As already stated, to prove Theorem 1.1, we use the method of successive approximations.
3.1 Approximating Linear Problems
We construct approximate solutions, inductively, as follows:
-
(i)
first define \({{\varvec{u}}^{-1}}=0\), \({{P}^{-1}}=0\), \({{\varvec{B}}^{-1}}={{\varvec{B}}_{ini}}\) and \({{T}^{-1}}=0\), where \({{\varvec{B}}_{ini}}\) is a given nonzero constants vector, and
-
(ii)
assuming that \(({{\varvec{u}}^{m-1}}, {{P}^{m-1}}, {{\varvec{B}}^{m-1}}, {{T}^{m-1}})\) was defined for \(m\ge 1\), let \({{\varvec{u}}^{m}}\), \({{P}^{m}}\), \({{\varvec{B}}^{m}}\) and \({{T}^{m}}\) be the unique solution to the following boundary problems:
$$\begin{aligned} \left\{ \begin{aligned}&-\textrm{div}[{{(1+\left| D{{\varvec{u}}^{m-1}} \right| )}^{p-2}}D{{\varvec{u}}^{m}}]+\nabla {{P}^{m}} \\&\hspace{1.5cm}=[1-\beta ({{T}^{m}}-{{T}_{r}})]\varvec{g} +(\textrm{curl}~{{\varvec{B}}^{m-1}})\times {{\varvec{B}}^{m-1}}-({{\varvec{u}}^{m-1}}\cdot \nabla ){{\varvec{u}}^{m-1}}, \quad&in~\Omega , \\&-\Delta {{\varvec{B}}^{m}}=({{\varvec{B}}^{m-1}}\cdot \nabla ){{\varvec{u}}^{m}}-({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{B}}^{m-1}}, \quad&in~\Omega , \\&-\Delta {{T}^{m}}={{\left| \textrm{curl}{{\varvec{B}}^{m-1}} \right| }^{2}}+2{{\left| D({{\varvec{u}}^{m-1}}) \right| }^{2}} +\psi -({{\varvec{u}}^{m-1}}\cdot \nabla ){{T}^{m-1}}, \quad&in~\Omega , \\&\textrm{div}~{{\varvec{u}}^{m}}=0,\qquad \textrm{div}~{{\varvec{B}}^{m}}=0, \quad&in~\Omega , \\&{{\varvec{u}}^{m}}{{|}_{\partial \Omega }}=0,\quad {{\varvec{B}}^{m}}\cdot \varvec{n}{{|}_{\partial \Omega }}=0, \quad \textrm{curl}{{\varvec{B}}^{m}}\times \varvec{n}{{|}_{\partial \Omega }}=0,\quad {{T}^{m}}{{|}_{\partial \Omega }}=0. \end{aligned} \right. \end{aligned}$$
(8)
The following result holds true.
Proposition 3.1
Assume that \(p\in (1,2)\), \(q>3\), \(r>3\) and let \({{\gamma }_{0}}=1-\frac{3}{q}\). Let \({\Omega }\) be a bounded smooth domain, and let be \(\varvec{g}\in {{L}^{q}}(\Omega )\). Then for any \(m\in \mathbb {N}\), there exists a weak solution \(({{\varvec{u}}^{m}},{{P}^{m}},{{\varvec{B}}^{m}},{{T}^{m}})\) of problem (8) such that
$${{\varvec{u}}^{m}}\in {{C}^{1,{{\gamma }_{0}}}}(\overline{\Omega }),\quad {{P}^{m}}\in {{C}^{0,{{\gamma }_{0}}}}({\overline{\Omega }}), \quad {{\varvec{B}}^{m}}\in {{W}^{2,r}}(\Omega ),\quad {{T}^{m}}\in {{W}^{2,2}}(\Omega ).$$
Moreover, if \({{\left\| \varvec{g} \right\| }_{q}}\), \({{\left\| \psi \right\| }_{2}}\) satisfies the assumption
$$\begin{aligned} {{\left\| \varvec{g} \right\| }_{q}}<1, \quad {{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+CK\beta {{c}_{2}}{{\left\| \varvec{g} \right\| }_{q}}{{\left\| \psi \right\| }_{2}} +{{c}_{2}}{{\left\| \psi \right\| }_{2}}<\frac{1}{4{{K}_{1}}}, \end{aligned}$$
(9)
then
$$\begin{aligned}{} & {} {{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}({\overline{\Omega }})}} +{{\left\| {{P}^{m}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}} (\overline{\Omega })}}+{{\left\| {{\varvec{B}}^{m}} \right\| }_{2,r}}+{{\left\| {{T}^{m}} \right\| }_{2,2}}\nonumber \\\le & {} 2{{K}_{2}}{{\left\| \varvec{g}\right\| }_{q}}+2CK\beta {{c}_{2}}{{\left\| \varvec{g} \right\| }_{q}}{{\left\| \psi \right\| }_{2}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}}, \end{aligned}$$
(10)
uniformly in \(m\in \mathbb {N}\).
Proof
Setting
$${{I}_{m}}={{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}} +{{\left\| {{P}^{m}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}}+{{\left\| {{\varvec{B}}^{m}} \right\| }_{2,r}} +{{\left\| {{T}^{m}} \right\| }_{2,2}}.$$
Let be \(m=0\), first of all, we consider the following boundary-value problem
$$\begin{aligned} \left\{ \begin{aligned}&-\Delta {{T}^{0}}=\psi , \\&{{T}^{0}}{{|}_{\partial \Omega }}=0, \end{aligned} \right. \end{aligned}$$
where \(\psi \in {{L}^{2}}(\Omega )\). By Lemma 2.3, there exists a weak solution \({{T}^{0}}\in {{W}^{2,2}}(\Omega )\), and
$$\begin{aligned} {{\left\| {{T}^{0}} \right\| }_{2,2}}\le {{c}_{2}}{{\left\| \psi \right\| }_{2}}. \end{aligned}$$
(11)
Then, we consider the following boundary-value problem
$$\begin{aligned} \left\{ \begin{aligned}&-\Delta {{\varvec{u}}^{0}}+\nabla {{P}^{0}}=[1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g}, \\&\textrm{div}~{{\varvec{u}}^{0}}=0, \\&{{\varvec{u}}^{0}}{{|}_{\partial \Omega }}=0, \end{aligned} \right. \end{aligned}$$
where \(\varvec{g}\in {{L}^{q}}(\Omega )\). Since \({{\left\| [1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g} \right\| }_{q}} \le {{\left\| (1+\beta {{T}_{r}})\varvec{g} \right\| }_{q}}+{{\left\| \beta {{T}^{0}}\varvec{g} \right\| }_{q}} \le (1+\beta {{T}_{r}}){{\left\| \varvec{g}\right\| }_{q}} +C\beta {{\left\| {{T}^{0}} \right\| }_{2,2}}{{\left\| \varvec{g} \right\| }_{q}}\), thus, by Lemma 2.1, we can find a solution \(({{\varvec{u}}^{0}},{{P}^{0}})\in {{C}^{1,{{\gamma }_{0}}}}({\overline{\Omega }})\times {{C}^{0,{{\gamma }_{0}}}}({\overline{\Omega }})\) (see [27], Theorem 3.2) and
$$\begin{aligned} {{\left\| {{\varvec{u}}^{0}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}} +{{\left\| {{P}^{0}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}} \le {\widetilde{c}}({{\left\| {{\varvec{u}}_{0}} \right\| }_{1,2}}+{{\left\| [1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g} \right\| }_{q}}). \end{aligned}$$
(12)
where \({\widetilde{c}}>1\). By writing the definition of weak solution with the test function \(\varphi\) replaced by \({{\varvec{u}}_{0}}\), we get
$${{\int _{\Omega }{\left| D{{\varvec{u}}^{0}} \right| ^{2}}}}dx=\int _{\Omega }{[1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g}\cdot {{\varvec{u}}^{0}}}dx.$$
By Lemma 2.4 and the Hölder inequality, we have
$$\begin{aligned} {{\int _{\Omega }{\left| D{{\varvec{u}}^{0}} \right| ^{2}}}}dx&=\left\| D{{\varvec{u}}^{0}} \right\| _{2}^{2}\ge \frac{1}{c_{3}^{2}}\left\| {{\varvec{u}}^{0}} \right\| _{1,2}^{2}, \\ \int _{\Omega }{[1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g}\cdot {{\varvec{u}}^{0}}}dx&\le {{\left\| [1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g} \right\| }_{2}}{{\left\| {{\varvec{u}}^{0}} \right\| }_{2}} \\&\le \left[ (1+\beta {{T}_{r}}){{\left\| \varvec{g} \right\| }_{q}} +C\beta {{\left\| {{T}^{0}} \right\| }_{2,2}}{{\left\| \varvec{g} \right\| }_{q}}\right] \left\| {{\varvec{u}}_{0}} \right\| _{1,2}^{{}}, \end{aligned}$$
which obviously implies
$$\begin{aligned} {{\left\| {{\varvec{u}}^{0}} \right\| }_{1,2}}\le c_{3}^{2}\left[ (1+\beta {{T}_{r}}) +C\beta {{\left\| {{T}^{0}} \right\| }_{2,2}}\right] {{\left\| \varvec{g} \right\| }_{q}}. \end{aligned}$$
(13)
\(\square\)
By (11)–(13), we get
$$\begin{aligned} \begin{aligned} {{\left\| {{\varvec{u}}^{0}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}+{{\left\| {{P}^{0}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}}&\le {\widetilde{c}}\big ({{\left\| {{\varvec{u}}_{0}} \right\| }_{1,2}}+{{\left\| [1-\beta ({{T}^{0}}-{{T}_{r}})]\varvec{g} \right\| }_{q}}\big ) \\&\le {{c}_{0}}(c_{3}^{2}+1)(1+\beta {{T}_{r}}+C\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}){{\left\| \varvec{g} \right\| }_{q}}. \end{aligned} \end{aligned}$$
(14)
Then we consider the following boundary-value problem
$$\begin{aligned} \left\{ \begin{aligned}&-\Delta {{\varvec{B}}^{0}}=({{\varvec{B}}_{ini}}\cdot \nabla ){{\varvec{u}}^{0}}, \\&\textrm{div}~{{\varvec{B}}^{0}}=0, \\&{{\varvec{B}}^{0}}\cdot \varvec{n}{{|}_{\partial \Omega }}=0, \\&\textrm{curl}{{\varvec{B}}^{0}}\times \varvec{n}{{|}_{\partial \Omega }}=0. \end{aligned} \right. \end{aligned}$$
Obviously, by (14), we have \(({{\varvec{B}}_{ini}}\cdot \nabla ){{\varvec{u}}^{0}}\in {{L}^{r}}(\Omega )\). Noting that \(\textrm{div}{{\varvec{B}}^{0}}=0\), it follows that \({\mathrm{curl\,curl}}{{\varvec{B}}^{0}}=-\Delta {{\varvec{B}}^{0}}\). Since \(({{\varvec{B}}_{ini}}\cdot \nabla ){{\varvec{u}}^{0}}=\textrm{curl}({{\varvec{u}}^{0}}\times {{\varvec{B}}_{ini}})\) and \(\varvec{n}\cdot \textrm{curl}({{\varvec{u}}^{0}}\times {{\varvec{B}}_{ini}})\) is the operator of a tangential derivative along the vector \(\varvec{n}\times {{\varvec{u}}^{0}}\times {{\varvec{B}}_{ini}}\) on \(\partial \Omega\), we can follow from \({{\varvec{u}}^{0}}{{|}_{\partial \Omega }}=0\) that \([({{\varvec{B}}_{ini}}\cdot \nabla ){{\varvec{u}}^{0}}]\cdot \varvec{n}]{{|}_{\partial \Omega }} =\textrm{curl}({{\varvec{u}}^{0}}\times {{\varvec{B}}_{ini}})\cdot \varvec{n}{{|}_{\partial \Omega }}=0\) [29]. Moreover, obviously there have \(\textrm{div}[({{\varvec{B}}_{ini}}\cdot \nabla ){{\varvec{u}}^{0}}]=\textrm{div}[\textrm{curl}({{\varvec{u}}^{0}}\times {{\varvec{B}}_{ini}})]=0\), according to Lemma 2.2, we can find a unique solution \({{\varvec{B}}^{0}}\in {{W}^{2,r}}(\Omega )\) such that
$$\begin{aligned} {{\left\| {{\varvec{B}}^{0}} \right\| }_{2,r}}\le {{c}_{1}}{{\left\| ({{\varvec{B}}_{ini}}\cdot \nabla ){{\varvec{u}}^{0}} \right\| }_{r}} \le C{{c}_{1}}{{\left\| {{\varvec{u}}^{0}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}. \end{aligned}$$
(15)
So we can get from (11), (14) and (15) that
$$\begin{aligned} \begin{aligned} {{I}_{0}}&={{\left\| {{\varvec{u}}^{0}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}} +{{\left\| {{P}^{0}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}}+{{\left\| {{\varvec{B}}^{0}} \right\| }_{2,r}} +{{\left\| {{T}^{0}} \right\| }_{2,2}} \\&\le {{c}_{0}}(c_{3}^{2}+1)(1+C{{c}_{1}})(1+\beta {{T}_{r}} +C\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}){{\left\| \varvec{g} \right\| }_{q}}+{{c}_{2}}{{\left\| \psi \right\| }_{2}}. \end{aligned} \end{aligned}$$
(16)
Let be \(m\ge 1\), assuming that \(({{\varvec{u}}^{m}},{{P}^{m}},{{\varvec{B}}^{m}},{{T}^{m}})\in {{C}^{1,{{\gamma }_{0}}}}({\overline{\Omega }}) \times {{C}^{0,{{\gamma }_{0}}}}({\overline{\Omega }})\times {{W}^{2,r}}(\Omega )\times {{W}^{2,2}}(\Omega )\) is a solution of (8). We consider the case where \(m+1\).
