Leray’s Problem for the Nonstationary Micropolar Fluid Flow

Abstract

Motivated by the pipe network problems, in this paper, we consider the Leray’s problem for the nonstationary flow of a micropolar fluid. We prove that in an unbounded domain with cylindrical outlets to infinity, there exists a unique solution to the nonlinear micropolar equations which exponentially tends to the generalized nonstationary micropolar Poiseuille solution in each cylindrical outlet.

Appendices

Proof of Theorem 5.2

Step 1. Let us first take in the integral identity (5.2) test functions $\varvec{\varphi }(x,t)=E_{\varvec{\beta }}^{(k)}(x){\mathbf {u}}(x,t)+{\mathbf {Z}}^{(k)}(x,t)$ and $\varvec{\psi }(x,t)=E_{\varvec{\beta }}^{(k)}(x){\mathbf {w}}(x,t)$, where $E_{\varvec{\beta }}^{(k)}(x)$ is the exponentially decaying “step” weight function given with (see Sect. 2.2):

$$\begin{aligned} E_{\varvec{\beta }}^{(k)}(x)={\left\{ \begin{array}{ll} 1, &{}\quad x \in \Omega _{(0)},\\ E_{\beta _{j}}(x_{1}^{(j)}), &{}\quad x \in \Omega _{jk},\ j=1,\dots ,m, \\ E_{\beta _{j}}(k), &{}\quad x \in \Omega _{j} {\setminus } \Omega _{jk},\ j=1,\dots ,m, \end{array}\right. } \end{aligned}$$

while $E_{\beta _{j}}$ is a weight function satisfying (2.1) and ${\mathbf {Z}}^{(k)}(x,t)$ is the vector field constructed in Lemma 2.4. It can be easily verified that $\text{ div } \varvec{\varphi }(x,t)=\text{ div } (E_{\varvec{\beta }}^{(k)}(x){\mathbf {u}}(x,t))+\text{ div } {\mathbf {Z}}^{(k)}(x,t)=0$ and $\varvec{\varphi }\vert _{\partial \Omega }=0$. We thus obtain the relations:

$$\begin{aligned} \begin{aligned}&\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \frac{\partial {\mathbf {u}}}{\partial \tau } \cdot {\mathbf {u}}\mathrm{d}x\mathrm{d}\tau +\int _{0}^{t} \int _{\Omega } \frac{\partial {\mathbf {u}}}{\partial \tau } \cdot {\mathbf {Z}}^{(k)}\mathrm{d}x\mathrm{d}\tau \\&\qquad +\mu \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot \nabla (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}+{\mathbf {Z}}^{(k)}))\mathrm{d}x\mathrm{d}\tau \\&\quad =a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}+{\mathbf {Z}}^{(k)})\mathrm{d}x\mathrm{d}\tau +\int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}+{\mathbf {Z}}^{(k)})\mathrm{d}x\mathrm{d}\tau ,\\&\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \frac{\partial {\mathbf {w}}}{\partial \tau } \cdot {\mathbf {w}}\mathrm{d}x\mathrm{d}\tau +\alpha \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {w}} \cdot \nabla (E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d}\tau \\&\qquad +\beta \int _{0}^{t} \int _{\Omega } \text{ div } {\mathbf {w}}\ \text{ div }(E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d}\tau \\&\qquad +2a \int _{0}^{t} \int _{\Omega } {\mathbf {w}} \cdot (E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d} \tau =a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {u}} \cdot (E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d}\tau \\&\qquad +\int _{0}^{t} \int _{\Omega } {\mathbf {g}} \cdot (E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d}\tau . \end{aligned}\nonumber \\ \end{aligned}$$

(A.1)

There hold the following relations:

$$\begin{aligned} \begin{aligned} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}(x) \frac{\partial {\mathbf {u}}(x,\tau )}{\partial \tau }\cdot {\mathbf {u}}(x,\tau )\mathrm{d}x\mathrm{d}\tau&=\frac{1}{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)}(x) \vert {\mathbf {u}}(x,t) \vert ^{2}\mathrm{d}x\\&\quad -\,\frac{1}{2}\int _{\Omega } E_{\varvec{\beta }}^{(k)}(x) \vert {\mathbf {a}}(x) \vert ^{2}\mathrm{d}x,\\ \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot \nabla (E_{\varvec{\beta }}^{(k)} {\mathbf {u}})\mathrm{d}x\mathrm{d}\tau&=\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}\mathrm{d}x\mathrm{d}\tau \\&\quad +\,\int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot (\nabla E_{\varvec{\beta }}^{(k)} {\mathbf {u}})\mathrm{d}x\mathrm{d}\tau ,\\ \int _{0}^{t} \int _{\Omega } \text{ div } {\mathbf {w}} \text{ div }(E_{\varvec{\beta }}^{(k)} {\mathbf {w}})\mathrm{d}x\mathrm{d}\tau&=\int _{0}^{t} \int _{\Omega } \text{ div } {\mathbf {w}} \nabla E_{\varvec{\beta }}^{(k)} \cdot {\mathbf {w}}\mathrm{d}x\mathrm{d}\tau \\&\quad +\,\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {w}})^{2}\mathrm{d}x\mathrm{d}\tau . \end{aligned}\nonumber \\ \end{aligned}$$

