert g(t,u_{t})-g(t,v_{t}) \bigr\vert ^{2} \\ &\quad = \frac{\nu}{2}\Vert w\Vert ^{2}+\frac{C^{2}}{\nu} \vert w\vert ^{2}\Vert u\Vert ^{2}+\frac {2L^{2}(\beta )}{\nu\lambda_{1}} \bigl\vert w \bigl(t-\rho(t) \bigr) \bigr\vert ^{2}\\ &\qquad {}+ \frac{2}{\nu\lambda _{1}} \bigl\vert g(t,u_{t})-g(t,v_{t}) \bigr\vert ^{2}, \end{aligned}$$
which implies
$$\begin{aligned} &\frac{d}{dt} \bigl(\vert w\vert ^{2}+\alpha^{2} \Vert w\Vert ^{2} \bigr)+\nu \Vert w\Vert ^{2} \\ &\quad \leq\frac{2C^{2}}{\nu} \vert w\vert ^{2}\Vert u\Vert ^{2}+\frac{4L^{2}(\beta)}{\nu\lambda _{1}} \bigl\vert w \bigl(t-\rho(t) \bigr) \bigr\vert ^{2}+\frac{4}{\nu\lambda _{1}} \bigl\vert g(t,u_{t})-g(t,v_{t}) \bigr\vert ^{2}. \end{aligned}$$
(3.44)
Noting
$$\begin{aligned}& \int_{\tau}^{t} \bigl\vert w \bigl(s-\rho(s) \bigr) \bigr\vert ^{2}\,ds\leq\frac{1}{1-M} \int_{\tau -h}^{t} \bigl\vert w(s) \bigr\vert ^{2}\,ds, \\& \int_{\tau}^{t} \bigl\vert g(s,u_{s})-g(s,v_{s}) \bigr\vert ^{2}\,ds\leq C_{g}^{2} \int_{\tau -h}^{t} \bigl\vert u(s)-v(s) \bigr\vert ^{2}\,ds=C_{g}^{2} \int_{\tau-h}^{t} \bigl\vert w(s) \bigr\vert ^{2}\,ds, \end{aligned}$$
and integrating (3.44) over \([\tau, t]\), we obtain
$$\begin{aligned} &\vert w\vert ^{2}+\alpha^{2}\Vert w\Vert ^{2}+\nu \int^{t}_{\tau} \Vert w\Vert ^{2}\,ds \\ &\quad \leq \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau) \bigr\Vert ^{2} +\frac{2C^{2}}{\nu} \int _{\tau }^{t}\vert w\vert ^{2}\Vert u\Vert ^{2}\,ds+\frac{4L^{2}(\beta)}{\nu\lambda_{1}} \int_{\tau }^{t} \bigl\vert w \bigl(s-\rho (s) \bigr) \bigr\vert ^{2}\,ds \\ &\qquad {} +\frac{4}{\nu\lambda_{1}} \int_{\tau }^{t} \bigl\vert g(t,u_{s})-g(t,v_{s}) \bigr\vert ^{2}\,ds \\ &\quad \leq \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau) \bigr\Vert ^{2} +\frac{2C^{2}}{\nu} \int _{\tau }^{t}\vert w\vert ^{2}\Vert u\Vert ^{2}\,ds+\frac{4L^{2}(\beta)}{\nu\lambda_{1}(1-M)} \int_{\tau -h}^{t} \bigl\vert w(s) \bigr\vert ^{2}\,ds \\ &\qquad {} +\frac{4C_{g}^{2}}{\nu\lambda_{1}} \int_{\tau -h}^{t} \bigl\vert w(s) \bigr\vert ^{2}\,ds \\ &\quad \leq \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau) \bigr\Vert ^{2}+ \biggl(\frac{4L^{2}(\beta)}{\nu \lambda _{1}(1-M)}+ \frac{4C_{g}^{2}}{\nu\lambda_{1}} \biggr) \int_{\tau-h}^{\tau} \bigl\vert w(s) \bigr\vert ^{2}\,ds \\ &\qquad {} + \int_{\tau}^{t} \biggl(\frac{2C^{2}}{\nu} \Vert u \Vert ^{2}+\frac{4L^{2}(\beta )}{\nu \lambda_{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda_{1}} \biggr) \bigl\vert w(s) \bigr\vert ^{2}\,ds \\ &\quad \leq \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau) \bigr\Vert ^{2}+ \biggl(\frac{4L^{2}(\beta)}{\nu \lambda _{1}(1-M)}+ \frac{4C_{g}^{2}}{\nu\lambda_{1}} \biggr)\Vert \phi_{1}-\phi_{2}\Vert ^{2}_{L_{H}^{2}} \\ &\qquad {} + \int_{\tau}^{t} \biggl(\frac{2C^{2}}{\nu} \Vert u \Vert ^{2}+\frac{4L^{2}(\beta )}{\nu \lambda_{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda_{1}} \biggr) \bigl\vert w(s) \bigr\vert ^{2}\,ds. \end{aligned}$$
(3.45)
Since \(u(t)\in L^{\infty}(\tau,T;V)\cap L^{2}(\tau,T;V)\), neglecting the integrating term on left side of (3.45), putting \(s\in (t-h,t)\) instead of t and using the Gronwall inequality to (3.45), we see
$$\begin{aligned} \bigl\vert w(s) \bigr\vert ^{2} \leq& \biggl( \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau) \bigr\Vert ^{2}+ \biggl(\frac {4L^{2}(\beta )}{\nu\lambda_{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda_{1}} \biggr) \Vert \phi_{1}-\phi _{2}\Vert ^{2}_{L_{H}^{2}} \biggr) \\ &{}\times e^{\int_{\tau}^{t}(\frac{2C^{2}}{\nu} \Vert u\Vert ^{2}+\frac{4L^{2}(\beta )}{\nu \lambda_{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda_{1}})\,ds}. \end{aligned}$$
Similarly, using the Poincaré inequality, we get
$$\begin{aligned} \bigl\Vert w(s) \bigr\Vert ^{2} \leq& \frac{1}{\alpha^{2}} \biggl( \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau ) \bigr\Vert ^{2}+ \biggl(\frac{4L^{2}(\beta)}{\nu\lambda_{1}(1-M)}+ \frac{4C_{g}^{2}}{\nu \lambda_{1}} \biggr)\Vert \phi_{1}-\phi_{2}\Vert ^{2}_{L_{H}^{2}} \biggr) \\ &{}\times e^{\frac{1}{\lambda_{1}}\int_{\tau}^{t}(\frac{2C^{2}}{\nu} \Vert u\Vert ^{2}+\frac{4L^{2}(\beta)}{\nu\lambda_{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda _{1}})\,ds}. \end{aligned}$$
Moreover,
$$\begin{aligned} &\int^{t}_{\tau} \bigl\Vert w(s) \bigr\Vert ^{2}\,ds \\ &\quad \leq \frac{1}{\nu\alpha ^{2}} \biggl( \bigl\vert w(\tau) \bigr\vert ^{2}+\alpha^{2} \bigl\Vert w(\tau) \bigr\Vert ^{2}+ \biggl(\frac{4L^{2}(\beta)}{\nu \lambda _{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda_{1}} \biggr)\Vert \phi_{1}-\phi_{2}\Vert ^{2}_{L_{H}^{2}} \biggr) \\ &\qquad {}\times \biggl(1+e^{\int_{\tau}^{t}(\frac{2C^{2}}{\nu} \Vert u\Vert ^{2}+\frac {4L^{2}(\beta )}{\nu\lambda_{1}(1-M)}+\frac{4C_{g}^{2}}{\nu\lambda_{1}})\,ds} \int_{\tau }^{t} \biggl(\frac {2C^{2}}{\nu} \Vert u \Vert ^{2}+\frac{4L^{2}(\beta)}{\nu\lambda_{1}(1-M)}+\frac {4C_{g}^{2}}{\nu\lambda_{1}} \biggr)\,ds \biggr). \end{aligned}$$
This implies the continuous dependence on the initial data for the solution which generates a continuous process \(\{\tilde{U}(\cdot ,\cdot )\}\).