Firstly, we consider the following boundary-value problem:
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta {{T}^{m+1}}&{}={{\left| {\textrm{curl}}{{\varvec{B}}^{m}} \right| }^{2}} +2{{\left| D({{\varvec{u}}^{m}}) \right| }^{2}}-({{\varvec{u}}^{m}}\cdot \nabla ){{T}^{m}}+\psi , \\ {{T}^{m+1}}{{|}_{\partial \Omega }}&{}=0. \end{array}\right. } \end{aligned}$$
Since \({{\varvec{u}}^{m}}\in {{C}^{1,{{\gamma }_{0}}}}(\overline{\Omega })\), \({{\varvec{B}}^{m}}\in {{W}^{2,r}}(\Omega )\), \({{T}^{m}}\in {{W}^{2,2}}(\Omega )\), thus,
$$\begin{aligned} {{\left\| ({{\varvec{u}}^{m}}\cdot \nabla ){{T}^{m}} \right\| }_{2}}&\le C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}}, \\ {{\left\| {{\left| \textrm{curl}{{\varvec{B}}^{m}} \right| }^{2}} \right\| }_{2}}&=\left\| \textrm{curl}{{\varvec{B}}^{m}} \right\| _{4}^{2}\le C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}, \\ {{\left\| D{{({{\varvec{u}}^{m}})}^{2}} \right\| }_{2}}&=\left\| D({{\varvec{u}}^{m}}) \right\| _{4}^{2}\le C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}. \end{aligned}$$
Then we get \({{\left| \textrm{curl}{{\varvec{B}}^{m}} \right| }^{2}}+2{{\left| D({{\varvec{u}}^{m}}) \right| }^{2}} +\psi -({{\varvec{u}}^{m}}\cdot \nabla ){{T}^{m}}\in {{L}^{2}}(\Omega )\). By Lemma 2.3, there exists a weak solution \({{T}^{m+1}}\in {{W}^{2,2}}(\Omega )\) such that
$$\begin{aligned} \begin{aligned} {{\left\| {{T}^{m+1}} \right\| }_{2,2}}&\le {{c}_{2}}{{\left\| {{\left| \textrm{curl}{{\varvec{B}}^{m}} \right| }^{2}} +2{{\left| D({{\varvec{u}}^{m}}) \right| }^{2}}+\psi -({{\varvec{u}}^{m}}\cdot \nabla ){{T}^{m}} \right\| }_{2}} \\&\le {{c}_{2}}(C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2} +{{\left\| \psi \right\| }_{2}}+C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}}). \end{aligned} \end{aligned}$$
(17)
Then we consider the following boundary-value problem:
$$\begin{aligned} {\left\{ \begin{array}{ll} -\textrm{div}[(1+&{}\Big | D{{\varvec{u}}^{m}} \Big |)^{p-2}D{{\varvec{u}}^{m+1}}]+\nabla {{P}^{m+1}} \\ &{}\qquad =[1-\beta ({{T}^{m+1}}-{{T}_{r}})]\varvec{g} +\textrm{curl}{{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}}-({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{u}}^{m}}, \\ \textrm{div}~{{\varvec{u}}^{m+1}}&{} =0, \\ {{\varvec{u}}^{m+1}}{{|}_{\partial \Omega }}&{} =0, \end{array}\right. } \end{aligned}$$
(18)
where \(\varvec{g}\in {{L}^{q}}(\Omega )\). Since the assumption implies that
$$\begin{aligned} {\Vert [1-\beta ({{T}^{m+1}}-{{T}_{r}})]\varvec{g}\Vert }_{q}&\le (1+\beta {{T}_{r}}){{\left\| \varvec{g} \right\| }_{q}}+C\beta {{\left\| {{T}^{m+1}} \right\| }_{2,2}}{{\left\| \varvec{g} \right\| }_{q}} \\&\le \big [1+\beta {{T}_{r}}+C\beta {{c}_{2}}(C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2} +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}+{{\left\| \psi \right\| }_{2}} \\&\qquad +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}})\big ]{{\left\| \varvec{g} \right\| }_{q}}, \\ {{\left\| ({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{u}}^{m}} \right\| }_{q}}&\le {{\left\| {{\varvec{u}}^{m}} \right\| }_{\infty }}{{\left\| \nabla {{\varvec{u}}^{m}} \right\| }_{q}} \le C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}, \\ {{\left\| \textrm{curl} {{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}} \right\| }_{q}}&\le {{\left\| \textrm{curl}{{\varvec{B}}^{m}} \right\| }_{q}}{{\left\| {{\varvec{B}}^{m}} \right\| }_{\infty }} \le {{\left\| \textrm{curl}{{\varvec{B}}^{m}} \right\| }_{\infty }}{{\left\| {{\varvec{B}}^{m}} \right\| }_{\infty }} \le C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}, \end{aligned}$$
then \([1-\beta ({{T}^{m+1}}-{{T}_{r}})]\varvec{g}+\textrm{curl}{{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}} -({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{u}}^{m}}\in {{L}^{q}}(\Omega )\). Then by Lemma 2.1, we can get a solution \(({{\varvec{u}}^{m+1}},{{P}^{m+1}})\in {{C}^{1,{{\gamma }_{0}}}}({\overline{\Omega }}) \times {{C}^{0,{{\gamma }_{0}}}}({\overline{\Omega }})\) (see [27], theorem 3.2) such that
$$\begin{aligned} \begin{aligned} \big \Vert {{\varvec{u}}^{m+1}} \big \Vert&_{{{C}^{1,{{\gamma }_{0}}}}}+{{\left\| {{P}^{m+1}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}} \\&\le {\widetilde{c}}({{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}} +{{\left\| \varvec{g}[1-\beta ({{T}^{m+1}}-{{T}_{r}})] +\textrm{curl}{{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}}-({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{u}}^{m}} \right\| }_{q}}) \\&\le {\widetilde{c}}({{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}}+\left[ 1+\beta {{T}_{r}} +C\beta {{c}_{2}}\left( C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2} +{{\left\| \psi \right\| }_{2}}\right. \right. \\&\quad \left. \left. +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}} {{\left\| {{T}^{m}} \right\| }_{2,2}}\right) \right] {{\left\| \varvec{g} \right\| }_{q}} +C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}), \end{aligned} \end{aligned}$$
(19)
where \({\widetilde{c}}={{c}_{0}}{{(1+{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}})}^{\alpha }}\), \({{c}_{0}}>1\), \(\alpha =\alpha ({{\gamma }_{0}},n,p)>2\).
Remark 3.1
As in [27] (see also [23]), we emphasize here that the way in which depends by the norm of is important in our proof.
By writing the definition of weak solution of (18) with the test function \(\varphi\) replaced by \(\varvec{u}^{m+1}\), we get
$$\begin{aligned} \begin{aligned} \int _{\Omega }(1+\left| D{{\varvec{u}}^{m}} \right| )^{p-2}&{{\left| D{{\varvec{u}}^{m+1}} \right| }^{2}}dx =\int _{\Omega }{[1-\beta ({{T}^{m+1}}-{{T}_{r}})]\varvec{g}\cdot {{\varvec{u}}^{m+1}}dx} \\&-\int _{\Omega }{({{\varvec{u}}^{m}}\cdot \nabla ){\varvec{u}}^{m}\cdot {{\varvec{u}}^{m+1}}dx} +\int _{\Omega }\textrm{curl}{{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}}\cdot {\varvec{u}}^{m+1}dx. \end{aligned} \end{aligned}$$
Since \(1<p<2\), by Lemma 2.4 and Hölder inequality, there follows
$$\begin{aligned} \begin{aligned} \int _{\Omega }{{{(1+\left| D{{\varvec{u}}^{m}} \right| )}^{p-2}}{{\left| D{{\varvec{u}}^{m+1}} \right| }^{2}}dx}&\ge {{(1+{{\left\| D{{\varvec{u}}^{m}} \right\| }_{\infty }})}^{p-2}}\left\| D{{\varvec{u}}^{m+1}} \right\| _{2}^{2} \\&\ge \frac{1}{c_{3}^{2}}{{(1+{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}})}^{p-2}}\left\| {{\varvec{u}}^{m+1}} \right\| _{1,2}^{2}, \\&\Big | \int _{\Omega }[1-\beta ({T}^{m+1}-{{T}_{r}})]\varvec{g}{\varvec{u}}^{m+1}dx\\&\quad -\int _{\Omega }{({{\varvec{u}}^{m}}\cdot \nabla ) {{\varvec{u}}^{m}}\cdot {{\varvec{u}}^{m+1}}dx}+\int _{\Omega }\textrm{curl} {{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}}\cdot {{\varvec{u}}^{m+1}}\textrm{dx} \Big | \\&\le {{\left\| (1+\beta {{T}_{r}})\varvec{g} \right\| }_{2}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{2}} +{{\left\| \beta {{T}^{m+1}}\varvec{g} \right\| }_{2}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{2}} \\&\quad +{{\left\| ({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{u}}^{m}} \right\| }_{2}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{2}} \\&\quad +{{\left\| \textrm{curl} {{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}} \right\| }_{2}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{2}} \\&\le (1+\beta {{T}_{r}}){{\left\| \varvec{g} \right\| }_{q}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}} +C\beta {{c}_{2}}(C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2} +{{\left\| \psi \right\| }_{2}} \\&\quad +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}}) \cdot {{\left\| \varvec{g} \right\| }_{q}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}}\\&\quad +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}}\\&\quad +C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}}, \end{aligned} \end{aligned}$$
which implies
$$\begin{aligned} \begin{aligned} {{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}}&\le c_{3}^{2} {{(1+{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}})}^{2-p}} \left[ (1+\beta {{T}_{r}}){{\left\| \varvec{g} \right\| }_{q}}+C\beta {{c}_{2}}(C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2} +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}\right. \\&\quad \left. +{{\left\| \psi \right\| }_{2}}+C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}} {{\left\| {{T}^{m}} \right\| }_{2,2}})\cdot {{\left\| \varvec{g} \right\| }_{q}} +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}+C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}\right] . \end{aligned} \end{aligned}$$
(20)
Inserting (20) into (19), we obtain
$$\begin{aligned} \begin{aligned}&{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}+{{\left\| {{P}^{m+1}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}} \\&\le {\widetilde{c}}\left( {{\left\| {{\varvec{u}}^{m+1}} \right\| }_{1,2}} +[1+\beta {{T}_{r}}+C\beta {{c}_{2}} (C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2} +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}+{{\left\| \psi \right\| }_{2}}\right. \\&\quad \left. +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}})] {{\left\| \varvec{g} \right\| }_{q}}+C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}\right) \\&\le {{c}_{0}}{{(1+{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}})}^{\alpha }}\left( c_{3}^{2} {{(1+{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}})}^{2-p}} \cdot \left[ (1+\beta {{T}_{r}}){{\left\| \varvec{g} \right\| }_{q}} +C\beta {{c}_{2}}\left( C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}\right. \right. \right. \\&\quad \left. \left. +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}+{{\left\| \psi \right\| }_{2}} +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}}\right) \cdot {{\left\| \varvec{g} \right\| }_{q}}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2} +C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}\right] \\&\quad +\left[ 1+\beta {{T}_{r}}+C{{c}_{2}}\beta \left( C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2} +C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}+{{\left\| \psi \right\| }_{2}} +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}}\right) \right] {{\left\| \varvec{g} \right\| }_{q}} \\&\quad \left. +C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}\right) \\&\le {{c}_{0}}(c_{3}^{2}+1){{(1+{{I}_{m}})}^{\alpha +2-p}} \left( [1+\beta {{T}_{r}}+C\beta {{c}_{2}}(CI_{m}^{2}+{{\left\| \psi \right\| }_{2}})] {{\left\| \varvec{g} \right\| }_{q}}+CI_{m}^{2}\right) . \end{aligned} \end{aligned}$$
(21)
Then, we consider the boundary-value problem
$$\begin{aligned} \left\{ \begin{aligned}&-\Delta {{\varvec{B}}^{m+1}}=({{\varvec{B}}^{m}}\cdot \nabla ){{\varvec{u}}^{m+1}}-({{\varvec{u}}^{m+1}}\cdot \nabla ){{\varvec{B}}^{m}}, \\&\textrm{div}{{\varvec{B}}^{m+1}}=0, \\&{{\varvec{B}}^{m+1}}\cdot \varvec{n}{{|}_{\partial \Omega }}=0, \\&\textrm{curl} {{\varvec{B}}^{m+1}}\times \varvec{n}{{|}_{\partial \Omega }}=0. \\ \end{aligned} \right. \end{aligned}$$
The assumption implies that
$$\begin{aligned} {{\left\| ({{\varvec{B}}^{m}}\cdot \nabla ){{\varvec{u}}^{m+1}} \right\| }_{r}}\le {{\left\| {{\varvec{B}}^{m}} \right\| }_{\infty }} {{\left\| \nabla {{\varvec{u}}^{m+1}} \right\| }_{r}}\le C{{\left\| {{\varvec{B}}^{m}} \right\| }_{2,r}} {{\left\| {{\varvec{u}}^{m+1}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}, \\ {{\left\| ({{\varvec{u}}^{m+1}}\cdot \nabla ){{\varvec{B}}^{m}} \right\| }_{r}}\le {{\left\| {{\varvec{u}}^{m+1}} \right\| }_{r}} {{\left\| \nabla {{\varvec{B}}^{m}} \right\| }_{\infty }}\le C{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}} {{\left\| {{\varvec{B}}^{m}} \right\| }_{2,r}}, \end{aligned}$$
then \(({{\varvec{B}}^{m}}\cdot \nabla ){{\varvec{u}}^{m+1}}-({{\varvec{u}}^{m+1}}\cdot \nabla ){{\varvec{B}}^{m}}\) belongs to \({{L}^{r}}(\Omega )\). Since \(\varvec{n}\cdot \textrm{curl}({{\varvec{u}}^{m+1}}\times {{\varvec{B}}^{m}})\) is the operator of a tangential derivative along the vector \(\varvec{n}\times {{\varvec{u}}^{m+1}}\times {{\varvec{B}}^{m}}\) on \(\partial \Omega\), we can follow from \({{\varvec{u}}^{m+1}}{{|}_{\partial \Omega }}=0\) that \([({{\varvec{B}}^{m}}\cdot \nabla ){{\varvec{u}}^{m+1}}-({{\varvec{u}}^{m+1}}\cdot \nabla ){{\varvec{B}}^{m}}] \cdot \varvec{n}]{{|}_{\partial \Omega }}=[\textrm{curl}({{\varvec{u}}^{m+1}}\times {{\varvec{B}}^{m}})]\cdot \varvec{n}{{|}_{\partial \Omega }}=0\) (see [29]). Moreover, since \(\textrm{div}[({{\varvec{B}}^{m}}\cdot \nabla ){{\varvec{u}}^{m+1}}-({{\varvec{u}}^{m+1}}\cdot \nabla ){{\varvec{B}}^{m}}] =\textrm{div}[\textrm{curl}({{\varvec{u}}^{m+1}}\times {{\varvec{B}}^{m}})]=0\), using Lemma 2.2, we can find a unique solution \({{\varvec{B}}^{m+1}}\in {{W}^{2,r}}(\Omega )\) such that
$$\begin{aligned} {{\left\| {{\varvec{B}}^{m+1}} \right\| }_{2,r}}\le {{c}_{1}}({{\left\| ({{\varvec{B}}^{m}}\cdot \nabla ){{\varvec{u}}^{m+1}} \right\| }_{r}} +{{\left\|
({{\varvec{u}}^{m+1}}\cdot \nabla ){{\varvec{B}}^{m}} \right\| }_{r}})\le 2C{{c}_{1}} {{\left\| {{\varvec{u}}^{m+1}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{\varvec{B}}^{m}} \right\| }_{2,r}}. \end{aligned}$$
(22)
Combining (17), (21) and (22), we obtain
$$\begin{aligned} \begin{aligned} {{I}_{m+1}}&={\left\| {\varvec{u}}^{m+1} \right\| }_{{C}^{1,{{\gamma }_{0}}}} +\big \Vert {{P}^{^{m+1}}} \big \Vert _{{C}^{0,{{\gamma }_{0}}}}+{{\big \Vert {{\varvec{B}}^{^{m+1}}} \big \Vert }_{2,r}}+{{\big \Vert {{T}^{^{m+1}}} \big \Vert }_{2,2}} \\&\le {{c}_{0}}(c_{3}^{2}+1){{(1+{{I}_{m}})}^{\alpha +2-p}}\left( [1+\beta {{T}_{r}} +C\beta {{c}_{2}}(CI_{m}^{2}+{{\left\| \psi \right\| }_{2}})]{{\left\| \varvec{g} \right\| }_{q}}+CI_{m}^{2}\right) \\&\quad +2C{{c}_{1}}{{\left\| {{\varvec{u}}^{m+1}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{\varvec{B}}^{m}} \right\| }_{2,r}} +{{c}_{2}}(C\left\| {{\varvec{B}}^{m}} \right\| _{2,r}^{2}+C\left\| {{\varvec{u}}^{m}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}+{{\left\| \psi \right\| }_{2}} \\&\quad +C{{\left\| {{\varvec{u}}^{m}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}{{\left\| {{T}^{m}} \right\| }_{2,2}}) \\&\le {{c}_{0}}(1+C{{c}_{1}})(1+c_{3}^{2}) \cdot (1+2{{I}_{m}}){{(1+{{I}_{m}})}^{\alpha +2-p}}\cdot \left( [1+\beta {{T}_{r}}+C\beta {{c}_{2}} (CI_{m}^{2}\right. \\&\quad \left. +{{\left\| \psi \right\| }_{2}})]{{\left\| \varvec{g} \right\| }_{q}}+CI_{m}^{2}\right) +{{c}_{2}}(CI_{m}^{2}+{{\left\| \psi \right\| }_{2}}). \end{aligned} \end{aligned}$$
(23)
We shall prove the boundedness of the sequence \(\{{{I}_{m}}\}\) by a fixed point argument. Setting, for any \(t\ge 0\),
$$\begin{aligned} \begin{aligned} F(t)&={{c}_{0}}(1+C{{c}_{1}})(1+c_{3}^{2})\cdot (1+2t)\cdot {{(1+t)}^{\alpha +2-p}} \cdot \\&\quad \left( \left[ 1+\beta {{T}_{r}}+C\beta {{c}_{2}}(C{{t}^{2}}+{{\left\| \psi \right\| }_{2}})\right] {{\left\| \varvec{g} \right\| }_{q}} +C{{t}^{2}}\right) +{{c}_{2}}(C{{t}^{2}}+{{\left\| \psi \right\| }_{2}})-t. \end{aligned} \end{aligned}$$
We look for a root of F(t). Let us observe that if \(0\le t\le 1\), then
$$\begin{aligned} \begin{aligned} F(t)&\le K\left( [1+\beta {{T}_{r}}+C\beta {{c}_{2}}(C{{t}^{2}} +{{\left\| \psi \right\| }_{2}})]{{\left\| \varvec{g} \right\| }_{q}}+C{{t}^{2}}\right) +{{c}_{2}}(C{{t}^{2}}+{{\left\| \psi \right\| }_{2}})-t \\&\le (K\beta C{{c}_{2}}+KC+C{{c}_{2}}){{t}^{2}}-t+K(1+\beta {{T}_{r}} +C\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}){{\left\| \varvec{g} \right\| }_{q}}+{{c}_{2}}{{\left\| \psi \right\| }_{2}}, \end{aligned} \end{aligned}$$
where \(K={{c}_{0}}(1+C{{c}_{1}})(1+c_{3}^{2})\cdot 3\cdot {{2}^{\alpha +2-p}}>1\). For convenience, we write \({{K}_{1}}=K\beta C{{c}_{2}}+KC+C{{c}_{2}}\), \({{K}_{2}}=K(1+\beta {{T}_{r}})\), then
$$F(t)\le {{K}_{1}}{{t}^{2}}-t+{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+{{c}_{2}}{{\left\| \psi \right\| }_{2}} +CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}\triangleq G(t).$$
We note that the function G(t) has two positive roots \({{s}_{1}}<{{s}_{2}}\), if and only if the discriminant \(\Delta =1-4{{K}_{1}}({{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+{{c}_{2}} {{\left\| \psi \right\| }_{2}}+CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})>0\), namely
$${{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+{{c}_{2}}{{\left\| \psi \right\| }_{2}} +CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}<\frac{1}{4{{K}_{1}}},$$
and we have that
$$0<{{s}_{1}}=\frac{1-\sqrt{\Delta }}{2{{K}_{1}}}<1.$$
Since \(F(0)>0\) and \(F(t)\le G(t)\), where \(t\in [0,1]\), there exists \({{t}_{1}}\), with \(0<{{t}_{1}}<{{s}_{1}}\) such that \(F({{t}_{1}})=0\), i.e.