(A.2)

Using relations (A.2) in (A.1), we obtain:

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}} \vert ^{2} \mathrm{d}x+\mu \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \mathrm{d}x\mathrm{d}\tau \\&\quad =\frac{1}{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {a}} \vert ^{2} \mathrm{d}x-\int _{0}^{t} \int _{\Omega } \frac{\partial {\mathbf {u}}}{\partial \tau } \cdot {\mathbf {Z}}^{(k)}\mathrm{d}x\mathrm{d}\tau \\&\qquad -\mu \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot (\nabla E_{\varvec{\beta }}^{(k)} {\mathbf {u}}\\&\qquad +\nabla {\mathbf {Z}}^{(k)})\mathrm{d}x\mathrm{d}\tau +a\int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}\\&\qquad +{\mathbf {Z}}^{(k)})\mathrm{d}x\mathrm{d}\tau +\int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}+{\mathbf {Z}}^{(k)})\mathrm{d}x \mathrm{d}\tau \\&\qquad \frac{1}{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2} \mathrm{d}x+\alpha \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}\mathrm{d}x\mathrm{d}\tau +\beta \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {w}})^{2}\mathrm{d}x\mathrm{d}\tau \\&\qquad +2a\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2}\mathrm{d}x\mathrm{d}\tau \\&\quad =\frac{1}{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {b}} \vert ^{2}\mathrm{d}x-\alpha \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {w}} \cdot (\nabla E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d}\tau \\&\qquad -\beta \int _{0}^{t} \int _{\Omega } \text{ div } {\mathbf {w}} \nabla E_{\varvec{\beta }}^{(k)} \cdot {\mathbf {w}}\mathrm{d}x\mathrm{d}\tau \\&\qquad +a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {u}} \cdot (E_{\varvec{\beta }}^{(k)}{\mathbf {w}})\mathrm{d}x\mathrm{d}\tau +\int _{0}^{t} \int _{\Omega } {\mathbf {g}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {w}})\mathrm{d}x\mathrm{d}\tau . \end{aligned}\nonumber \\ \end{aligned}$$

(A.3)

We estimate the terms on the right-hand side of (A.3) using estimate (2.1)$_{3}$ for the weight function, Lemma 2.1 and 2.4 in the following way:

$$\begin{aligned} \int _{0}^{t} \int _{\Omega } \frac{\partial {\mathbf {u}}}{\partial \tau } \cdot {\mathbf {Z}}^{(k)}\mathrm{d}x\mathrm{d}\tau\le & {} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2} \right) ^{1/2} \left( \int _{0}^{t} \int _{\Omega } E_{-\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {Z}}^{(k)} \vert ^{2} \right) ^{1/2} \nonumber \\\le & {} c \gamma _{*} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2}+\int _{0}^{t}\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \right) ,\nonumber \\ \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot \nabla E_{\varvec{\beta }}^{(k)} {\mathbf {u}}\mathrm{d}x\mathrm{d}\tau\le & {} c \gamma _{*} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \right) ^{1/2} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \right) ^{1/2}\nonumber \\\le & {} c\gamma _{*} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \right) ,\nonumber \\ \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot \nabla {\mathbf {Z}}^{(k)} \mathrm{d}x\mathrm{d}\tau\le & {} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \right) ^{1/2} \left( \int _{0}^{t} \int _{\Omega } E_{-\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {Z}}^{(k)} \vert ^{2} \right) ^{1/2} \nonumber \\\le & {} c \gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2},\nonumber \\ \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {u}}\mathrm{d}x\mathrm{d}\tau= & {} \int _{0}^{t} \int _{\Omega } [E_{\varvec{\beta }}^{(k)}]^{1/2} \text{ rot } {\mathbf {w}} \cdot [E_{\varvec{\beta }}^{(k)}]^{1/2} {\mathbf {u}} \nonumber \\\le & {} \frac{\varepsilon _{1}}{2}\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}+\frac{1}{2\varepsilon _{1}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}} \vert ^{2}, \nonumber \\ \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot {\mathbf {Z}}^{(k)}\mathrm{d}x\mathrm{d}\tau\le & {} c \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2} \right) ^{1/2} \left( \int _{0}^{t} \int _{\Omega } E_{-\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {Z}}^{(k)} \vert ^{2} \right) ^{1/2}\nonumber \\\le & {} c \gamma _{*} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}+\int _{0}^{t}\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2} \right) , \nonumber \\ \int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {u}}\mathrm{d}x\mathrm{d}\tau= & {} \int _{0}^{t} \int _{\Omega } [E_{\varvec{\beta }}^{(k)}]^{1/2} {\mathbf {f}} \cdot [E_{\varvec{\beta }}^{(k)}]^{1/2} {\mathbf {u}}\nonumber \\\le & {} \frac{c}{\varepsilon _{3}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {f}} \vert ^{2}+c\varepsilon _{3}\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}, \nonumber \\ \int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot {\mathbf {Z}}^{(k)}\mathrm{d}x\mathrm{d}\tau= & {} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}{\mathbf {f}} \cdot E_{-\varvec{\beta }}^{(k)}{\mathbf {Z}}^{(k)} \nonumber \\\le & {} \frac{c}{\varepsilon _{3}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {f}} \vert ^{2}+c \varepsilon _{3} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}, \end{aligned}$$

(A.4)

and

$$\begin{aligned} \begin{aligned} \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {w}}\cdot \nabla E_{\varvec{\beta }}^{(k)}{\mathbf {w}}\mathrm{d}x\mathrm{d}\tau&\le c \gamma _{*} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2} \right) ^{1/2} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2} \right) ^{1/2}\\&\le c \gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2},\\ \int _{0}^{t} \int _{\Omega } \text{ div } {\mathbf {w}} \nabla E_{\varvec{\beta }}^{(k)} \cdot {\mathbf {w}}\mathrm{d}x\mathrm{d}\tau&\le c \gamma _{*} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {w}})^{2} \right) ^{1/2} \left( \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2} \right) ^{1/2}\\&\le c \gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2},\\ \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {u}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {w}})\mathrm{d}x\mathrm{d}\tau&\le \int _{0}^{t} \int _{\Omega } \vert E_{\varvec{\beta }}^{(k)}]^{1/2} {\mathbf {w}} \cdot \vert E_{\varvec{\beta }}^{(k)}]^{1/2} \text{ rot } {\mathbf {u}} \\&\le \frac{1}{2\varepsilon _{2}}\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2}+\frac{\varepsilon _{2}}{2} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}, \\ \int _{0}^{t} \int _{\Omega } {\mathbf {g}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {w}}\mathrm{d}x\mathrm{d}\tau&\le \frac{c}{\varepsilon _{3}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {g}} \vert ^{2}+c\varepsilon _{3} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.5)

Now, applying estimates (A.4)–(A.5) into (A.3) and taking $\varepsilon _{1}$ and $\varepsilon _{2}$ small enough, we obtain:

$$\begin{aligned} \begin{aligned}&\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}} \vert ^{2}\mathrm{d}x+\ \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2}\mathrm{d}x+ \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}\mathrm{d}x\mathrm{d}\tau \\&\qquad + \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}\mathrm{d}x\mathrm{d}\tau +\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {w}})^{2}\mathrm{d}x\mathrm{d}\tau +\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2}\mathrm{d}x\mathrm{d}\tau \\&\quad \le c \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {a}} \vert ^{2}\mathrm{d}x+c \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {b}} \vert ^{2}\mathrm{d}x +c(\varepsilon _{3}+\gamma _{*}) \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}+c(\varepsilon _{3}+\gamma _{*})\\&\qquad \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}+c\gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2}+c\gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}}_{\tau } \vert ^{2} \\&\qquad +\frac{c}{\varepsilon _{3}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {f}} \vert ^{2}+\frac{c}{\varepsilon _{3}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {g}} \vert ^{2}+c\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert {\mathbf {u}} \vert ^{2}+c\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert {\mathbf {w}} \vert ^{2}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.6)