$$\begin{aligned} \begin{aligned}&{{c}_{0}}(1+C{{c}_{1}})(1+c_{3}^{2})\cdot (1+2{{t}_{1}})\cdot {{(1+{{t}_{1}})}^{\alpha +2-p}}\cdot \Big ([1+\beta {{T}_{r}}+C\beta {{c}_{2}}(Ct_{1}^{2}+{{\left\| \psi \right\| }_{2}})]{{\left\| g \right\| }_{q}} \\&+Ct_{1}^{2}\Big )+{{c}_{2}}(Ct_{1}^{2}+{{\left\| \psi \right\| }_{2}})-{{t}_{1}}=0. \end{aligned} \end{aligned}$$
Since \({{t}_{1}}>0\), it follows that \({{c}_{0}}(1+C{{c}_{1}})(1+Cc_{3}^{2})\cdot (1+\beta {{T}_{r}} +C\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}){{\left\| \varvec{g} \right\| }_{q}}+{{c}_{2}}{{\left\| \psi \right\| }_{2}}-{{t}_{1}}\le 0\). If we suppose that \({{I}_{m}}\le {{t}_{1}}\), by inequality (23) and the fact that \(F({{t}_{1}})=0\), we obtain
$$\begin{aligned} \begin{aligned} {{I}_{m+1}}&\le {{c}_{0}}(1+C{{c}_{1}})(1+c_{3}^{2}) \cdot (1+2{{I}_{m}}){{(1+{{I}_{m}})}^{\alpha +2-p}}\cdot \left( [1+\beta {{T}_{r}}+C{{c}_{2}}\beta (CI_{m}^{2}\right. \\&\quad \left. +{{\left\| \psi \right\| }_{2}})]{{\left\| \varvec{g} \right\| }_{q}}+CI_{m}^{2}\right) +{{c}_{2}}(CI_{m}^{2}+{{\left\| \psi \right\| }_{2}}) \\&\le {{c}_{0}}(1+C{{c}_{1}})(1+c_{3}^{2})\cdot (1+2{{t}_{1}}) \cdot {{(1+{{t}_{1}})}^{\alpha +2-p}}\cdot \left( [1+\beta {{T}_{r}}+C\beta {{c}_{2}}(Ct_{1}^{2}\right. \\&\quad \left. +{{\left\| \psi \right\| }_{2}})]{{\left\| \varvec{g} \right\| }_{q}}+Ct_{1}^{2}\right) + {{c}_{2}}(Ct_{1}^{2}+{{\left\| \psi \right\| }_{2}}) \\&\le F({{t}_{1}})+{{t}_{1}} \\&={{t}_{1}}, \end{aligned} \end{aligned}$$
which proves our claim. Therefore,
$${{I}_{m}}\le {{t}_{1}}<{{s}_{1}}<\frac{1}{2{{K}_{1}}}<2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}} +2{{c}_{2}}{{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}\le 1~,~\forall ~m\in \mathbb {N}.$$
3.2 Convergence of Approximate Solutions
For any \(j\in \mathbb {N}\), set \({{\varvec{Q}}^{j+1}}(x)={{\varvec{u}}^{j+1}}(x)-{{\varvec{u}}^{j}}(x)\), \({{R}^{j+1}}(x)={{P}^{j+1}}(x)-{{P}^{j}}(x)\), \({{\varvec{J}}^{j+1}}(x)={{\varvec{B}}^{j+1}}(x)-{{\varvec{B}}^{j}}(x)\), \({{W}^{j+1}}(x)={{T}^{j+1}}(x)-{{T}^{j}}(x)\). Taking \(m=j\) and \(m=j+1\), respectively, in the weak formula of (8), then subtracting one from the other, we can get for \(\forall \varphi \in C_{0}^{\infty }(\Omega )\)
$$\begin{aligned} \begin{aligned} \int _{\Omega }{{(1+\left| D{{\varvec{u}}^{j}} \right| )}^{p-2}}&D{{\varvec{u}}^{j+1}}D{\varvec{\varphi }} dx =\int _{\Omega }{{{(1+\left| D{{\varvec{u}}^{j-1}} \right| )}^{p-2}}D{{\varvec{u}}^{j}}D{\varvec{\varphi }} dx} -\int _{\Omega }{\beta {{W}^{j+1}}\varvec{g}{\varvec{\varphi }} }dx \\&-\int _{\Omega }{({{\varvec{Q}}^{j}}\cdot \nabla ){{\varvec{u}}^{j}}{\varvec{\varphi }} dx-} \int _{\Omega }{({{\varvec{u}}^{j-1}}\cdot \nabla ){{\varvec{Q}}^{j}}{\varvec{\varphi }} dx} +\int _{\Omega }{\textrm{curl}{{\varvec{B}}^{j}}\times {{\varvec{J}}^{j}}{\varvec{\varphi }} dx} \\&+\int _{\Omega }{\textrm{curl}{{\varvec{J}}^{j}}\times {{\varvec{B}}^{j-1}}{\varvec{\varphi }} dx}+\int _{\Omega }{{{R}^{j+1}}\textrm{div}{\varvec{\varphi }} dx}. \end{aligned} \end{aligned}$$
Nextly, by subtracting \(\int _{\Omega }{(1+\left| D{{\varvec{u}}^{j}} \right| }{{)}^{p-2}}D{{\varvec{u}}^{j}}D{\varvec{\varphi }} dx\) from both sides of the above equality, we could obtain
$$\begin{aligned} \begin{aligned}&\int _{\Omega }{{{(1+\left| D{{\varvec{u}}^{j}} \right| )}^{p-2}}D{{\varvec{Q}}^{j+1}}D{\varvec{\varphi }} dx} \\&=\int _{\Omega }{\left[ {{\left( 1+\left| D{{\varvec{u}}^{j-1}} \right| \right) }^{p-2}}-{{\left( 1+\left| D{{\varvec{u}}^{j}} \right| \right) }^{p-2}}\right] D{{u}^{j}}D{\varvec{\varphi }} dx} \\&-\int _{\Omega }{\beta {{W}^{j+1}}\varvec{g}{\varvec{\varphi }} dx}-\int _{\Omega }{({{\varvec{Q}}^{j}}\cdot \nabla ){{\varvec{u}}^{j}}{\varvec{\varphi }} dx} -\int _{\Omega }({{\varvec{u}}^{j-1}}\cdot \nabla ){{\varvec{Q}}^{j}}{\varvec{\varphi }} dx \\&\quad +\int _{\Omega }\textrm{curl}{{\varvec{B}}^{j}}\times {{\varvec{J}}^{j}}{\varvec{\varphi }} dx \\&+\int _{\Omega }\textrm{curl}{{J}^{j}}\times {{\varvec{B}}^{j-1}}{\varvec{\varphi }} dx+\int _{\Omega }{{R}^{j+1}}\textrm{div}{\varvec{\varphi }} dx. \end{aligned} \end{aligned}$$
(24)
This identity, by a continuity argument, still holds with \(\forall ~{\varvec{\varphi }} \in V(\Omega )\), in which case the last term of (24) vanishes. Here, we recall that \({{\varvec{u}}^{j-1}}=0\), \({{\varvec{B}}^{j-1}}={{\varvec{B}}_{ini}}\) for \(j=0\) and then \({{\varvec{Q}}^{0}}={{\varvec{u}}^{0}}\), \({{\varvec{J}}^{0}}={{\varvec{B}}^{0}}-{{\varvec{B}}_{ini}}\).
Similarly, by taking \(m=j\) and \(m=j+1\), respectively, in the week formula of (8), then subtracting one from the other, we can get for \(\forall ~ {\varvec{\eta }} \in H_{0}^{1}(\Omega )\)
$$\begin{aligned} \begin{aligned} \int _{\Omega }\nabla {{\varvec{J}}^{j+1}}\nabla {\varvec{\eta }} dx&=\int _{\Omega }{({{\varvec{B}}^{j}}\cdot \nabla ){{\varvec{Q}}^{j+1}}{\varvec{\eta }} dx}\\&\quad +\int _{\Omega }{({{\varvec{J}}^{j}}\cdot \nabla ){{\varvec{u}}^{j}}{\varvec{\eta }} dx} -\int _{\Omega }{({{\varvec{Q}}^{j+1}}\cdot \nabla ){{\varvec{B}}^{j}}{\varvec{\eta }} dx}\\&\quad -\int _{\Omega }{({{\varvec{u}}^{j}}\cdot \nabla ){{\varvec{J}}^{j}}{\varvec{\eta }} dx}. \end{aligned} \end{aligned}$$
(25)
Then, for \(\forall \phi \in C_{0}^{\infty }(\Omega )\), we get
$$\begin{aligned} \begin{aligned} \int _{\Omega }{\nabla {{W}^{j+1}}\cdot \nabla \phi dx}&=\int _{\Omega }{(\nabla {{\varvec{B}}^{j}} +\nabla {{\varvec{B}}^{j-1}})\nabla {{\varvec{J}}^{j}}\cdot \phi dx}\\&\quad +\int _{\Omega }{2D({{\varvec{Q}}^{j}}) \cdot (D({{\varvec{u}}^{j}})+D({{\varvec{u}}^{j-1}}))\cdot \phi dx} \\&-\int _{\Omega }{{{\varvec{u}}^{j}}\cdot \nabla {{W}^{j}}\cdot \phi dx} -\int _{\Omega }{{{\varvec{Q}}^{j}}\cdot \nabla {{T}^{j-1}}\cdot \phi dx}. \end{aligned} \end{aligned}$$
(26)
Proposition 3.2
Assume that all the assumptions of Proposition 3.1 are satisfied and let \(\{{{\varvec{u}}^{m}}\}\), \(\{{{p}^{m}}\}\), \(\{{{\varvec{B}}^{m}}\}\) and \(\{{{T}^{m}}\}\) be the corresponding sequence. Then if
$$\begin{aligned} \begin{aligned}&(2-p+2c_{3}^{2}+2C{{c}_{3}})(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}) \\&\quad \cdot [1+2C{{c}_{3}}(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})] \\&\quad \cdot {{(1+2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})}^{2-p}}<1, \end{aligned} \end{aligned}$$
(27)
with K given by Proposition 3.1, the series \(\sum \limits _{m}{{{\varvec{Q}}^{m}}}\) converges to a function \({\varvec{Q}}\) in \({{W}^{1,2}}(\Omega )\), the series \(\sum \limits _{m}{{{R}^{m}}}\) converges to a function R in \({{L}^{2}}(\Omega )\), the series \(\sum \limits _{m}{{{\varvec{J}}^{m}}}\) converges to a function \({\varvec{J}}\) in \({{W}^{1,2}}(\Omega )\), the series \(\sum \limits _{m}{{{W}^{m}}}\) converges to a function W in \({{W}^{1,2}}(\Omega )\).
Proof
First, let us verify that the following estimates hold
$$\begin{aligned} \begin{aligned} (a).~~&{{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}}+{{\left\| {{\varvec{J}}^{1}} \right\| }_{1,2}}+{{\left\| {{W}^{1}} \right\| }_{1,2}} \\&\le (2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} \\&\quad +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}) [1+2C{{c}_{3}}(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} \\&\quad +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})] {{(1+2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})}^{2-p}} \\&\quad \cdot (2-p+2C{{c}_{3}}+3C{{c}_{3}}\beta {{\left\| \varvec{g} \right\| }_{q}}+6C)(1+2{{K}_{2}} {{\left\| \varvec{g} \right\| }_{q}}\\&\quad +2{{c}_{2}}{{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}} {{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}), \\ (b).~~&\text {if~,~for~} j\ge 1~,~\text {it~holds~that} \\&{{\left\| D{{\varvec{Q}}^{j}} \right\| }_{2}}+{{\left\| {{\varvec{J}}^{j}} \right\| }_{1,2}}+{{\left\| {{W}^{j}} \right\| }_{1,2}} \\&\le \frac{2-p+2C{{c}_{3}}+3C{{c}_{3}}\beta {{\left\| \varvec{g} \right\| }_{q}}+6C}{2-p+2c_{3}^{2} +2C{{c}_{3}}}\\&\quad (1+2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}} {{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}) \\&\quad \cdot \{(2-p+2c_{3}^{2}+2C{{c}_{3}})(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}} {{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}) \\&\quad {}[1+2C{{c}_{3}}(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}} \\&\quad +2{{c}_{2}}{{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}} {{\left\| \varvec{g} \right\| }_{q}})](1+2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}} +2{{c}_{2}}{{\left\| \psi \right\| }_{2}}\\&\quad +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})^{2-p}{{\}}^{j}}, \\ \text {then} \\&{{\left\| D{{\varvec{Q}}^{j+1}} \right\| }_{2}}+{{\left\| {{\varvec{J}}^{j+1}} \right\| }_{1,2}}+{{\left\| {{W}^{j+1}} \right\| }_{1,2}} \\&\le \frac{2-p+2C{{c}_{3}}+3C{{c}_{3}}\beta {{\left\| \varvec{g} \right\| }_{q}}+6C}{2-p+2c_{3}^{2} +2C{{c}_{3}}}\\&\quad (1+2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}}{{\left\| \psi \right\| }_{2}} +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}}) \\&\quad \cdot \{(2-p+2c_{3}^{2}+2C{{c}_{3}})(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}}+2{{c}_{2}} {{\left\| \psi \right\| }_{2}}\\&\quad +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}} {{\left\| \varvec{g} \right\| }_{q}})[1+2C{{c}_{3}}(2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}} \\&\quad +2{{c}_{2}}{{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}} {{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})](1+2{{K}_{2}} {{\left\| \varvec{g} \right\| }_{q}}\\&\quad +2{{c}_{2}}{{\left\| \psi \right\| }_{2}} +2CK\beta {{c}_{2}}{{\left\| \psi \right\| }_{2}}{{\left\| \varvec{g} \right\| }_{q}})^{2-p}{{\}}^{j+1}}. \end{aligned} \end{aligned}$$
(28)
\(\square\)
By the above arguments, setting \(j=0\) and testing with \({\varvec{Q}}^{1}\) in (24), we get
$$\begin{aligned} \begin{aligned} \int _{\Omega }{{{(1+\left| D{{\varvec{u}}^{0}} \right| )}^{p-2}}{{\left| D{{\varvec{Q}}^{1}} \right| }^{2}}dx}&=\int _{\Omega }{[1-{{(1+\left| D{{\varvec{u}}^{0}} \right| )}^{p-2}}]D{{\varvec{u}}^{0}}D{{\varvec{Q}}^{1}}dx} -\int _{\Omega }{\beta {{W}^{1}}\varvec{g}\cdot {{\varvec{Q}}^{1}}dx} \\&-\int _{\Omega }{({{\varvec{u}}^{0}}\cdot \nabla ){{\varvec{u}}^{0}}{{Q}^{1}}dx} +\int _{\Omega }{\textrm{curl}{{\varvec{B}}^{0}}\times {{\varvec{B}}^{0}}\cdot {{Q}^{1}}dx}. \end{aligned} \end{aligned}$$
Since \(p<2\), then
$$\begin{aligned} \int _{\Omega }{{{\bigg (1+\left| D{{\varvec{u}}^{0}} \right| \bigg )}^{p-2}}{{\left| D{{\varvec{Q}}^{1}} \right| }^{2}}dx}\ge & {} {{\bigg (1+{{\left\| D{{\varvec{u}}^{0}} \right\| }_{\infty }}\bigg )}^{p-2}}\left\| D{{\varvec{Q}}^{1}} \right\| _{2}^{2} \\\ge & {} {{\bigg (1+{{\left\| {{\varvec{u}}^{0}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}\bigg )}^{p-2}}\left\| D{\varvec{Q}}^{1} \right\| _{2}^{2}, \end{aligned}$$
using the Hölder inequality, Lemma 2.4 and 2.7, we obtain
$$\begin{aligned} \begin{aligned}&\left| {\int _\Omega {[1 - {{\bigg (1 + \left| {D{\varvec{u}^0}} \right| \bigg )}^{p - 2}}]D{\varvec{u}^0}D{{\varvec{Q}}^1}dx} } \right. - \int _\Omega \beta {W^1}\varvec{g} \cdot {{\varvec{Q}}^1}dx - \int _\Omega {\bigg ({\varvec{u}^0} \cdot \nabla \bigg ){\varvec{u}^0}{{\varvec{Q}}^1}dx} \\&\qquad \left. { + \int _\Omega {\textrm{curl}{\varvec{B}^0} \times {\varvec{B}^0} \cdot {{\varvec{Q}}^1}dx} } \right| \\&\le \bigg (2-p\bigg ){{\left\| D{{\varvec{u}}^{0}} \right\| }_{2}}{{\left\| D{{\varvec{u}}^{0}} \right\| }_{\infty }} {{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}}+\beta {{\left\| \varvec{g} \right\| }_{2}}{{\left\| {{\varvec{Q}}^{1}} \right\| }_{2}} {{\left\| {{W}^{1}} \right\| }_{\infty }}+{{\left\| {{\varvec{u}}^{0}} \right\| }_{\infty }} {{\left\| \nabla {{\varvec{u}}^{0}} \right\| }_{2}}{{\left\| {{\varvec{Q}}^{1}} \right\| }_{2}} \\&\qquad +{{\left\| \textrm{curl}{{\varvec{B}}^{0}}\times {{\varvec{B}}^{0}} \right\| }_{2}}{{\left\| {{\varvec{Q}}^{1}} \right\| }_{2}} \\&\le \bigg (2-p\bigg )\left\| {{\varvec{u}}^{0}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2}{{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}} +C{{c}_{3}}\beta {{\left\| \varvec{g} \right\| }_{2}}{{\left\| {{W}^{1}} \right\| }_{1,2}} {{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}}+C{{c}_{3}}\left\| {{\varvec{u}}^{0}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2} {{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}} \\&\qquad +C{{c}_{3}}\left\| {{\varvec{B}}^{0}} \right\| _{2,r}^{2}{{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}}, \end{aligned} \end{aligned}$$
hence
$$\begin{aligned} {{\left\| D{{\varvec{Q}}^{1}} \right\| }_{2}}\le & {} {{\bigg (1+{{\left\| {{\varvec{u}}^{0}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}\bigg )}^{2-p}} \cdot \left[ \bigg (2-p+C{{c}_{3}}\bigg )\left\| {{\varvec{u}}^{0}} \right\| _{{{C}^{1,{{\gamma }_{0}}}}}^{2} \right. \nonumber \\{} & {} \left. +C{{c}_{3}}\beta {{\left\| \varvec{g} \right\| }_{q}}{{\left\| {{W}^{1}} \right\| }_{1,2}} +C{{c}_{3}}\left\| {{\varvec{B}}^{0}} \right\| _{2,r}^{2}\right] . \end{aligned}$$
(29)
Next, setting \(j=0\) and testing with \({{\varvec{J}}^1}\) in (25), there have