Step 2 We now take in the integral identity (5.2) the test functions $\varvec{\varphi }(x,t)=E_{\varvec{\beta }}^{(k)}(x){\mathbf {u}}_{t}(x,t)+{\mathbf {Z}}_{t}^{(k)}(x,t)$, $\varvec{\psi }(x,t)=E_{\varvec{\beta }}^{(k)}(x){\mathbf {w}}_{t}(x,t)$ to obtain:

$$\begin{aligned} \begin{aligned}&\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2}+\int _{0}^{t} \int _{\Omega } {\mathbf {u}}_{\tau } \cdot {\mathbf {Z}}_{\tau }^{(k)}+\mu \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}}\cdot \nabla (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}_{\tau }+{\mathbf {Z}}_{\tau }^{(k)})\\&\quad =a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}_{\tau }+{\mathbf {Z}}_{\tau }^{(k)})+\int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {u}}_{\tau }+{\mathbf {Z}}_{\tau }^{(k)})\\&\qquad \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}}_{\tau } \vert ^{2}+\alpha \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {w}} \cdot \nabla (E_{\varvec{\beta }}^{(k)}{\mathbf {w}}_{\tau })+\beta \int _{0}^{t} \int _{\Omega } \text{ div } {\mathbf {w}} \text{ div }( E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau })\\&\qquad +2a \int _{0}^{t} \int _{\Omega } {\mathbf {w}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau })=a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {u}} \cdot (E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau })+\int _{0}^{t} \int _{\Omega } {\mathbf {g}} \cdot (E_{\varvec{\beta }}^{(k)}{\mathbf {w}}_{\tau }). \end{aligned}\nonumber \\ \end{aligned}$$

(A.7)

Rewriting the system of Eq. (A.7) yields:

$$\begin{aligned} \begin{aligned}&\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2}+\frac{\mu }{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}\\&\quad =\frac{\mu }{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {a}} \vert ^{2}-\int _{0}^{t} \int _{\Omega } {\mathbf {u}}_{\tau } \cdot {\mathbf {Z}}_{\tau }^{(k)}-\mu \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}} \cdot (\nabla E_{\varvec{\beta }}^{(k)} {\mathbf {u}}_{\tau }) \\&\qquad -\mu \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {u}}\cdot \nabla {\mathbf {Z}}_{\tau }^{(k)}+a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {u}}_{\tau }\\&\qquad +a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {w}} \cdot {\mathbf {Z}}_{\tau }^{(k)}+\int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {u}}_{\tau }+\int _{0}^{t} \int _{\Omega } {\mathbf {f}} \cdot {\mathbf {Z}}_{\tau }^{(k)} \\&\qquad \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}}_{\tau } \vert ^{2}+\frac{\alpha }{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}+\frac{\beta }{2}\int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {w}})^{2}\\&\qquad +a\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2}=\frac{\alpha }{2}\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {b}} \vert ^{2}+\frac{\beta }{2} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {b}})^{2} \\&\qquad +a \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {b}} \vert ^{2}-\alpha \int _{0}^{t} \int _{\Omega } \nabla {\mathbf {w}}\cdot (\nabla E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau })-\beta \int _{\Omega } \text{ div } {\mathbf {w}} \nabla E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau } \\&\qquad +a \int _{0}^{t} \int _{\Omega } \text{ rot } {\mathbf {u}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau }+\int _{0}^{t} \int _{\Omega } {\mathbf {g}} \cdot E_{\varvec{\beta }}^{(k)} {\mathbf {w}}_{\tau }. \end{aligned}\nonumber \\ \end{aligned}$$

(A.8)

Estimating the right-hand of (A.8) in a similar manner as in Step 1 yields:

$$\begin{aligned} \begin{aligned}&\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2}+\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}}_{\tau } \vert ^{2}+\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}+\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2} \\&\quad +\int _{\Omega } E_{\varvec{\beta }}^{(k)} (\text{ div } {\mathbf {w}})^{2}+\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}} \vert ^{2} \le c\int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {a}} \vert ^{2}+c \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {b}} \vert ^{2} \\&\quad +c(\gamma _{*}+\varepsilon _{3})\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {u}}_{\tau } \vert ^{2}+c\gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {u}} \vert ^{2}+c(\gamma _{*}+\varepsilon _{3})\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert {\mathbf {w}}_{\tau } \vert ^{2} \\&\quad +c\gamma _{*} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} \vert \nabla {\mathbf {w}} \vert ^{2}+c\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert \nabla {\mathbf {u}} \vert ^{2} +c\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert \nabla {\mathbf {w}} \vert ^{2} \\&\quad +\frac{c}{\varepsilon _{3}}\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert {\mathbf {f}} \vert ^{2}+\frac{c}{\varepsilon _{3}}\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert {\mathbf {g}} \vert ^{2}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.9)

Step 3 Summing the obtained inequalities (A.6) and (A.9) yields:

$$\begin{aligned} \begin{aligned}&\int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {u}} \vert ^{2}+\vert \nabla {\mathbf {u}} \vert ^{2})+\int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {w}} \vert ^{2}+\vert \nabla {\mathbf {w}} \vert ^{2})\\&\qquad +\int _{0}^{t} \int _{\Omega } E_{\beta }^{(k)} (\vert {\mathbf {u}}_{\tau } \vert ^{2}+\vert \nabla {\mathbf {u}} \vert ^{2})+\int _{0}^{t} \int _{\Omega } E_{\beta }^{(k)} (\vert {\mathbf {w}}_{\tau } \vert ^{2}+\vert \nabla {\mathbf {w}} \vert ^{2})\\&\qquad +\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}((\text{ div } {\mathbf {w}})^{2}+\vert {\mathbf {w}} \vert ^{2})+\int _{\Omega }E_{\varvec{\beta }}^{(k)}((\text{ div } {\mathbf {w}})^{2}+\vert {\mathbf {w}} \vert ^{2}) \\&\quad \le c(\gamma _{*}+\varepsilon _{3}) \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\vert {\mathbf {u}}_{\tau } \vert ^{2}+\vert \nabla {\mathbf {u}} \vert ^{2})+c(\gamma _{*}+\varepsilon _{3}) \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)} (\vert {\mathbf {w}}_{\tau } \vert ^{2}+\vert \nabla {\mathbf {w}} \vert ^{2}) \\&\qquad +c\int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {a}} \vert ^{2}+\vert \nabla {\mathbf {a}} \vert ^{2})+c\int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {b}} \vert ^{2}+\vert \nabla {\mathbf {b}} \vert ^{2})+\frac{c}{\varepsilon _{3}} \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {f}} \vert ^{2}+\vert {\mathbf {g}} \vert ^{2})\\&\qquad +c \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert {\mathbf {u}} \vert ^{2}+c \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert {\mathbf {w}} \vert ^{2}+c \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert \nabla {\mathbf {u}} \vert ^{2}+c \int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}\vert \nabla {\mathbf {w}} \vert ^{2}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.10)

Taking $\varepsilon _{3}$ and $\gamma _{*}$ sufficiently small in (A.10) and using Gronwall’s inequality, we obtain:

$$\begin{aligned} \begin{aligned}&\int _{\Omega } E_{\beta }^{(k)}(\vert {\mathbf {u}} \vert ^{2}+ \vert \nabla {\mathbf {u}} \vert ^{2})+\int _{\Omega } E_{\beta }^{(k)}(\vert {\mathbf {w}} \vert ^{2}+ \vert \nabla {\mathbf {w}} \vert ^{2}) \\&\qquad +\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {u}}_{\tau } \vert ^{2}+ \vert \nabla {\mathbf {u}} \vert ^{2})+\int _{0}^{t} \int _{\Omega } E_{\varvec{\beta }}^{(k)}(\vert {\mathbf {w}}_{\tau } \vert ^{2}+ \vert \nabla {\mathbf {w}} \vert ^{2}) \\&\quad \le c(\vert \vert {\mathbf {a}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert {\mathbf {b}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert {\mathbf {f}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}+\vert \vert {\mathbf {g}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}). \end{aligned}\nonumber \\ \end{aligned}$$

(A.11)

Finally, passing to the limit $k \rightarrow \infty$ in (A.11), we obtain (right-hand side does not depend on k):

$$\begin{aligned} \begin{aligned}&\sup _{t \in [0,T]}\vert \vert {\mathbf {u}}(\cdot ,t) \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert {\mathbf {u}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1,1}(\Omega ^{T})}+\sup _{t \in [0,T]}\vert \vert {\mathbf {w}}(\cdot ,t) \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert {\mathbf {w}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1,1}(\Omega ^{T})} \\&\quad \le c(\vert \vert {\mathbf {a}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert {\mathbf {b}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert {\mathbf {f}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}+\vert \vert {\mathbf {g}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}). \end{aligned}\nonumber \\ \end{aligned}$$

(A.12)

It is important to note that assuming $\frac{2\nu _{r}^{2}(\alpha +4c_{p})}{\alpha }<\frac{\mu }{2}$ and $\nu _{r}^{2}< \frac{\alpha }{8}$, we can derive the above inequality (A.12) without using Gronwall’s inequality and thus obtain that the constant c is independent of T.

Step 4 Higher regularity.