$$\begin{aligned} \begin{aligned} \int _\Omega {{\left| {\nabla {{\varvec{J}}^1}} \right| }^2}dx&= \int _\Omega {({\varvec{B}^0} \cdot \nabla ){{\varvec{Q}}^1}{J^1}dx} + \int _\Omega {({\varvec{B}^0} \cdot \nabla ){\varvec{u}^0}{{\varvec{J}}^1}dx} \\&- \int _\Omega {({\varvec{Q}}^1} \cdot \nabla ){\varvec{B}^0}{{\varvec{J}}^1}dx - \int _\Omega ({\varvec{B}_{ini}} \cdot \nabla ){\varvec{u}^0}{{\varvec{J}}^1}dx - \int _\Omega ({\varvec{u}^0} \cdot \nabla ){\varvec{B}^0}{{\varvec{J}}^1}dx, \end{aligned} \end{aligned}$$
by the Poincáre inequality, it yields
$$\begin{aligned} \int _\Omega {{\left| {\nabla {{\varvec{J}}^1}} \right| }^2}dx = \left\| {\nabla {{\varvec{J}}^1}} \right\| _2^2 \ge C\left\| {{{\varvec{J}}^1}} \right\| _{1,2}^2, \end{aligned}$$
and by the Hölder inequality and embedding theorem, we get
$$\begin{aligned} \left| {\int _\Omega {({\varvec{u}^0} \cdot \nabla ){\varvec{B}^0}{{\varvec{J}}^1}dx} } \right| \le {\left\| {{\varvec{u}^0}} \right\| _2}{\left\| {\nabla {\varvec{B}^0}} \right\| _\infty }{\left\| {{{\varvec{J}}^1}} \right\| _2} \le C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{\varvec{B}^0}} \right\| _{2,r}}{\left\| {{{\varvec{J}}^1}} \right\| _{1,2}}, \end{aligned}$$
and
$$\begin{aligned} \begin{aligned}&\Big | \int _\Omega ({\varvec{B}^0} \cdot \nabla ){{\varvec{Q}}^1}{{\varvec{J}}^1}dx + \int _\Omega {({\varvec{B}^0} \cdot \nabla ){\varvec{u}^0}{{\varvec{J}}^1}dx} - \int _\Omega {({{\varvec{Q}}^1} \cdot \nabla ){\varvec{B}^0}{{\varvec{J}}^1}dx}\\&\quad - \int _\Omega {({\varvec{B}_{ini}} \cdot \nabla ){\varvec{u}^0}{{\varvec{J}}^1}dx} \Big | \\&\le {\left\| {{\varvec{B}^0}} \right\| _\infty }{\left\| {\nabla {{\varvec{Q}}^1}} \right\| _2} {\left\| {{{\varvec{J}}^1}} \right\| _2} + {\left\| {{\varvec{B}^0}} \right\| _\infty }{\left\| {\nabla {\varvec{u}^0}} \right\| _2} {\left\| {{{\varvec{J}}^1}} \right\| _2} + {\left\| {{{\varvec{Q}}^1}} \right\| _2}{\left\| {\nabla {\varvec{B}^0}} \right\| _3}{\left\| {{{\varvec{J}}^1}} \right\| _6} \\&\qquad + {\left\| {{\varvec{B}_{ini}}} \right\| _\infty }{\left\| {\nabla {\varvec{u}^0}} \right\| _2}{\left\| {{{\varvec{J}}^1}} \right\| _{1,2}} \\&\le 2C{c_3}{\left\| {{\varvec{B}^0}} \right\| _{2,r}}{\left\| {D{{\varvec{Q}}^1}} \right\| _2} {\left\| {{{\varvec{J}}^1}} \right\| _{1,2}} + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {{\varvec{B}^0}} \right\| _{2,r}}{\left\| {{{\varvec{J}}^1}} \right\| _{1,2}} + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{{\varvec{J}}^1}} \right\| _{1,2}}, \end{aligned} \end{aligned}$$
which obviously implies
$$\begin{aligned} \begin{aligned} {\left\| {{{\varvec{J}}^1}} \right\| _{1,2}} \le 2C{c_3}{\left\| {{\varvec{B}^0}} \right\| _{2,r}}{\left\| {D{{\varvec{Q}}^1}} \right\| _2} + 2C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{\varvec{B}^0}} \right\| _{2,r}} + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}. \end{aligned} \end{aligned}$$
(30)
Then, setting \(j=0\) and testing with \({W^1}\) in (26), there is
$$\int _\Omega {{{\left| {\nabla {W^1}} \right| }^2}dx} = \int _\Omega {\nabla {\varvec{B}^0}\nabla {{\varvec{J}}^0}{W^1}dx} + \int _\Omega {2D({{\varvec{Q}}^0})D({\varvec{u}^0}){W^1}dx} - \int _\Omega {{\varvec{u}^0}\nabla {T^0}{W^1}dx},$$
by the Poincáre inequality, it yields
$$\int _\Omega {{{\left| {\nabla {W^1}} \right| }^2}dx} = \left\| {\nabla {W^1}} \right\| _2^2 \ge C\left\| {{W^1}} \right\| _{1,2}^2,$$
and by the Hölder inequality and embedding theorem, we get
$$\begin{aligned} \begin{aligned}&\Big | \int _\Omega \nabla {\varvec{B}^0}\nabla {{\varvec{J}}^0}{W^1}dx + \int _\Omega {2D({{\varvec{Q}}^0})D({\varvec{u}^0}){W^1}dx} - \int _\Omega {{\varvec{u}^0}\nabla {T^0}{W^1}dx} \Big | \\&\le {\left\| {\nabla {\varvec{B}^0}} \right\| _\infty }{\left\| {\nabla {\varvec{B}^0}} \right\| _2}{\left\| {{W^1}} \right\| _2} + 2{\left\| {D({\varvec{u}^0})} \right\| _2}{\left\| {D({\varvec{u}^0})} \right\| _\infty }{\left\| {{W^1}} \right\| _2} + {\left\| {{\varvec{u}^0}} \right\| _\infty }{\left\| {\nabla {T^0}} \right\| _2}{\left\| {{W^1}} \right\| _2}\\&\le C\left\| {{\varvec{B}^0}} \right\| _{2,r}^2{\left\| {{W^1}} \right\| _{1,2}} + C\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}^2 {\left\| {{W^1}} \right\| _{1,2}} + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {{T^0}} \right\| _{2,2}}{\left\| {{W^1}} \right\| _{1,2}}, \end{aligned} \end{aligned}$$
which obviously implies
$$\begin{aligned} {\left\| {{W^1}} \right\| _{1,2}} \le C\left\| {{\varvec{B}^0}} \right\| _{2,r}^2 + C\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{T^0}} \right\| _{2,2}}.
\end{aligned}$$
(31)
Combining (29), (30) and (31) and using estimate (10), we obtain
$$\begin{aligned} \begin{aligned}&{\left\| {D{{\varvec{Q}}^1}} \right\| _2} + {\left\| {{{\varvec{J}}^1}} \right\| _{1,2}} + {\left\| {{W^1}} \right\| _{1,2}} \\&\le {\left( 1 + {\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}\right) ^{2 - p}} \cdot \left[ (2 - p + C{c_3}) \left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C\beta {c_3}{\left\| \varvec{g} \right\| _q} {\left\| {{W^1}} \right\| _{1,2}} + C{c_3}\left\| {{\varvec{B}^0}} \right\| _{2,r}^2\right] \\&\quad + 2C{c_3}{\left\| {{\varvec{B}^0}} \right\| _{2,r}}{\left\| {D{{\varvec{Q}}^1}} \right\| _2} + 2C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{\varvec{B}^0}} \right\| _{2,r}} + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}} + C\left\| {{\varvec{B}^0}} \right\| _{2,r}^2 \\&\quad + C\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C{\left\| {{\varvec{u}^0}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{T^0}} \right\| _{2,2}} \\&\le (2 - p + 2C{c_3} + 3C\beta {c_3}{\left\| \varvec{g} \right\| _q} + 6C)(1 + 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}) \\&\quad \cdot (2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})[1 + 2C{c_3}(2{K_2} {\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} \\&\quad + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})] \cdot {(1 + 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})^{2 - p}}, \end{aligned} \end{aligned}$$
this completes the proof of (a).
Now we prove (b). Assume that the hypothesis in (b) holds. As for (a), by setting \(j \ge 1\) and \({\varvec{\varphi }} = {\varvec{Q}}^{j + 1} \in V(\Omega )\) in (24), we get
$$\begin{aligned} \begin{aligned}&\int _\Omega {{\bigg (1 + \left| {D{\varvec{u}^j}} \right| \bigg )}^{p - 2}}{{\left| {D{{\varvec{Q}}^{j + 1}}} \right| }^2}dx \\&= \int _\Omega {[{{\bigg (1 + \left| {D{\varvec{u}^{j - 1}}} \right| \bigg )}^{p - 2}} - {{\bigg (1 + \left| {D{\varvec{u}^j}} \right| \bigg )}^{p - 2}}]D{\varvec{u}^j}D{{\varvec{Q}}^{j + 1}}dx} \\&- \int _\Omega {\beta {W^{j + 1}}\varvec{g}{{\varvec{Q}}^{j + 1}}dx} - \int _\Omega \bigg ({\varvec{Q}}^j \cdot \nabla \bigg ){\varvec{u}^j}{{\varvec{Q}}^{j + 1}}dx \\&\quad - \int _\Omega {\bigg ({\varvec{u}^{j - 1}} \cdot \nabla \bigg ){{\varvec{Q}}^j}{{\varvec{Q}}^{j + 1}}dx} \\&+ \int _\Omega {\textrm{curl}{\varvec{B}^j} \times {{\varvec{J}}^j}{{\varvec{Q}}^{j + 1}}dx} + \int _\Omega {\textrm{curl}{{\varvec{J}}^j} \times {\varvec{B}^{j - 1}}{{\varvec{Q}}^{j + 1}}dx}. \end{aligned} \end{aligned}$$
Since \(p<2\), we get
$$\begin{aligned} \int _\Omega {{{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}{{\left| {D{{\varvec{Q}}^{j + 1}}} \right| }^2}dx}\ge & {} {\left( 1 + {\left\| {D{\varvec{u}^j}} \right\| _\infty }\right) ^{p - 2}}\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2^2 \\\ge & {} {\left( 1 + {\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}\right) ^{p - 2}}\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2^2, \end{aligned}$$
then the Hölder inequality, and Lemmas 2.4 and 2.7 yield that
$$\begin{aligned}{} & {} \begin{aligned}&\left| \int _\Omega {\left[ {{\left( 1 + \left| {D{\varvec{u}^{j - 1}}} \right| \right) }^{p - 2}} - {{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}\right] D{\varvec{u}^j}D{{\varvec{Q}}^{j + 1}}dx}\right. \\&\quad \left. - \int _\Omega {\beta {W^{j + 1}}\varvec{g}{{\varvec{Q}}^{j + 1}}dx} \right. \left. { - \int _\Omega {({{\varvec{Q}}^j} \cdot \nabla ){\varvec{u}^j}{{\varvec{Q}}^{j + 1}}dx} } \right| \\&\le (2 - p){\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {D{\varvec{u}^j}} \right\| _\infty }{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} + \beta {\left\| \varvec{g} \right\| _2}{\left\| {{W^{j + 1}}} \right\| _\infty }{\left\| {{{\varvec{Q}}^{j + 1}}} \right\| _2}\\&\quad + {\left\| {{{\varvec{Q}}^j}} \right\| _2}{\left\| {\nabla {\varvec{u}^j}} \right\| _\infty }{\left\| {{{\varvec{Q}}^{j + 1}}} \right\| _2} \\&\le (2 - p){\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} + C\beta {c_3} {\left\| \varvec{g} \right\| _q}{\left\| {{W^{j + 1}}} \right\| _{1,2}}{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} \\&\quad + c_3^2{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2}, \end{aligned} \\{} & {} \begin{aligned}&\left| { - \int _\Omega {({\varvec{u}^{j - 1}} \cdot \nabla ){{\varvec{Q}}^j}{{\varvec{Q}}^{j + 1}}dx} + \int _\Omega {\textrm{curl}{\varvec{B}^j} \times {{\varvec{J}}^j}{{\varvec{Q}}^{j + 1}}dx} + \int _\Omega {\textrm{curl}{J^j} \times {B^{j - 1}}{{\varvec{Q}}^{j + 1}}dx} } \right| \\&\le {\left\| {{\varvec{u}^{j - 1}}} \right\| _\infty }{\left\| {\nabla {{\varvec{Q}}^j}} \right\| _2} {\left\| {{{\varvec{Q}}^{j + 1}}} \right\| _2} + {\left\| {\textrm{curl}{\varvec{B}^j}} \right\| _3}{\left\| {{{\varvec{J}}^j}} \right\| _6} {\left\| {{{\varvec{Q}}^{j + 1}}} \right\| _2} \\&\quad + {\left\| {\textrm{curl}{{\varvec{J}}^j}} \right\| _2} {\left\| {{\varvec{B}^{j - 1}}} \right\| _\infty }{\left\| {{{\varvec{Q}}^{j + 1}}} \right\| _2} \\&\le c_3^2{\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {D{{\varvec{Q}}^j}} \right\| _2} {\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} \\&\quad + C{c_3}\left( {\left\| {{\varvec{B}^j}} \right\| _{2,r}} + {\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}}\right) {\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2}, \end{aligned} \end{aligned}$$
hence,
$$\begin{aligned} \begin{aligned}&{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} \le {\left( 1 + {\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}\right) ^{2 - p}} \cdot \left[ \left( 2 - p + c_3^2\right) {\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} + C\beta {c_3}{\left\| \varvec{g} \right\| _q}{\left\| {{W^{j + 1}}} \right\| _{1,2}}\right. \\&\quad \left. + c_3^2{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}} + C{c_3} \left( {\left\| {{\varvec{B}^j}} \right\| _{2,r}} + {\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}}\right) {\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}\right] . \end{aligned} \end{aligned}$$
(32)
Then setting \(j \ge 1\) and testing with \({{\varvec{J}}^{j + 1}}\) in (25), we get
$$\begin{aligned} \begin{aligned} \int _\Omega {{{\left| {\nabla {{\varvec{J}}^{j + 1}}} \right| }^2}dx}&= \int _\Omega {({\varvec{B}^j} \cdot \nabla ){{\varvec{Q}}^{j + 1}}{{\varvec{J}}^{j + 1}}dx} + \int _\Omega {({{\varvec{J}}^j} \cdot \nabla ){\varvec{u}^j}{{\varvec{J}}^{j + 1}}dx} \\&- \int _\Omega {({{\varvec{Q}}^{j + 1}} \cdot \nabla ){\varvec{B}^j}{{\varvec{J}}^{j + 1}}dx} - \int _\Omega {({\varvec{u}^j} \cdot \nabla ){{\varvec{J}}^j}{{\varvec{J}}^{j + 1}}dx}, \end{aligned} \end{aligned}$$
using the Poincaáre inequality, Hölder inequality and Lemma 2.4, we get
$$\begin{aligned}{} & {} \int _{\Omega }{{{\left| \nabla {{J}^{j+1}} \right| }^{2}}dx}=\left\| \nabla {{J}^{j+1}} \right\| _{2}^{2}\ge C\left\| {{J}^{j+1}} \right\| _{1,2}^{2}, \\{} & {} \begin{aligned}&\Big | \int _\Omega {({\varvec{B}^j} \cdot \nabla ){{\varvec{Q}}^{j + 1}}{{\varvec{J}}^{j + 1}}dx} + \int _\Omega {({{\varvec{J}}^j} \cdot \nabla ){\varvec{u}^j}{{\varvec{J}}^{j + 1}}dx} \\&\quad - \int _\Omega {({{\varvec{Q}}^{j + 1}} \cdot \nabla ){\varvec{B}^j}{{\varvec{J}}^{j + 1}}dx} { - \int _\Omega {({\varvec{u}^j} \cdot \nabla ){{\varvec{J}}^j}{{\varvec{J}}^{j + 1}}dx} } \Big | \\&\le {\left\| {{\varvec{B}^j}} \right\| _\infty }{\left\| {\nabla {{\varvec{Q}}^{j + 1}}} \right\| _2}{\left\| {{{\varvec{J}}^{j + 1}}} \right\| _2} + {\left\| {{J^j}} \right\| _2}{\left\| {\nabla {\varvec{u}^j}} \right\| _\infty }{\left\| {{{\varvec{J}}^{j + 1}}} \right\| _2} + {\left\| {{{\varvec{Q}}^{j + 1}}} \right\| _2}{\left\| {\nabla {\varvec{B}^j}} \right\| _3}{\left\| {{{\varvec{J}}^{j + 1}}} \right\| _6} \\&\quad + {\left\| {{\varvec{u}^j}} \right\| _\infty }{\left\| {\nabla {{\varvec{J}}^j}} \right\| _2}{\left\| {{{\varvec{J}}^{j + 1}}} \right\| _2} \\&\le 2C{c_3}{\left\| {{\varvec{B}^j}} \right\| _{2,r}}{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2}{\left\| {{{\varvec{J}}^{j + 1}}} \right\| _{1,2}} + 2C{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{{\varvec{J}}^{j + 1}}} \right\| _{1,2}}, \end{aligned} \end{aligned}$$
hence,
$$\begin{aligned} {\left\| {{{\varvec{J}}^{j + 1}}} \right\| _{1,2}} \le 2C{c_3}{\left\| {{\varvec{B}^j}} \right\| _{2,r}} {\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} + 2C{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}. \end{aligned}$$
(33)
Next, setting \(j \ge 1\) and testing with \({W^{j + 1}}\) in (26), we get