For almost all $t \in [0,T]$, there holds the identity:

We can thus consider as a weak solution of the following stationary problem:

$$\begin{aligned} \begin{aligned}&-\mu \Delta {\mathbf {u}}(x,t)+\nabla p(x,t)=a\text{ rot } {\mathbf {w}}(x,t)+{\mathbf {f}}(x,t)-{\mathbf {u}}_{t}(\cdot ,t), \\&\text{ div } {\mathbf {u}}(x,t)=0, \\&-\alpha \Delta {\mathbf {w}}(x,t)-\beta \nabla \text{ div } {\mathbf {w}}(x,t)=a\text{ rot } {\mathbf {u}}(x,t)+{\mathbf {g}}(x,t)-2a {\mathbf {w}}(x,t)-{\mathbf {w}}_{t}(x,t), \\&{\mathbf {u}}(x,t)\vert _{\partial \Omega }=0,\quad {\mathbf {w}}(x,t)\vert _{\partial \Omega }=0, \\&\int _{\sigma _{j}} {\mathbf {u}}(x,t) \cdot {\mathbf {n}}(x)\mathrm{d}x^{(j)'}=0, \qquad j=1,\dots ,m. \end{aligned} \end{aligned}$$

Since we have $a\text{ rot } {\mathbf {w}}(\cdot ,t)+{\mathbf {f}}(\cdot ,t)-{\mathbf {u}}_{t}(\cdot ,t) \in {\mathcal {L}}_{2,\varvec{\beta }}(\Omega )$, $a\text{ rot } {\mathbf {u}}(\cdot ,t)+{\mathbf {g}}(\cdot ,t)-2a{\mathbf {w}}(\cdot ,t)-{\mathbf {w}}_{t}(\cdot ,t) \in {\mathcal {L}}_{2,\varvec{\beta }}(\Omega )$, $\partial \Omega \in C^{2}$, assuming that $\gamma _{*}$ is sufficiently small and using estimates (A.12), we obtain the following:

$$\begin{aligned}&\vert \vert {\mathbf {u}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{2,1}(\Omega ^{T})}+\vert \vert {\mathbf {w}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{2,1}(\Omega ^{T})}+\vert \vert \nabla p \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})} \le c(\vert \vert {\mathbf {a}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}^{2}\nonumber \\&\quad +\vert \vert {\mathbf {b}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}^{2}+\vert \vert {\mathbf {f}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}+\vert \vert {\mathbf {g}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}). \end{aligned}$$

(A.13)

As in Step 3, assuming $\frac{2\nu _{r}^{2}(\alpha +4c_{p})}{\alpha }<\frac{\mu }{2}$ and $\nu _{r}^{2}< \frac{\alpha }{8}$, we can derive the above inequality (A.13) without using Gronwall’s inequality and thus obtain that the constant c is independent of T.

Proof of Theorem 5.3

We seek the solution $({\mathbf {u}}(x,t),{\mathbf {w}}(x,t),p(x,t))$ of problem:

$$\begin{aligned} \begin{aligned}&\frac{\partial {\mathbf {u}}(x,t)}{\partial t}-\mu \Delta {\mathbf {u}}(x,t)+\nabla p(x,t)=a \text{ rot } {\mathbf {w}}(x,t)+{\mathbf {f}}(x,t), \\&\text{ div }\ {\mathbf {u}}(x,t)=0, \\&\frac{\partial {\mathbf {w}}(x,t)}{\partial t}-\alpha \Delta {\mathbf {w}}(x,t)-\beta \nabla \text{ div } {\mathbf {w}}(x,t)+2a {\mathbf {w}}(x,t)=a \text{ rot } {\mathbf {u}}(x,t)+{\mathbf {g}}(x,t), \\&{\mathbf {u}}, {\mathbf {w}}=0\ \text{ on }\ \partial \Omega , \\&{\mathbf {u}}(x,0)={\mathbf {a}}(x),\quad {\mathbf {w}}(x,0)={\mathbf {b}}(x), \\&\int _{\sigma _{j}} {\mathbf {u}}(x,t) \cdot {\mathbf {n}}(x)\mathrm{d}x^{(j)'}=F_{j}(t),\ j=1,\dots ,m, \end{aligned} \end{aligned}$$

in the form:

$$\begin{aligned} \begin{aligned} {\mathbf {u}}(x,t)&={\mathbf {v}}(x,t)+{\mathbf {V}}(x,t)={\mathbf {v}}(x,t)+{\mathbf {U}}(x,t)+{\mathbf {Z}}(x,t) \\&={\mathbf {v}}(x,t)+\sum _{j=1}^{m} \zeta (x_{1}^{(j)}){\mathbf {U}}^{(j)} (x^{(j)'},t)+{\mathbf {Z}}(x,t)\\&={\mathbf {v}}(x,t)+\sum _{j=1}^{m} \zeta (x_{1}^{(j)})(U_{1}^{(j)}(x^{(j)'},t),0,0)+{\mathbf {Z}}(x,t), \\ {\mathbf {w}}(x,t)&={\mathbf {w}}_{\mathrm{int}}(x,t)+{\mathbf {W}}(x,t)={\mathbf {w}}_{\mathrm{int}}(x,t)+\sum _{j=1}^{m} \zeta (x_{1}^{(j)}){\mathbf {W}}^{(j)}(x^{(j)'},t) \\&={\mathbf {w}}_{\mathrm{int}}(x,t)+\sum _{j=1}^{m} \zeta (x_{1}^{(j)})(0,W_{2}^{(j)}(x^{(j)'},t),W_{3}^{(j)}(x^{(j)'},t)), \\ p(x,t)&={\tilde{p}}(x,t)+P(x,t)={\tilde{p}}(x,t)+\sum _{j=1}^{m} \zeta (x_{1}^{(j)})P^{(j)}(x^{(j)},t)\\&={\tilde{p}}(x,t)+\sum _{j=1}^{m}\zeta (x_{1}^{(j)})(-q^{(j)}(t)x_{1}^{(j)}+p_{0}^{(j)}(t)). \end{aligned} \end{aligned}$$

Now, for $({\mathbf {v}}(x,t),{\mathbf {w}}_{\mathrm{int}}(x,t),{\tilde{p}}(x,t))$, we get the problem:

$$\begin{aligned} \begin{aligned}&\frac{\partial {\mathbf {v}}(x,t)}{\partial t}-\mu \Delta {\mathbf {v}}(x,t)+\nabla {\tilde{p}}(x,t)=a\text{ rot } {\mathbf {w}}_{\mathrm{int}}+\tilde{{\mathbf {f}}}(x,t),\\&\text{ div } {\mathbf {v}}(x,t)=0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\frac{\partial {\mathbf {w}}_{\mathrm{int}}(x,t)}{\partial t}-\alpha \Delta {\mathbf {w}}_{\mathrm{int}}(x,t)-\beta \nabla \text{ div } {\mathbf {w}}_{\mathrm{int}}(x,t)+2a {\mathbf {w}}_{\mathrm{int}}=a \text{ rot } {\mathbf {v}}(x,t)+\tilde{{\mathbf {g}}}(x,t), \end{aligned} \end{aligned}$$

with the right-hand sides $\tilde{{\mathbf {f}}}(x,t)=\hat{{\mathbf {f}}}(x,t)+{\mathbf {f}}_{(1)}(x,t)+{\mathbf {f}}_{(2)}(x,t)$, where:

$$\begin{aligned} \begin{aligned}&{\mathbf {f}}_{(1)}(x,t)=\mu \sum _{j=1}^{m} \zeta ''(x_{1}^{(j)}) \left( U_{1}^{(j)}(x^{(j)'},t),0,0 \right) \\&\quad -\sum _{j=1}^{m} \zeta '(x_{1}^{(j)}) \left( -q^{(j)}(t)x_{1}^{(j)}+p_{0}^{(j)}(t),0,0 \right) \\&a \left( 0,-\zeta '(x_{1}^{j)})W_{3}^{(j)}(x^{(j)'},t)),\zeta '(x_{1}^{(j)})W_{2}^{(j)}(x^{(j)'},t) \right) ,\\&{\mathbf {f}}_{(2)}(x,t)=-\frac{\partial {\mathbf {Z}}(x,t)}{\partial t}+\mu \Delta {\mathbf {Z}}(x,t), \end{aligned} \end{aligned}$$

and $\tilde{{\mathbf {g}}}(x,t)=\hat{{\mathbf {g}}}(x,t)+{\mathbf {g}}_{(1)}(x,t)+{\mathbf {g}}_{(2)}(x,t)$, where:

$$\begin{aligned} \begin{aligned} {\mathbf {g}}_{(1)}(x,t)&=\alpha \sum _{j=1}^{m} \zeta ''(x_{1}^{(j)})(0,W_{2}^{(j)}(x^{(j)'},t),W_{3}^{(j)}(x^{(j)'},t))\\&\quad +\beta \sum _{j=1}^{m} \left( \zeta '(x_{1}^{(j)}) \left( \frac{\partial W_{2}^{(j)}}{\partial x_{2}^{(j)}}+\frac{\partial W_{3}^{(j)}}{\partial x_{3}^{(j)}} \right) ,0,0 \right) ,\\ {\mathbf {g}}_{(2)}(x,t)&=a \text{ rot } {\mathbf {Z}}(x,t), \end{aligned} \end{aligned}$$

and the new initial data:

$$\begin{aligned} \begin{aligned}&{\mathbf {v}}(x,0):=\tilde{{\mathbf {a}}}(x)=\hat{{\mathbf {a}}}(x)-{\mathbf {Z}}(x,0), \quad {\mathbf {w}}_{\mathrm{int}}(x,0):=\tilde{{\mathbf {b}}}(x)=\hat{{\mathbf {b}}}(x). \end{aligned} \end{aligned}$$