$$\begin{aligned} \begin{aligned} \int _\Omega {{{\left| {\nabla {W^{j + 1}}} \right| }^2}dx}&= \int _\Omega {(\nabla {\varvec{B}^j} + \nabla {\varvec{B}^{j - 1}})\nabla {{\varvec{J}}^j}{W^{j + 1}}dx} - \int _\Omega {{\varvec{u}^j} \cdot \nabla {W^j}{W^{j + 1}}dx} \\&+ \int _\Omega {2D({{\varvec{Q}}^j})(D({\varvec{u}^j}) + D({\varvec{u}^{j - 1}})){W^{j + 1}}dx}- \int _\Omega {{{\varvec{Q}}^j} \cdot \nabla {T^{j - 1}}{W^{j + 1}}dx}, \end{aligned} \end{aligned}$$
using the Poincáre inequality, Hölder inequality and embedding theorem, we get
$$\begin{aligned}{} & {} \int _\Omega {{{\left| {\nabla {W^{j + 1}}} \right| }^2}dx} = \left\| {\nabla {W^{j + 1}}} \right\| _2^2 \ge C\left\| {{W^{j + 1}}} \right\| _{1,2}^2, \\{} & {} \begin{aligned}&\left| {\int _\Omega {(\nabla {\varvec{B}^j} + \nabla {\varvec{B}^{j - 1}})\nabla {{\varvec{J}}^j}{W^{j + 1}}dx} + \int _\Omega {2D({{\varvec{Q}}^j})(D({\varvec{u}^j}) + D({\varvec{u}^{j - 1}})){W^{j + 1}}dx} } \right. \\&\quad \left. { - \int _\Omega {{\varvec{u}^j} \cdot
\nabla {W^j}{W^{j + 1}}dx} - \int _\Omega {{{\varvec{Q}}^j} \cdot \nabla {T^{j - 1}}{W^{j + 1}}dx} } \right| \\&\le {\left\| {\nabla {\varvec{B}^j}} \right\| _\infty }{\left\| {\nabla {{\varvec{J}}^j}} \right\| _2}{\left\| {{W^{j + 1}}} \right\| _2} + {\left\| {\nabla {\varvec{B}^{j - 1}}} \right\| _\infty }{\left\| {\nabla {{\varvec{J}}^j}} \right\| _2}{\left\| {{W^{j + 1}}} \right\| _2} + 2{\left\| {D({{\varvec{Q}}^j})} \right\| _2}({\left\| {D({\varvec{u}^j})} \right\| _\infty } \\&\quad + {\left\| {D({\varvec{u}^{j - 1}})} \right\| _\infty }){\left\| {{W^{j + 1}}} \right\| _2} + {\left\| {{\varvec{u}^j}} \right\| _\infty }{\left\| {\nabla {W^j}} \right\| _2}{\left\| {{W^{j + 1}}} \right\| _2} + {\left\| {{{\varvec{Q}}^j}} \right\| _\infty }{\left\| {\nabla {T^{j - 1}}} \right\| _2}{\left\| {{W^{j + 1}}} \right\| _2} \\&\le C\left\| {{\varvec{B}^j}} \right\| _{2,r}{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}{\left\| {{W^{j + 1}}} \right\| _{1,2}} + C\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}{\left\| {{W^{j + 1}}} \right\| _{1,2}} + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}({\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} \\&\quad + {\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}}){\left\| {{W^{j + 1}}} \right\| _{1,2}} + C{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{W^j}} \right\| _{1,2}}{\left\| {{W^{j + 1}}} \right\| _{1,2}} \\&\quad + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{T^{j - 1}}} \right\| _{2,2}}{\left\| {{W^{j + 1}}} \right\| _{1,2}}, \end{aligned} \end{aligned}$$
which obviously implies
$$\begin{aligned} \begin{aligned} {\left\| {{W^{j + 1}}} \right\| _{1,2}}&\le C\left\| {{\varvec{B}^j}} \right\| _{2,r}{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}} + C\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}} + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} \\&+ C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}} + C{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{W^j}} \right\| _{1,2}} + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{T^{j - 1}}} \right\| _{2,2}}. \end{aligned} \end{aligned}$$
(34)
Combining (32), (33) and (34) and appealing to (10), we get
$$\begin{aligned} \begin{aligned}&{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} + {\left\| {{{\varvec{J}}^{j + 1}}} \right\| _{1,2}} + {\left\| {{W^{j + 1}}} \right\| _{1,2}} \\&\le {(1 + {\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}})^{2 - p}}\left[ (2 - p + c_3^2){\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} + C{c_3}{\left\| \varvec{g} \right\| _q}{\left\| {{W^{j + 1}}} \right\| _{1,2}}\right. \\&\quad \left. + c_3^2{\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {D{{\varvec{Q}}^j}} \right\| _2} + C{c_3}({\left\| {{\varvec{B}^j}} \right\| _{2,r}} + {\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}}){\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}\right] \\&\quad + 2C{c_3}{\left\| {{\varvec{B}^j}} \right\| _{2,r}}{\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} \\&\quad + 2C{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} + C{\left\| {{\varvec{B}^j}} \right\| _{2,r}}{\left\| {{{\varvec{J}}^j}} \right\| _{1,2}} \\&\quad + C{\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}} {\left\| {{{\varvec{J}}^j}} \right\| _{1,2}} + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}} \\&\quad + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2}{\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}} + C{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{W^j}} \right\| _{1,2}} + C{\left\| {D{{\varvec{Q}}^j}} \right\| _2} {\left\| {{T^{j - 1}}} \right\| _{1,2}} \\&\le \left[ 1 + 2C{c_3}(2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})\right] \left( {1 + 2{K_2}{{\left\| \varvec{g} \right\| }_q} + 2{c_2}{{\left\| \psi \right\| }_2}} \right. \\&\quad {\left. { + 2CK\beta {c_2}{{\left\| \psi \right\| }_2}{{\left\| \varvec{g} \right\| }_q}} \right) ^{2 - p}} \cdot (2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2} {\left\| \varvec{g} \right\| _q})\\&\quad (2 - p + 2c_3^2 + C\beta {c_3}{\left\| \varvec{g} \right\| _q} \\&\quad + 2C{c_3} + C)\left( {\left\| {D{{\varvec{Q}}^j}} \right\| _2} + {\left\| {{J^j}} \right\| _{1,2}} + {\left\| {{W^j}} \right\| _{1,2}}\right) , \end{aligned} \end{aligned}$$
which gives (28) via the hypothesis in (b). Therefore, by induction, (28) holds for any given \(j \in \mathbb {N}\).
By assumption (27), the series \(\sum \limits _j {({{\left\| {D{{\varvec{Q}}^j}} \right\| }_2} + {{\left\| {{{\varvec{J}}^j}} \right\| }_{1,2}} + {{\left\| {{W^j}} \right\| }_{1,2}})}\) converges. Since \(\sum \limits _j {{{\left\| {D{{\varvec{Q}}^j}} \right\| }_2}}\), \(\sum \limits _j {{{\left\| {{{\varvec{J}}^j}} \right\| }_{1,2}}}\) and \(\sum \limits _j {{{\left\| {{W^j}} \right\| }_{1,2}}}\) are positive series, both \(\sum \limits _j {{{\left\| {D{{\varvec{Q}}^j}} \right\| }_2}}\), \(\sum \limits _j {{{\left\| {{{\varvec{J}}^j}} \right\| }_{1,2}}}\) and \(\sum \limits _j {{{\left\| {{W^j}} \right\| }_{1,2}}}\) converge. Therefore, by the completeness of \({W^{1,2}}(\Omega )\), there follows the convergence of the series \(\sum \limits _j {{{\varvec{Q}}^j}(x)}\), \(\sum \limits _j {{{\varvec{J}}^j}(x)}\) and \(\sum \limits _j {{W^j}(x)}\) in the norm \({\left\| \cdot \right\| _{1,2}}\) to a function \({\varvec{Q}}(x) \in {W^{1,2}}(\Omega )\), \({\varvec{J}}(x) \in {W^{1,2}}(\Omega )\) and \(W(x) \in {W^{1,2}}(\Omega )\), respectively.
By (24), the following identity holds in the distributional sense:
$$\begin{aligned} \begin{aligned} \nabla {R^{j + 1}}&= \nabla \cdot \left[ {\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) ^{p - 2}}D{{\varvec{Q}}^{j + 1}}\right] - \nabla \cdot \{ [{(1 + \left| {D{\varvec{u}^{j - 1}}} \right| )^{p - 2}} \\&\quad - {(1 + \left| {D{\varvec{u}^j}} \right| )^{p - 2}}]D{\varvec{u}^j}\} \\&+ \beta {W^{j + 1}}\varvec{g} - ({{\varvec{Q}}^j} \cdot \nabla ){\varvec{u}^j} - ({\varvec{u}^{j - 1}} \cdot \nabla ){{\varvec{Q}}^j} \\&\quad + \textrm{curl}{\varvec{B}^j} \times {{\varvec{J}}^j} + \textrm{curl}{{\varvec{J}}^j} \times {\varvec{B}^{j - 1}}. \end{aligned} \end{aligned}$$
(35)
To get estimates on the \({L^2}\) \(-\)norm of \({R^{j + 1}}\), by Lemma 2.5 it is sufficient to estimate the \({W^{-1,2}}\) \(-\)norm of the right-hand side of the previous equations.
$$\begin{aligned} \begin{aligned}&{\left\| {\nabla \cdot \left[ {{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}D{{\varvec{Q}}^{j + 1}}\right] - \nabla \cdot \left\{ \left[ {{(1 + \left| {D{\varvec{u}^{j - 1}}} \right| )}^{p - 2}} - {{(1 + \left| {D{\varvec{u}^j}} \right| )}^{p - 2}}\right] D{\varvec{u}^j} \right\} } \right\| _{ - 1,2}} \\&\le {\left\| {{{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}D{{\varvec{Q}}^{j + 1}} - \left[ {{\left( 1 + \left| {D{\varvec{u}^{j - 1}}} \right| \right) }^{p - 2}} - {{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}\right] D{\varvec{u}^j}} \right\| _2} \\&\le {\left\| {{{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}D{{\varvec{Q}}^{j + 1}}} \right\| _2} + {\left\| {\left[ {{\left( 1 + \left| {D{\varvec{u}^{j - 1}}} \right| \right) }^{p - 2}} - {{\left( 1 + \left| {D{\varvec{u}^j}} \right| \right) }^{p - 2}}\right] D{\varvec{u}^j}} \right\| _2} \\&\le {\left\| {D{{\varvec{Q}}^{j + 1}}} \right\| _2} + (2 - p){\left\| {D{{\varvec{Q}}^j}} \right\| _2} {\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}. \end{aligned} \end{aligned}$$
Then,
$$\begin{aligned} {\left\| \beta {W^{j + 1}}{\varvec{g}} \right\| _{ - 1,2}} \le {\left\| {\beta {W^{j + 1}}}\varvec{g} \right\| _2} \le \beta {\left\| \varvec{g} \right\| _q}{\left\| {{W^{j + 1}}} \right\| _\infty } \le C\beta {\left\| \varvec{g} \right\| _q}{\left\| {{W^{j + 1}}} \right\| _{1,2}}. \end{aligned}$$
Finally, using Lemma 2.4
$$\begin{aligned} \begin{aligned}&{\left\| { - ({\varvec{Q}^j} \cdot \nabla ){\varvec{u}^j} - ({\varvec{u}^{j - 1}} \cdot \nabla ){\varvec{Q}^j} + \textrm{curl}{\varvec{B}^j} \times {J^j} + \textrm{curl}{\varvec{J}^j} \times {\varvec{B}^{j - 1}}} \right\| _{-1,2}} \\&\le {\left\| { - ({\varvec{Q}^j} \cdot \nabla ){\varvec{u}^j} - ({\varvec{u}^{j - 1}} \cdot \nabla ){\varvec{Q}^j} + \textrm{curl}{\varvec{B}^j} \times {\varvec{J}^j} + \textrm{curl}{\varvec{J}^j} \times {\varvec{B}^{j - 1}}} \right\| _2} \\&\le {\left\| {{\varvec{Q}^j}} \right\| _2}{\left\| {\nabla {\varvec{u}^j}} \right\| _\infty } + {\left\| {{\varvec{u}^{j - 1}}} \right\| _\infty }{\left\| {\nabla {\varvec{Q}^j}} \right\| _2} + {\left\| {\textrm{curl}{\varvec{B}^j}} \right\| _3}{\left\| {{\varvec{J}^j}} \right\| _6} + {\left\| {\textrm{curl}{\varvec{J}^j}} \right\| _2}{\left\| {{\varvec{B}^{j - 1}}} \right\| _\infty } \\&\le {c_3}{\left\| {{\varvec{u}^j}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {D{\varvec{Q}^j}} \right\| _2} + {c_3}{\left\| {{\varvec{u}^{j - 1}}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {D{\varvec{Q}^j}} \right\| _2} + C{\left\| {{\varvec{B}^j}} \right\| _{2,r}}{\left\| {{\varvec{J}^j}} \right\| _{1,2}} \\&\quad + C{\left\| {{\varvec{B}^{j - 1}}} \right\| _{2,r}}{\left\| {{\varvec{J}^j}} \right\| _{1,2}}. \end{aligned} \end{aligned}$$
Combining all these above and taking into account estimates (28) and (10), straightforward calculations lead to
$$\begin{aligned} \begin{aligned} {\left\| {{R^{j + 1}}} \right\| _2}&\le C{\left\| {\nabla {R^{j + 1}}} \right\| _{ - 1,2}} \\&\le C\left[ {{\left\| {D{\varvec{Q}^{j + 1}}} \right\| }_2} + \left( {2 - p + {c_3}} \right) {{\left\| {{\varvec{u}^j}} \right\| }_{{C^{1,{\gamma _0}}}}}{{\left\| {D{\varvec{Q}^j}} \right\| }_2} + C\beta {{\left\| \varvec{g} \right\| }_q}{{\left\| {{W^{j + 1}}} \right\| }_{1,2}} \right. \\&\quad \left. + {c_3}{{\left\| {{\varvec{u}^{j - 1}}} \right\| }_{{C^{1,{\gamma _0}}}}}{{\left\| {D{\varvec{Q}^j}} \right\| }_2} \right. \\&\quad \left. + C{{\left\| {{\varvec{B}^j}} \right\| }_{2,r}}{{\left\| {{J^j}} \right\| }_{1,2}} + C{{\left\| {{\varvec{B}^{j - 1}}} \right\| }_{2,r}}{{\left\| {{\varvec{J}^j}} \right\| }_{1,2}} \right] \\&\left. \le C\left( {2 - p + 2c_3^2 + 6C\beta {c_3}{{\left\| \varvec{g} \right\| }_q} + 2C{c_3} + 2{c_3} + 2C + 6C\beta {{\left\| \varvec{g} \right\| }_q}} \right) \right. \\&\quad \left( {1 + 2{K_2}{{\left\| \varvec{g} \right\| }_q} + 2{c_2}{{\left\| \psi \right\| }_2}} \right. \\&\quad {\left. { + 2CK\beta {c_2}{{\left\| \psi \right\| }_2}{{\left\| \varvec{g} \right\| }_q}} \right) ^{2 - p}} \left( {2{K_2}{{\left\| \varvec{g} \right\| }_q} + 2{c_2}{{\left\| \psi \right\| }_2} + 2CK\beta {c_2}{{\left\| \psi \right\| }_2} {{\left\| \varvec{g} \right\| }_q}} \right) \cdot \\&\quad \left( {{{\left\| {D{\varvec{Q}^j}} \right\| }_2} + {{\left\| {{\varvec{J}^j}} \right\| }_{1,2}}} { + {{\left\| {{W^j}} \right\| }_{1,2}}} \right) \\&\le \frac{{\left( 2 - p + 2C{c_3} + 3C\beta {c_3}{{\left\| \varvec{g} \right\| }_q} + 6C\right) \left( 2 - p + 2c_3^2 + 6C\beta {c_3}{{\left\| \varvec{g} \right\| }_q} + 2C{c_3} + 2{c_3} + 2C + 6C\beta {{\left\| \varvec{g} \right\| }_q} \right) }}{{2 - p + 2c_3^2 + 2C{c_3}}} \\&\quad \times {\left( 1 + 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}\right) ^{3 - p}}\\&\quad \left( 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}\right) \\&\quad \cdot \left\{ {\left( 2 - p + 2c_3^2 + 2C{c_3}\right) \left( 2{K_2}{{\left\| \varvec{g} \right\| }_q} + 2{c_2}{{\left\| \psi \right\| }_2} + 2CK\beta {c_2}{{\left\| \psi \right\| }_2} {{\left\| \varvec{g} \right\| }_q}\right) } \right. \\&\quad {}\left[ 1 + 2C{c_3}\left( 2{K_2}{\left\| \varvec{g} \right\| _q} \right. \right. \\&\quad \left. \left. + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2} {\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})\right) \right] \\&\quad \times \left( {1 + 2{K_2}{{\left\| \varvec{g} \right\| }_q} + 2{c_2}{{\left\| \psi \right\| }_2}} \right. {\left. { + 2CK\beta {c_2} {{\left\| \psi \right\| }_2}{{\left\| \varvec{g} \right\| }_q}} \right) ^{2 - p}}{\} ^j}. \end{aligned} \end{aligned}$$
Hence, using again bound (27), we can state that there exists a function \(R(x) \in {L^2}(x)\) to which the series \(\sum \limits _j {{R^j}(x)}\) converges in the \({L^2}\) \(-\)norm.
3.3 Existence Results
Set \({\left\| \varvec{g} \right\| _q}\) and \({\left\| \psi \right\| _2}\) are small enough, and the following conditions are satisfied
$$\begin{aligned} \begin{aligned}&{\left\| \varvec{g} \right\| _q}< 1, \\&2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2} {\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}< \frac{1}{{2{K_1}}}, \\&\left( 2 - p + 2c_3^2 + 2C{c_3}\right) \left( 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}\right) \left[ 1 + 2C{c_3}(2{K_2}{\left\| \varvec{g} \right\| _q}\right. \\&\left. + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2} {\left\| \varvec{g} \right\| _q})\right] \\&\quad {\left( 1 + 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}\right) ^{2 - p}} < 1. \end{aligned} \end{aligned}$$
(36)
Since the sequences \(\{ {\varvec{u}^m}\}\), \(\{ {P^m}\}\), \(\{ {\varvec{B}^m}\}\) and \(\{ {T^m}\}\) constructed in Proposition 3.1 satisfy the following relations: \({\varvec{u}^m}(x) = \sum \limits _{j = 1}^m {{Q^j}(x)} + {\varvec{u}^0}(x)\), \({P^m}(x) = \sum \limits _{j = 1}^m {{R^j}(x)} + {P^0}(x)\), \({\varvec{B}^m}(x) = \sum \limits _{j = 1}^m {{J^j}(x)} + {\varvec{B}^0}(x)\), \({T^m}(x) = \sum \limits _{j = 1}^m {{W^j}(x)} + {T^0}(x)\), setting \(\varvec{u}(x) = \varvec{Q}(x) + {\varvec{u}^0}(x)\), \(P(x) = R(x) + {P^0}(x)\), \(\varvec{B}(x) = \varvec{J}(x) + {\varvec{B}^0}(x)\) and \(T(x) = W(x) + {T^0}(x)\) with \(\varvec{Q}(x)\), R(x), \(\varvec{J}(x)\) and W(x), as in Proposition 3.2, the sequences \(\{ {\varvec{u}^m}(x)\}\), \(\{ {P^m}(x)\}\), \(\{ {\varvec{B}^m}(x)\}\) and \(\{ {T^m}(x)\}\) converge to the functions \(\varvec{u}(x)\), P(x), \(\varvec{B}(x)\) and T(x), respectively, in the \({W^{1,2}}\), \({L^2}\), \({W^{1,2}}\) and \({W^{1,2}}\) \(-\)norm. On the other hand, recalling Proposition 3.1, by Arzelà−Ascoli theorem, there exists a subsequence \(\{ {\varvec{u}^{{k_m}}}\}\) converging in \({C^{1,\gamma }}({{\overline{\Omega }}} )\), hence in \({W^{1,2}}(\Omega )\), to a function \(\tilde{\varvec{u}}\). Since all the sequence \(\{ {\varvec{u}^m}\}\) converges to \(\varvec{u}\) in \({W^{1,2}}(\Omega )\), then \(\varvec{u} = \tilde{\varvec{u}}\). In the same way, one can prove that \(P \in {C^{0,\gamma }}({{\overline{\Omega }}} )\), \(\varvec{B} \in {W^{2,r}}(\Omega )\) and \(T \in {W^{2,2}}(\Omega )\). Since \(\nabla \cdot {\varvec{u}^{{k_m}}} = 0\), \({\varvec{u}^{{k_m}}}{|_{\partial \Omega }} = 0\), \(\nabla \cdot {\varvec{B}^{{k_m}}} = 0\), \({\varvec{B}^{{k_m}}} \cdot \varvec{n}{|_{\partial \Omega }} = 0\), \(\textrm{curl}{\varvec{B}^{{k_m}}} \times \varvec{n}{|_{\partial \Omega }} = 0\) and \({T^{{k_m}}}{|_{\partial \Omega }} = 0\), for any \(m \in \mathbb {N}\), it follows that \(\nabla \cdot \varvec{u} = 0\), \(\varvec{u}{|_{\partial \Omega }} = 0\), \(\nabla \cdot \varvec{B} = 0\), \(\varvec{B} \cdot \varvec{n}{|_{\partial \Omega }} = 0\), \(\textrm{curl}\varvec{B} \times \varvec{n}{|_{\partial \Omega }} = 0\) and \(T{|_{\partial \Omega }} = 0\).
Let us prove that for all \(\varvec{\varphi } \in C_0^\infty (\Omega )\),
$$\begin{aligned} \begin{aligned}&\int _\Omega {{{(1 + \left| {D\varvec{u}} \right| )}^{p - 2}}D\varvec{u}D\varvec{\varphi } dx} - \int _\Omega {P \textrm{div}\varvec{\varphi } dx} + \int _\Omega {(\varvec{u} \cdot \nabla )u\varvec{\varphi } dx} - \int _\Omega {\textrm{curl}\varvec{B} \times \varvec{B}\varvec{\varphi } dx} \\&+ \int _\Omega {\beta T\varvec{g}\varvec{\varphi } dx} = \mathop {\lim }\limits _{m \rightarrow \infty } \{ \int _\Omega {{{(1 + \left| {D{\varvec{u}^{m - 1}}} \right| )}^{p - 2}}D{\varvec{u}^m}D\varvec{\varphi } dx - } \int _\Omega {{P^m}\textrm{div}\varvec{\varphi } dx} \\&+ \int _\Omega {({\varvec{u}^{m - 1}} \cdot \nabla ){\varvec{u}^{m - 1}}\varvec{\varphi } dx} - \int _\Omega {\textrm{curl}{\varvec{B}^{m - 1}} \times {\varvec{B}^{m - 1}}\varvec{\varphi } dx} + \int _\Omega {\beta {T^m}\varvec{g}\varvec{\varphi } dx}\}, \end{aligned} \end{aligned}$$
(37)
and for all \({\varvec{\eta }} \in C_0^\infty (\Omega )\),
$$\begin{aligned} \begin{aligned}&\int _\Omega {\nabla \varvec{B} \cdot \nabla {\varvec{\eta }} dx} - \int _\Omega {(\varvec{B} \cdot \nabla )\varvec{u} \cdot {\varvec{\eta }} dx} + \int _\Omega {(\varvec{u} \cdot \nabla )\varvec{B} \cdot {\varvec{\eta }}dx} \\&= \mathop {\lim }\limits _{m \rightarrow \infty } \{ \int _\Omega {\nabla {\varvec{B}^m} \cdot \nabla {\varvec{\eta }} dx} - \int _\Omega {({\varvec{B}^{m - 1}} \cdot \nabla ){\varvec{u}^m} \cdot {\varvec{\eta }} dx} + \int _\Omega {({\varvec{u}^m} \cdot \nabla ){\varvec{B}^{m - 1}} \cdot {\varvec{\eta }} dx} \}, \end{aligned} \end{aligned}$$
(38)
and for al \(\phi \in C_0^\infty (\Omega )\),
$$\begin{aligned} \begin{aligned}&\int _\Omega {\nabla T\nabla \phi dx} - \int _\Omega {{{\left| {\textrm{curl}\varvec{B}} \right| }^2} \cdot \phi dx} - \int _\Omega {2{{\left| {D(\varvec{u})} \right| }^2} \cdot \phi dx} + \int _\Omega {(\varvec{u} \cdot \nabla )T \cdot \phi dx} \\&= \mathop {\lim }\limits _{m \rightarrow \infty } \{ \int _\Omega {\nabla {T^m}\nabla \phi dx} - \int _\Omega {{{\left| {\textrm{curl}{\varvec{B}^{m - 1}}} \right| }^2} \cdot \phi dx} - \int _\Omega {2{{\left| {D({\varvec{u}^m})} \right| }^2} \cdot \phi dx} \\&\quad + \int _\Omega {({\varvec{u}^{m - 1}} \cdot \nabla ){T^{m - 1}} \cdot \phi dx} \}. \end{aligned} \end{aligned}$$
(39)
First, using the Hölder inequality, we get
$$\begin{aligned}{} & {} \begin{aligned}&\Big | \int _\Omega (1 + \left| {D{\varvec{u}^{m - 1}}} \right| )^{p - 2}D{\varvec{u}^m}D{\varvec{\varphi }} dx - \int _\Omega {{{(1 + \left| {D\varvec{u}} \right| )}^{p - 2}}D\varvec{u}D{\varvec{\varphi }} dx} \Big | \\&= \left| {\int _\Omega {{{(1 + \left| {D{\varvec{u}^{m - 1}}} \right| )}^{p - 2}}(D{\varvec{u}^m} - D\varvec{u})D{\varvec{\varphi }} dx} + \int _\Omega {[{{(1 + \left| {D{\varvec{u}^{m - 1}}} \right| )}^{p - 2}}} } \right. \\&\quad \left. { - {{(1 + \left| {D\varvec{u}} \right| )}^{p - 2}}]D\varvec{u}D{\varvec{\varphi }} dx} \right| \\&\le {\left\| {D{\varvec{u}^m} - D\varvec{u}} \right\| _2}{\left\| {D{\varvec{\varphi }} } \right\| _2} + (2 - p) {\left\| {D{\varvec{u}^{m - 1}} - D\varvec{u}} \right\| _2}{\left\| {D{\varvec{u}}} \right\| _2}{\left\| {D{\varvec{\varphi }} } \right\| _\infty }, \\&\left| {\int _\Omega {{P^m}\textrm{div}{\varvec{\varphi }} dx} - \int _\Omega {P\textrm{div}{\varvec{\varphi }} dx} } \right| = \left| {\int _\Omega {({P^m} - P)\textrm{div}{\varvec{\varphi }} dx} } \right| \le {\left\| {{P^m} - P} \right\| _2} {\left\| {\textrm{div}{\varvec{\varphi }} } \right\| _2}, \end{aligned} \\{} & {} \begin{aligned}&\Big | \int _\Omega ({\varvec{u}^{m - 1}} \cdot \nabla ){\varvec{u}^{m - 1}}{\varvec{\varphi }} dx - \int _\Omega {(\varvec{u} \cdot \nabla )\varvec{u}{\varvec{\varphi }} dx} \Big | \\&= \left| {\int _\Omega {[({\varvec{u}^{m - 1}} - \varvec{u}) \cdot \nabla ]{\varvec{u}^{m - 1}}{\varvec{\varphi }} dx} + \int _\Omega {(\varvec{u} \cdot \nabla )({\varvec{u}^{m - 1}} - \varvec{u}){\varvec{\varphi }} dx} } \right| \\&\le {\left\| {{\varvec{u}^{m - 1}} - \varvec{u}} \right\| _2}{\left\| {\nabla {\varvec{u}^{m - 1}}} \right\| _2}{\left\| {\varvec{\varphi }} \right\| _\infty } + {\left\| \varvec{u} \right\| _2}{\left\| {\nabla {\varvec{u}^{m - 1}} - \nabla \varvec{u}} \right\| _2}{\left\| {\varvec{\varphi }} \right\| _\infty }, \end{aligned} \\{} & {} \begin{aligned}&\Big | \int _\Omega \textrm{curl}{\varvec{B}^{m - 1}} \times {\varvec{B}^{m - 1}}{\varvec{\varphi }} dx - \int _\Omega {\textrm{curl}\varvec{B} \times \varvec{B}{\varvec{\varphi }} dx} \Big | \\&\le \left| {\int _\Omega {(\textrm{curl}{\varvec{B}^{m - 1}} - \textrm{curl}\varvec{B}) \times {\varvec{B}^{m - 1}}{\varvec{\varphi }} dx} + \int _\Omega {\textrm{curl}\varvec{B} \times ({\varvec{B}^{m - 1}} - \varvec{B}){\varvec{\varphi }} dx} } \right| \\&\le {\left\| {\textrm{curl}{\varvec{B}^{m - 1}} - \textrm{curl}\varvec{B}} \right\| _2}{\left\| {{\varvec{B}^{m - 1}}} \right\| _\infty } {\left\| {\varvec{\varphi }} \right\| _2} + {\left\| {\textrm{curl}\varvec{B}} \right\| _r}{\left\| {{\varvec{B}^{m - 1}} - \varvec{B}} \right\| _6} {\left\| {\varvec{\varphi }} \right\| _{\frac{{6r}}{{5r - 6}}}} \\&\le C{\left\| {\nabla {\varvec{B}^{m - 1}} - \nabla \varvec{B}} \right\| _2}{\left\| {{\varvec{B}^{m - 1}}} \right\| _{2,r}} {\left\| {\varvec{\varphi }} \right\| _2} + C{\left\| {\nabla \varvec{B}} \right\| _r}{\left\| {{\varvec{B}^{m - 1}} - \varvec{B}} \right\| _{1,2}} {\left\| {\varvec{\varphi }} \right\| _{\frac{{6r}}{{5r - 6}}}}, \end{aligned} \\{} & {} \begin{aligned}&\left| {\int _\Omega {\beta {T^m}\varvec{g}{\varvec{\varphi }} dx} - \int _\Omega {\beta T\varvec{g}{\varvec{\varphi }} dx} } \right| \le \beta {\left\| \varvec{g} \right\| _2}{\left\| {{T^m}} \right\| _2}{\left\| {\varvec{\varphi }} \right\| _\infty } + \beta {\left\| \varvec{g} \right\| _2}{\left\| T \right\| _2}{\left\| {\varvec{\varphi }} \right\| _\infty } \\&\le C\beta {\left\| \varvec{g} \right\| _q}{\left\| {{T^m}} \right\| _{2,2}}{\left\| {\varvec{\varphi }} \right\| _\infty } + C\beta {\left\| \varvec{g} \right\| _q}{\left\| T \right\| _{2,2}}{\left\| {\varvec{\varphi }} \right\| _\infty }, \end{aligned} \end{aligned}$$
and such quantities tend to zero as m goes to infinity, thanks to the \({W^{1,2}}(\Omega )\) convergence of \({\varvec{u}^m}\), \({\varvec{B}^m}\), and \({T^m}\), the \({L^2}\) convergence of \({P^m}\) and the boundedness of the norms \({\left\| {D\varvec{u}} \right\| _2}\), \({\left\| {\nabla {\varvec{u}^{m - 1}}} \right\| _2}\), \({\left\| \varvec{u} \right\| _2}\), \({\left\| {{\varvec{B}^{m - 1}}} \right\| _{2,r}}\), \({\left\| {\nabla \varvec{B}} \right\| _r}\), \({\left\| {D{\varvec{\varphi }} } \right\| _\infty }\), \({\left\| {\textrm{div}{\varvec{\varphi }} } \right\| _2}\) and \({\left\| {\varvec{\varphi }} \right\| _\infty }\). Observing that the right-hand side of (37) is equal to \(\int _\Omega {(1 + \beta {T_r})\varvec{g}{\varvec{\varphi }} dx}\), we have that for any \({\varvec{\varphi }} \in C_0^\infty (\Omega )\)
$$\begin{aligned} \begin{aligned}&\int _\Omega {{{(1 + \left| {D\varvec{u}} \right| )}^{p - 2}}D\varvec{u}D{\varvec{\varphi }} dx} - \int _\Omega {P\nabla \cdot {\varvec{\varphi }} dx} \\&= \int _\Omega {(1 + \beta {T_r})\varvec{g}{\varvec{\varphi }} dx} - \int _\Omega {\beta T\varvec{g}{\varvec{\varphi }} dx} - \int _\Omega {(\varvec{u} \cdot \nabla )\varvec{u}{\varvec{\varphi }} dx} + \int _\Omega {\textrm{curl}\varvec{B} \times \varvec{B}{\varvec{\varphi }} dx}. \end{aligned} \end{aligned}$$
Second, we have
$$\begin{aligned}{} & {} \left| {\int _\Omega {\nabla {\varvec{B}^m} \cdot