Due to the fact that $\text{ supp }_{x} {\mathbf {Z}}(x,t) \subset {\overline{\Omega }}_{(3)}$, we now easily obtain:

$$\begin{aligned} \begin{aligned}&\text {supp}_{x}({\mathbf {f}}_{(1)}(x,t)+{\mathbf {f}}_{(2)}(x,t)) \subset {\overline{\Omega }}_{(3)}, \\&\text {supp}_{x}({\mathbf {g}}_{(1)}(x,t)+{\mathbf {g}}_{2}(x,t)) \subset {\overline{\Omega }}_{(3)}. \end{aligned} \end{aligned}$$

It now follows that:

$$\begin{aligned} \begin{aligned}&\vert \vert {\mathbf {f}}_{(1)}+{\mathbf {f}}_{(2)} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}+\vert \vert {\mathbf {g}}_{(1)}+{\mathbf {g}}_{(2)} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})} \\&\quad \le c(\vert \vert {\mathbf {f}}_{(1)}+{\mathbf {f}}_{(2)} \vert \vert _{L_{2}(\Omega ^{T})}+\vert \vert {\mathbf {g}}_{(1)}+{\mathbf {g}}_{(2)} \vert \vert _{L_{2}(\Omega ^{T})})\\&\quad \le c \sum _{j=1}^{m}( \vert \vert a_{1}^{(j)} \vert \vert _{W_{2}^{1}(\sigma _{j})}+\vert \vert {\mathbf {b}}^{(j)'} \vert \vert _{W_{2}^{1}(\sigma _{j})}+\vert \vert f_{1}^{(j)} \vert \vert _{L_{2}(\Sigma _{j}^{T})}\\&\qquad +\vert \vert {\mathbf {g}}^{(j)'} \vert \vert _{L_{2}(\Sigma _{j}^{T})}+\vert \vert F_{j} \vert \vert _{W_{2}^{1}(\langle 0,T \rangle )}). \end{aligned} \end{aligned}$$

Furthermore, there hold the conditions:

$$\begin{aligned} \text{ div } \tilde{{\mathbf {a}}}(x)=\text{ div } \hat{{\mathbf {a}}}(x)+\sum _{j=1}^{m} \zeta '(x_{1}^{(j)})a_{1}^{(j)}(x^{(j)'})=0, \end{aligned}$$

and

$$\begin{aligned} \int _{\sigma _{j}} \tilde{{\mathbf {a}}}(x) \cdot {\mathbf {n}}(x)\mathrm{d}x^{(j)'}=0,\quad j=1,\dots ,m, \end{aligned}$$

with $\tilde{{\mathbf {a}}}(x)\vert _{\partial \Omega }=0$. Therefore, from Theorem 5.2, we conclude that there exists a unique solution $({\mathbf {v}}(x,t),{\mathbf {w}}_{\mathrm{int}}(x,t),{\tilde{p}}(x,t))$ of problem (5.1) with the right-hand sides $\tilde{{\mathbf {f}}}(x,t)$, $\tilde{{\mathbf {g}}}(x,t)$ and initial data $\tilde{{\mathbf {a}}}(x)$, $\tilde{{\mathbf {b}}}(x)$ satisfying estimates:

$$\begin{aligned} \begin{aligned}&\vert \vert {\mathbf {v}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{2,1}(\Omega ^{T})}+\vert \vert {\mathbf {w}}_{\mathrm{int}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{2,1}(\Omega ^{T})}+\vert \vert \nabla {\tilde{p}}\vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }(\Omega ^{T})}}\\&\quad \le c \left( \vert \vert \hat{{\mathbf {a}}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert \hat{{\mathbf {b}}} \vert \vert _{{\mathcal {W}}_{2,\varvec{\beta }}^{1}(\Omega )}+\vert \vert \hat{{\mathbf {f}}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})}+\vert \vert \hat{{\mathbf {g}}} \vert \vert _{{\mathcal {L}}_{2,\varvec{\beta }}(\Omega ^{T})} \right) \\&\qquad +c\sum _{j=1}^{m}\left( \vert \vert a_{1}^{(j)} \vert \vert _{W_{2}^{1}(\sigma _{j})}+\vert \vert {\mathbf {b}}^{(j)'} \vert \vert _{W_{2}^{1}(\sigma _{j})}+\vert \vert f_{1}^{(j)} \vert \vert _{L_{2}(\Sigma _{j}^{T})}\right. \\&\qquad \left. +\vert \vert {\mathbf {g}}^{(j)'} \vert \vert _{L_{2}(\Sigma _{j}^{T})}+\vert \vert F_{j} \vert \vert _{W_{2}^{1}(\langle 0,T \rangle )}\right) . \end{aligned} \end{aligned}$$