\nabla {\varvec{\eta }} dx} - \int _\Omega {\nabla \varvec{B} \cdot \nabla {\varvec{\eta }} dx} } \right| \\= & {} \left| {\int _\Omega {(\nabla {\varvec{B}^m} - \nabla \varvec{B}) \cdot \nabla {\varvec{\eta }} dx} } \right| \le {\left\| {\nabla {\varvec{B}^m} - \nabla \varvec{B}} \right\| _2}{\left\| {\nabla {\varvec{\eta }} } \right\| _2}, \\{} & {} \begin{aligned}&\Big | \int _\Omega ({\varvec{B}^{m - 1}} \cdot \nabla ){\varvec{u}^m}{\varvec{\eta }} dx - \int _\Omega {(\varvec{B} \cdot \nabla )\varvec{u}{\varvec{\eta }} dx} \Big | \\&= \left| {\int _\Omega {[({\varvec{B}^{m - 1}} - \varvec{B}) \cdot \nabla ]{\varvec{u}^m}{\varvec{\eta }} dx} + \int _\Omega {(\varvec{B} \cdot \nabla )({\varvec{u}^m} - \varvec{u}){\varvec{\eta }} dx} } \right| \\&\le {\left\| {{\varvec{B}^{m - 1}} - \varvec{B}} \right\| _2}{\left\| {\nabla {\varvec{u}^m}} \right\| _\infty } {\left\| {\varvec{\eta }} \right\| _2} + {\left\| \varvec{B} \right\| _r} {\left\| {\nabla {\varvec{u}^m} - \nabla \varvec{u}} \right\| _3}{\left\| {\varvec{\eta }} \right\| _{\frac{{3r}}{{2r - 3}}}} \\&\le {\left\| {{\varvec{B}^{m - 1}} - \varvec{B}} \right\| _2}{\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {\varvec{\eta }} \right\| _2} + C{\left\| \varvec{B} \right\| _r}{\left\| {{\varvec{u}^m} - \varvec{u}} \right\| _{1,2}} {\left\| {\varvec{\eta }} \right\| _{\frac{{3r}}{{2r - 3}}}}, \end{aligned} \\{} & {} \begin{aligned}&\Big | \int _\Omega ({\varvec{u}^m} \cdot \nabla ){\varvec{B}^{m - 1}}{\varvec{\eta }} dx - \int _\Omega {(\varvec{u} \cdot \nabla )\varvec{B}{\varvec{\eta }} dx} \Big | \\&= \left| {\int _\Omega {[({\varvec{u}^m} - \varvec{u}) \cdot \nabla ]{\varvec{B}^{m - 1}}{\varvec{\eta }} dx} + \int _\Omega {(\varvec{u} \cdot \nabla )({\varvec{B}^{m - 1}} - \varvec{B}){\varvec{\eta }} dx} } \right| \\&\le {\left\| {{\varvec{u}^m} - \varvec{u}} \right\| _3}{\left\| {\nabla {\varvec{B}^{m - 1}}} \right\| _r} {\left\| {\varvec{\eta }} \right\| _{\frac{{3r}}{{2r - 3}}}} + {\left\| \varvec{u} \right\| _\infty } {\left\| {\nabla {\varvec{B}^{m - 1}} - \nabla \varvec{B}} \right\| _2}{\left\| {\varvec{\eta }} \right\| _2} \\&\le C{\left\| {{\varvec{u}^m} - \varvec{u}} \right\| _{1,2}}{\left\| {\nabla {\varvec{B}^{m - 1}}} \right\| _r} {\left\| {\varvec{\eta }} \right\| _{\frac{{3r}}{{2r - 3}}}} + {\left\| \varvec{u} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {\nabla {\varvec{B}^{m - 1}} - \nabla \varvec{B}} \right\| _2}{\left\| {\varvec{\eta }} \right\| _2}. \end{aligned} \end{aligned}$$
Similarly as above, we get
$$\int _\Omega {\nabla \varvec{B} \cdot \nabla {\varvec{\eta }} dx} = \int _\Omega {(\varvec{B} \cdot \nabla )\varvec{u} \cdot {\varvec{\eta }} dx} - \int _\Omega {(\varvec{u} \cdot \nabla )\varvec{B} \cdot {\varvec{\eta }} dx},$$
for any \({\varvec{\eta }} \in C_0^\infty (\Omega )\).
Third, we have
$$\begin{aligned}{} & {} \left| {\int _\Omega {\nabla {T^m}\nabla \phi dx} - \int _\Omega {\nabla T\nabla \phi dx} } \right| = \left| \int _\Omega {(\nabla {T^m} - \nabla T)\nabla \phi dx} \right| \le {\left\| {\nabla {T^m} - \nabla T} \right\| _2} {\left\| {\nabla \phi } \right\| _2}, \\{} & {} \left| {\int _\Omega {{{\left| {\textrm{curl}{\varvec{B}^{m - 1}}} \right| }^2} \cdot \phi dx} - \int _\Omega {{{\left| {\textrm{curl}\varvec{B}} \right| }^2} \cdot \phi dx} } \right| = \left| {\int _\Omega {{{\left| {\nabla {\varvec{B}^{m - 1}}} \right| }^2} \cdot \phi dx} - \int _\Omega {{{\left| {\nabla \varvec{B}} \right| }^2} \cdot \phi dx} } \right| \\{} & {} \le {\left\| {{{\left| {\nabla {\varvec{B}^{m - 1}}} \right| }^2}} \right\| _2}{\left\| \phi \right\| _2} + {\left\| {{{\left| {\nabla \varvec{B}} \right| }^2}} \right\| _2}{\left\| \phi \right\| _2} \\{} & {} \le \left\| {{\varvec{B}^{m - 1}}} \right\| _{2,r}^2{\left\| \phi \right\| _2} + \left\| \varvec{B} \right\| _{2,r}^2{\left\| \phi \right\| _2}, \\{} & {} \left| {\int _\Omega {2{{\left| {D({\varvec{u}^m})} \right| }^2} \cdot \phi dx} - \int _\Omega {2{{\left| {D\varvec{u}} \right| }^2} \cdot \phi dx} } \right| \le 2{\left\| {{{\left| {D({\varvec{u}^m})} \right| }^2}} \right\| _2}{\left\| \phi \right\| _2} + 2{\left\| {{{\left| {D\varvec{u}} \right| }^2}} \right\| _2}{\left\| \phi \right\| _2} \\{} & {} \le C(\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}^2 + \left\| \varvec{u} \right\| _{{C^{1,{\gamma _0}}}}^2){\left\| \phi \right\| _2}, \\{} & {} \Big | \int _\Omega ({\varvec{u}^{m - 1}} \cdot \nabla ){T^{m - 1}} \cdot \phi dx - \int _\Omega {(\varvec{u} \cdot \nabla )T \cdot \phi dx} \Big | \\{} & {} = \left| {\int _\Omega {[({\varvec{u}^{m - 1}} - \varvec{u}) \cdot \nabla ]{T^{m - 1}} \cdot \phi dx} + \int _\Omega {(\varvec{u} \cdot \nabla )({T^{m - 1}} - T) \cdot \phi dx} } \right| \\{} & {} \le {\left\| {{\varvec{u}^{m - 1}} - \varvec{u}} \right\| _\infty }{\left\| {\nabla {T^{m - 1}}} \right\| _{2}}{\left\| \phi \right\| _2} + {\left\| \varvec{u} \right\| _\infty }{\left\| {\nabla {T^{m - 1}} - \nabla T} \right\| _2}{\left\| \phi \right\| _2} \\{} & {} \le C{\left\| {{\varvec{u}^{m - 1}} - \varvec{u}} \right\| _{1,2}}{\left\| {{T^{m - 1}}} \right\| _{1,2}}{\left\| \phi \right\| _2} + C{\left\| \varvec{u} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{T^{m - 1}} - T} \right\| _{1,2}}{\left\| \phi \right\| _2}. \end{aligned}$$
Similarly as above, we get
$$\begin{aligned} \int _\Omega {\nabla T\nabla \phi dx} = \int _\Omega {{{\left| {\textrm{curl}\varvec{B}} \right| }^2} \cdot \phi dx} + \int _\Omega {2{{\left| {D(\varvec{u})} \right| }^2} \cdot \phi dx} + \int _\Omega {\psi \phi dx} - \int _\Omega {(\varvec{u} \cdot \nabla )T \cdot \phi dx}, \end{aligned}$$
for any \(\phi \in C_0^\infty (\Omega )\).
By Definition 1.1 and Remark 1.1, we know \((\varvec{u},P,\varvec{B},T)\) is a solution of problem (3)−(4).
Finally, since
$$\begin{aligned} \begin{aligned}&{\left\| \varvec{u} \right\| _{{C^{1,\gamma }}}} + {\left\| P \right\| _{{C^{0,\gamma }}}} + {\left\| \varvec{B} \right\| _{2,r}} + {\left\| T \right\| _{2,2}} \\&\le {\left\| {\varvec{u} - {\varvec{u}^{{k_m}}}} \right\| _{{C^{1,\gamma }}}} + {\left\| {{\varvec{u}^{{k_m}}}} \right\| _{{C^{1,\gamma }}}} + {\left\| {P - {P^{{k_m}}}} \right\| _{{C^{0,\gamma }}}} + {\left\| {{P^{{k_m}}}} \right\| _{{C^{0,\gamma }}}} + {\left\| {{\varvec{B}^{{k_m}}}} \right\| _{2,r}} + {\left\| {{T^{{k_m}}}} \right\| _{2,2}}, \end{aligned} \end{aligned}$$
by taking \(m\rightarrow \infty\) and using the lower semi-continuity of the norms, it follows (10) that
$$\begin{aligned} \begin{aligned} {\left\| \varvec{u} \right\| _{{C^{1,\gamma }}}} + {\left\| P \right\| _{{C^{0,\gamma }}}} + {\left\| \varvec{B} \right\| _{2,r}} + {\left\| T \right\| _{2,2}} \le 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}. \end{aligned} \end{aligned}$$
3.4 Uniqueness Results
Let \(\frac{6}{5}< p < 2\) and assume that \(({\varvec{u}_1},{\varvec{B}_1},{T_1})\) and \(({\varvec{u}_2},{\varvec{B}_2},{T_2})\) are two solutions of problem (3)−(4). Let \(\bar{\varvec{u}} = {\varvec{u}_1} - {\varvec{u}_2}\), \(\bar{\varvec{B}} = {\varvec{B}_1} - {\varvec{B}_2}\) and \({\bar{T}} = {T_1} - {T_2}\). Using Definition 1.1, we bring \(({\varvec{u}_1},{\varvec{B}_1},{T_1})\) and \(({\varvec{u}_2},{\varvec{B}_2},{T_2})\) into (5) and subtract one from the other, then test with \({\varvec{\varphi }} = \bar{ \varvec{u}} = {\varvec{u}_1} - {\varvec{u}_2} \in {V_q}(\Omega )\), we get
$$\begin{aligned} \begin{aligned}&\int _\Omega {[S(D{\varvec{u}_1}) - S(D{\varvec{u}_2})] \cdot (D{\varvec{u}_1} - D{\varvec{u}_2})dx} \\&= - \int _\Omega {\beta {\bar{T}}\varvec{g} \bar{\varvec{u}} dx} + \int _\Omega {\textrm{curl}{\varvec{B}_2} \times \bar{\varvec{B}} \cdot \bar{\varvec{u}} dx} + \int _\Omega {\textrm{curl}\bar{\varvec{B}} \times {\varvec{B}_1} \cdot \bar{\varvec{u}} dx} + \int _\Omega {(\bar{\varvec{u}} \cdot \nabla )\bar{\varvec{u}} \cdot {\varvec{u}_1}dx}. \end{aligned} \end{aligned}$$
Using the Hölder inequality, we can write
$$\begin{aligned} \begin{aligned} \left\| {D\bar{\varvec{u}} } \right\| _p^p&= \int _\Omega {{{\left( \frac{{{{\left| {D\bar{\varvec{u}} } \right| }^2}}}{{{{(1 + \left| {D{\varvec{u}_1}} \right| + \left| {D{\varvec{u}_2}} \right| )}^{2 - p}}}}\right) }^{\frac{p}{2}}} \cdot {{\left( 1 + \left| {D{\varvec{u}_1}} \right| + \left| {D{\varvec{u}_2}} \right| \right) }^{\frac{{p(2 - p)}}{2}}}dx} \\&\le {\left( \int _\Omega {\frac{{{{\left| {D\bar{\varvec{u}} } \right| }^2}}}{{{{(1 + \left| {D{\varvec{u}_1}} \right| + \left| {D{\varvec{u}_2}} \right| )}^{2 - p}}}}dx} \right) ^{\frac{p}{2}}} \cdot {\left[ \int _\Omega {{{(1 + \left| {D{\varvec{u}_1}} \right| + \left| {D{\varvec{u}_2}} \right| )}^p}dx} \right] ^{\frac{{2 - p}}{2}}}. \end{aligned} \end{aligned}$$
Hence, recalling Lemma 2.6, we obtain
$$\begin{aligned} \begin{aligned} \left\| {D\bar{\varvec{u}} } \right\| _p^2&\le \int _\Omega {\frac{{{{\left| {D\bar{\varvec{u}} } \right| }^2}}}{{{{(1 + \left| {D{\varvec{u}_1}} \right| + \left| {D{\varvec{u}_2}} \right| )}^{2 - p}}}}dx} \cdot {\left[ \int _\Omega {{{(1 + \left| {D{\varvec{u}_1}} \right| + \left| {D{\varvec{u}_2}} \right| )}^p}dx} \right] ^{\frac{{2 - p}}{p}}} \\&\le C\left( \int _\Omega {(S(D{\varvec{u}_1}) - S(D{\varvec{u}_2})) \cdot (D{\varvec{u}_1} - D{\varvec{u}_2})dx} \right) \cdot (1 + \left\| {D{\varvec{u}_1}} \right\| _p^{2 - p} + \left\| {D{\varvec{u}_2}} \right\| _p^{2 - p}) \\&\le C\Big (- \int _\Omega {\beta {\bar{T}} \varvec{g}\cdot \bar{\varvec{u}} dx} + \int _\Omega {\textrm{curl}{\varvec{B}_2} \times \bar{\varvec{B}} \cdot \bar{\varvec{u}} dx} + \int _\Omega {\textrm{curl}\bar{\varvec{B}} \times {\varvec{B}_1} \cdot \bar{\varvec{u}} dx} \\&\quad + \int _\Omega {(\bar{\varvec{u}}\cdot \nabla )\bar{\varvec{u}} \cdot {\varvec{u}_1}dx} \Big ) \cdot (1 + {\left\| {D{\varvec{u}_1}} \right\| _p} + {\left\| {D{\varvec{u}_2}} \right\| _p}). \end{aligned} \end{aligned}$$
Using the Hölder and Sobolev’s inequalities, we have
$$\begin{aligned} \begin{aligned}&\left| {\int _\Omega {\beta {\bar{T}} \varvec{g} \bar{\varvec{u}} dx} + \int _\Omega {\textrm{curl}{\varvec{B}_2} \times \bar{\varvec{B}} \cdot \bar{\varvec{u}} dx} + \int _\Omega {\textrm{curl}\bar{\varvec{B}} \times {\varvec{B}_1} \cdot \bar{\varvec{u}} dx} + \int _\Omega {(\bar{\varvec{u}} \cdot \nabla )\bar{\varvec{u}} \cdot {\varvec{u}_1}dx} } \right| \\&\le \beta {\left\| \varvec{g} \right\| _{\frac{{3p}}{{5p - 6}}}}{\left\| {{\bar{T}} } \right\| _{\frac{{3p}}{{3 - p}}}} {\left\| {\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}} + {\left\| {\textrm{curl}{\varvec{B}_2}} \right\| _3}{\left\| {\bar{\varvec{B}} } \right\| _{\frac{{3p}}{{3p - 3}}}}{\left\| {\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}} + {\left\| {\textrm{curl}\bar{\varvec{B}} } \right\| _2}{\left\| {{\varvec{B}_1}} \right\| _{\frac{{6p}}{{5p - 6}}}} {\left\| {\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}} \\&\quad + \left\| {\bar{\varvec{u}}} \right\| _{\frac{{3p}}{{3 - p}}}^2{\left\| {\nabla {\varvec{u}_1}} \right\| _{\frac{{3p}}{{5p - 6}}}} \\&\le C\beta {\left\| \varvec{g} \right\| _q}{\left\| {{\bar{T}} } \right\| _{1,2}}{\left\| {\nabla \bar{\varvec{u}} } \right\| _p} + C{\left\| {\nabla {\varvec{B}_2}} \right\| _3}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}{\left\| {\nabla \bar{\varvec{u}} } \right\| _p} + C{\left\| {\nabla \bar{\varvec{B}} } \right\| _2}{\left\| {{\varvec{B}_1}} \right\| _{2,r}}{\left\| {\nabla \bar{\varvec{u}} } \right\| _p} \\&\quad + C\left\| {\nabla \bar{\varvec{u}}} \right\| _p^2{\left\| {{\varvec{u}_1}} \right\| _{{c^{1,{\gamma _0}}}}} \\&\le C\beta {c_3}{\left\| \varvec{g} \right\| _q}{\left\| {{\bar{T}} } \right\| _{1,2}}{\left\| {D\bar{\varvec{u}} } \right\| _p} + C{c_3}{\left\| {{\varvec{B}_2}} \right\| _{2,r}}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}{\left\| {D\bar{\varvec{u}} } \right\| _p} + C{c_3}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}{\left\| {{\varvec{B}_1}} \right\| _{2,r}}{\left\| {D\bar{\varvec{u}} } \right\| _p} \\&\quad + Cc_3^2\left\| {D\bar{\varvec{u}} } \right\| _p^2{\left\| {{\varvec{u}_1}} \right\| _{{c^{1,{\gamma _0}}}}}. \end{aligned} \end{aligned}$$
So we get
$$\begin{aligned} \begin{aligned}&{\left\| {D\bar{\varvec{u}} } \right\| _p} \le C\left[ {c_3}\beta {\left\| \varvec{g} \right\| _q}{\left\| {{\bar{T}} } \right\| _{1,2}} + {c_3}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}({\left\| {{\varvec{B}_1}} \right\| _{2,r}} + {\left\| {{\varvec{B}_2}} \right\| _{2,r}}) + c_3^2{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {{\varvec{u}_1}} \right\| _{{c^{1,{\gamma _0}}}}}\right] \\&\qquad \qquad \cdot (1 + {\left\| {D{\varvec{u}_1}} \right\| _p} + {\left\| {D{\varvec{u}_2}} \right\| _p}). \end{aligned} \end{aligned}$$
(40)
Inserting \(({\varvec{u}_1},{\varvec{B}_1},{T_1})\) and \(({\varvec{u}_2},{\varvec{B}_2},{T_2})\) into (6) and subtract one from the other, then test with \({\varvec{\eta }} = \bar{\varvec{B}} = {\varvec{B}_1} - {\varvec{B}_2} \in {W^{1,2}}(\Omega )\), we get
$$\int _\Omega {{{\left| {\nabla \bar{\varvec{B}} } \right| }^2}dx} = \int _\Omega {(\bar{\varvec{B}} \cdot \nabla ){\varvec{u}_1} \cdot \bar{\varvec{B}} dx} + \int _\Omega {({\varvec{B}_2} \cdot \nabla )\bar{\varvec{u}} \cdot \bar{\varvec{B}} dx} - \int _\Omega {(\bar{\varvec{u}} \cdot \nabla ){\varvec{B}_1} \cdot \bar{\varvec{B}} dx}.$$
Since
$$\int _\Omega {{{\left| {\nabla \bar{\varvec{B}} } \right| }^2}dx} = \left\| {\nabla \bar{\varvec{B}} } \right\| _2^2 \ge C\left\| {\bar{\varvec{B}} } \right\| _{1,2}^2,$$
and
$$\begin{aligned} \begin{aligned} \Big | \int _\Omega (\bar{\varvec{B}} \cdot \nabla ){\varvec{u}_1}&\cdot \bar{\varvec{B}} dx + \int _\Omega {({\varvec{B}_2} \cdot \nabla )\bar{\varvec{u}} \cdot \bar{\varvec{B}} dx} - \int _\Omega {(\bar{\varvec{u}} \cdot \nabla ){\varvec{B}_1} \cdot \bar{\varvec{B}}dx} \Big | \\&\le \left\| {\bar{\varvec{B}} } \right\| _2^2{\left\| {\nabla {\varvec{u}_1}} \right\| _\infty } + {\left\| {{\varvec{B}_2}} \right\| _{\frac{{6p}}{{5p - 6}}}}{\left\| {\bar{\varvec{u}}} \right\| _{\frac{{3p}}{{3 - p}}}} {\left\| {\nabla \bar{\varvec{B}} } \right\| _2} + {\left\| {\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}} {\left\| {\nabla {\varvec{B}_1}} \right\| _{\frac{{6p}}{{5p - 6}}}}{\left\| {\bar{\varvec{B}}} \right\| _2} \\&\le \left\| {\bar{\varvec{B}}} \right\| _{1,2}^2{\left\| {{\varvec{u}_1}} \right\| _{{C^{1,{\gamma _0}}}}} + C{c_3}{\left\| {{\varvec{B}_2}} \right\| _{2,r}}{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}} + C{c_3}{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {{\varvec{B}_1}} \right\| _{2,r}}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}, \end{aligned} \end{aligned}$$
we get
$$\begin{aligned} {\left\| {\bar{\varvec{B}} } \right\|
_{1,2}} \le C\left[ {\left\| {{\varvec{u}_1}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {\bar{\varvec{B}}} \right\| _{1,2}} + {c_3}({\left\| {{\varvec{B}_1}} \right\| _{2,r}} + {\left\| {{\varvec{B}_2}} \right\| _{2,r}}){\left\| {D\bar{\varvec{u}} } \right\| _p}\right] . \end{aligned}$$
(41)
Inserting \(({\varvec{u}_1},{\varvec{B}_1},{T_1})\) and \(({\varvec{u}_2},{\varvec{B}_2},{T_2})\) into (7) and subtract one from the other, then test with \(\phi = {\bar{T}} = {T_1} - {T_2} \in {W^{1,2}}(\Omega )\), we get
$$\begin{aligned} \begin{aligned} \int _\Omega {{{\left| {\nabla {\bar{T}} } \right| }^2}dx}&= \int _\Omega {\textrm{curl}\bar{\varvec{B}}\textrm{curl}{\varvec{B}_1} \cdot {\bar{T}} dx} + \int _\Omega {\textrm{curl}\bar{\varvec{B}} \textrm{curl}{\varvec{B}_2} \cdot {\bar{T}} dx} + 2\int _\Omega {D\bar{\varvec{u}} D{\varvec{u}_1} \cdot {\bar{T}} dx} \\&+ 2\int _\Omega {D\bar{\varvec{u}} D{\varvec{u}_2} \cdot {\bar{T}} dx} + \int _\Omega {(\bar{\varvec{u}} \cdot \nabla ){T_1} \cdot {\bar{T}} dx} + \int _\Omega {({\varvec{u}_2} \cdot \nabla ){\bar{T}} \cdot {\bar{T}} dx}, \end{aligned} \end{aligned}$$
hence
$$\int _\Omega {{{\left| {\nabla {\bar{T}} } \right| }^2}dx} = \left\| {\nabla {\bar{T}} } \right\| _2^2 \ge C\left\| {{\bar{T}} } \right\| _{1,2}^2,$$
and
$$\begin{aligned} \begin{aligned}&\left| {\int _\Omega {\textrm{curl}\bar{\varvec{B}} \textrm{curl}{\varvec{B}_1} \cdot {\bar{T}} dx} + \int _\Omega {\textrm{curl}\bar{\varvec{B}}\textrm{curl}{\varvec{B}_2} \cdot {\bar{T}} dx} + 2\int _\Omega {D\bar{\varvec{u}} D{\varvec{u}_1} \cdot {\bar{T}} dx} + 2\int _\Omega {D\bar{\varvec{u}} D{\varvec{u}_2} \cdot {\bar{T}} dx} } \right. \\&\quad \left. { + \int _\Omega {(\bar{\varvec{u}} \cdot \nabla ){T_1} \cdot {\bar{T}} dx} + \int _\Omega {({\varvec{u}_2} \cdot \nabla ){\bar{T}} \cdot {\bar{T}} dx} } \right| \\&\le \left| {\int _\Omega {\nabla \bar{\varvec{B}} \nabla {\varvec{B}_1} \cdot {\bar{T}} dx} + \int _\Omega {\nabla \bar{\varvec{B}} \nabla {\varvec{B}_2} \cdot {\bar{T}} dx} + 2\int _\Omega {D\bar{\varvec{u}} D{\varvec{u}_1} \cdot {\bar{T}} dx} + 2\int _\Omega {D\bar{\varvec{u}} D{\varvec{u}_2} \cdot {\bar{T}} dx} } \right. \\&\quad \left. { + \int _\Omega {(\bar{\varvec{u}} \cdot \nabla ){T_1} \cdot {\bar{T}} dx} + \int _\Omega {({\varvec{u}_2} \cdot \nabla ){\bar{T}} \cdot {\bar{T}} dx} } \right| \\&\le {\left\| {\nabla \bar{\varvec{B}}} \right\| _2}{\left\| {\nabla {\varvec{B}_1}} \right\| _{\frac{{6p}}{{5p - 6}}}} {\left\| {{\bar{T}} } \right\| _{\frac{{3p}}{{3 - p}}}} + {\left\| {\nabla \bar{\varvec{B}} } \right\| _2}{\left\| {\nabla {\varvec{B}_2}} \right\| _{\frac{{6p}}{{5p - 6}}}} {\left\| {{\bar{T}} } \right\| _{\frac{{3p}}{{3 - p}}}} + 2{\left\| {D\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}} {\left\| {D{\varvec{u}_1}} \right\| _{\frac{{3p}}{{5p - 6}}}}{\left\| {{\bar{T}} } \right\| _{\frac{{3p}}{{3 - p}}}} \\&\quad + 2{\left\| {D\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}}{\left\| {D{\varvec{u}_2}} \right\| _{\frac{{3p}}{{5p - 6}}}} {\left\| {{\bar{T}} } \right\| _{\frac{{3p}}{{3 - p}}}} + {\left\| {\bar{\varvec{u}} } \right\| _{\frac{{3p}}{{3 - p}}}} {\left\| {\nabla {T_1}} \right\| _{\frac{{6p}}{{5p - 6}}}}{\left\| {{\overline{T}} } \right\| _2} + {\left\| {{\varvec{u}_2}} \right\| _\infty }{\left\| {\nabla {\bar{T}} } \right\| _2}{\left\| {{\bar{T}} } \right\| _2} \\&\le C{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}{\left\| {{\varvec{B}_1}} \right\| _{2,r}}{\left\| {{\bar{T}}} \right\| _{1,2}} + C{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}{\left\| {{\varvec{B}_2}} \right\| _{2,r}}{\left\| {{\bar{T}} } \right\| _{1,2}} + C{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {{\varvec{u}_1}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{\bar{T}} } \right\| _{1,2}} \\&\quad + C{\left\| {D\bar{\varvec{u}}} \right\| _p}{\left\| {{\varvec{u}_2}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {{\bar{T}} } \right\| _{1,2}} + C{c_3}{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {{T_1}} \right\| _{2,2}} {\left\| {{\bar{T}} } \right\| _{1,2}} + C{\left\| {{\varvec{u}_2}} \right\| _{{C^{1,{\gamma _0}}}}}\left\| {{\bar{T}}} \right\| _{1,2}^2. \end{aligned} \end{aligned}$$
We get
$$\begin{aligned} \begin{aligned} {\left\| {{\bar{T}} } \right\| _{1,2}}&\le C{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}({\left\| {{\varvec{B}_1}} \right\| _{2,r}} + {\left\| {{\varvec{B}_2}} \right\| _{2,r}}) + C{\left\| {D\bar{\varvec{u}} } \right\| _p}({\left\| {{\varvec{u}_1}} \right\| _{{C^{1,{\gamma _0}}}}} + {\left\| {{\varvec{u}_2}} \right\| _{{C^{1,{\gamma _0}}}}}) \\&+ C{c_3}{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {{T_1}} \right\| _{2,2}} + C{\left\| {{\varvec{u}_2}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {{\bar{T}} } \right\| _{1,2}}. \end{aligned} \end{aligned}$$
(42)
Combining (40), (41) and (42), we finally obtain
$$\begin{aligned} \begin{aligned}&{\left\| {D\bar{\varvec{u}} } \right\| _p} + {\left\| {\bar{\varvec{B}} } \right\| _{1,2}} + {\left\| {{\bar{T}} } \right\| _{1,2}} \\&\le C[\beta {c_3}{\left\| \varvec{g} \right\| _q}{\left\| {{\bar{T}} } \right\| _{1,2}} + {c_3}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}}\left( {\left\| {{\varvec{B}_1}} \right\| _{2,r}} + {\left\| {{\varvec{B}_2}} \right\| _{2,r}}\right) + c_3^2{\left\| {D\bar{\varvec{u}}} \right\| _p}{\left\| {{\varvec{u}_1}} \right\| _{{c^{1,{\gamma _0}}}}}] \\&\quad \cdot (1 + {\left\| {D{\varvec{u}_1}} \right\| _p} + {\left\| {D{\varvec{u}_2}} \right\| _p}) + C[{\left\| {{\varvec{u}_1}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {\bar{\varvec{B}} } \right\| _{1,2}} + {c_3}({\left\| {{\varvec{B}_1}} \right\| _{2,r}} + {\left\| {{\varvec{B}_2}} \right\| _{2,r}}){\left\| {D\bar{\varvec{u}} } \right\| _p}] \\&\quad + C{\left\| {\bar{\varvec{B}}} \right\| _{1,2}}({\left\| {{\varvec{B}_1}} \right\| _{2,r}} + {\left\| {{\varvec{B}_2}} \right\| _{2,r}}) + C{\left\| {D\bar{\varvec{u}} } \right\| _p}({\left\| {{\varvec{u}_1}} \right\| _{{C^{1,{\gamma _0}}}}} + {\left\| {{\varvec{u}_2}} \right\| _{{C^{1,{\gamma _0}}}}}) \\&\quad + C{c_3}{\left\| {D\bar{\varvec{u}} } \right\| _p}{\left\| {{T_1}} \right\| _{2,2}} \\&\quad + C{\left\| {{\varvec{u}_2}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {{\bar{T}} } \right\| _{1,2}} \\&\le (2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2} {\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})[C(\beta {c_3}+ 2{c_3} + c_3^2)\\&\quad (1 + 4{K_2}{\left\| \varvec{g} \right\| _q} + 4{c_2}{\left\| \psi \right\| _2} \\&\quad + 4CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q}) + 3C{c_3} + 6C] \cdot ({\left\| {D\bar{\varvec{u}} } \right\| _p} + {\left\| {\bar{\varvec{B}} } \right\| _{1,2}} + {\left\| {{\bar{T}} } \right\| _{1,2}}). \end{aligned} \end{aligned}$$
So if \(2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2} {\left\| \varvec{g} \right\| _q}\) is sufficiently small, the uniqueness follows.