Existence of Regular Solutions for a Class of Incompressible Non-Newtonian MHD Equations Coupled to the Heat Equation

Abstract

We consider a system of PDE’s describing the steady flow of an electrically conducting fluid in the presence of a magnetic field. The system of governing equations composes of the stationary non-Newtonian incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy is taken into account in the momentum equation and the Joule heating and viscous heating terms are included. We proved the existence of $C^{1,\gamma }({\bar{\Omega }})\times W^{2,r}(\Omega )\times W^{2,2}{(\Omega )}$ solutions of the systems for $1< p<2$ corresponding to a small data and we show that this solution is unique in case $6/5< p < 2$. Moreover, we also proved the higher regularity properties of this solution.

1 Introduction and Main Result

The stationary incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy (arising from temperature differences in the fluid) is taken into account in the momentum equation could be described by

$$\begin{aligned} \left\{ \begin{aligned}&-\nu \textrm{div}~{\varvec{S}}+\rho _r\left( \varvec{u}\cdot \nabla \right) \varvec{u}+\nabla P =\rho _r[1-\beta (T-{{T}_{r}})]\varvec{g}+\frac{1}{\mu }\textrm{curl}\varvec{B}\times \varvec{B}, \\&-\frac{1}{\mu \sigma }\Delta \varvec{B}=\left( \varvec{B}\cdot \nabla \right) \varvec{u}-\left( \varvec{u}\cdot \nabla \right) \varvec{B}, \\&-\kappa \Delta T+\rho _rc_p\left( \varvec{u}\cdot \nabla \right) T=\frac{1}{\mu ^2\sigma }{{\left| \textrm{curl}\varvec{B} \right| }^{2}} +2\nu {\varvec{S}}: {D(\varvec{u})} +\psi , \\&\textrm{div}~\varvec{u}=0,\quad ~ \textrm{div}~\varvec{B}=0, \end{aligned} \right. \end{aligned}$$

(1)

where $\varvec{S}=\varvec{S}(D\varvec{u})$ is the stress tensor, $D\varvec{u}=\dfrac{1}{2}\left( \nabla {\varvec{u}}+(\nabla {\varvec{u}})^T\right)$ is the strain rate tensor. The unknowns $( \varvec{u},~ \varvec{B},~ P,~ T)$ denote the velocity of the fluid, the magnetic field, the pressure and the temprature, respectively. The given function $\varvec{g}$ is the acceleration of gravity, $\psi$ denotes a heat source. $\rho _r$ and $T_r$ are a reference density and temperature, respectively, and $\beta$ is the so-called thermal expansion coefficient. Other costant parameters appearing are the kinematic viscosity $\nu$, magnetic permeability $\mu$, electrical conductivity $\sigma$, heat conductivity $\kappa$, and $c_p$ the specific heat at constant pressure, $\lambda >0$ a positive constant. The terms $\dfrac{1}{{{\mu }^{2}}\sigma }{{\left| \textrm{curl}\varvec{B} \right| }^{2}}$ and $2\nu {\varvec{S}}: {D(\varvec{u})}$ model the Joule heating and the viscous heating, respectively.

System (1) describes the flow of an electrically conducting fluid in the presence of a magnetic field (e.g. flow of liquid metal and plasma). The modeling of the phenomenon and derivation of the equations are discussed, e.g. [1,2,3]. The research of this model is a significant interest not only in mathematical theory but also in practical applications, such as nuclear reactor technology, magnetic propulsion devices, design of electromagnetic pumps, continuous metal casting, crystal growth, semi-conductor manufacture and numerous other disciplines [4].

In the Newtonian case, the stress tensor $\varvec{S}(D\varvec{u})$ is a linear function of $D\varvec{u}$ ( i.e. $\varvec{S}(D\varvec{u})=\nu D\varvec{u}$, $\nu$ the kinematic viscosity coefficient), several papers are concerned with the existence of solutions to (1). Meir [5] proved the existence and uniqueness of a solution when the data of problem are small enough under mixed boundary conditions. Meir and Schmidt [6] proved the existence of solutions by applying the Leray–Schauder principle. Uniqueness was proven under certain smallness assumptions on the data. Alekseev [7] proved that if the value of ${\varvec{u}}$ is tangent on the boundary, there exists a solution, and if in addition the data of problem are small enough, then the solution is unique. Meir [8] proved the global solvability of the boundary value problem. The sufficient conditions on the data are established which provide a local uniqueness of the solution. Recently in Kim [9], the authors study the solvability of this model under mixed boundary conditions for temperature, magnetic field and velocity. It allowd that the thermal conductivity viscosity coefficient, electrical conductivity, magnetic permeability and specific heat of the fluid depended on the temperature. All the papers mentioned above have neglected the Joule heating and the viscous heating term in the model. In Bermúdez et al. [10], the authors study the solvability of the problem with the buoyancy effects due to temperature differences in the flow, Joule and viscous heating effects.

In the non-Newtonian case, the stress tensor $\varvec{S}(D\varvec{u})$ being a nonlinear function of $D\varvec{u}$, the reralted rusults on this subject are quite few. Mathematical expression for the constitutive law of non-Newtonian fluids is defined by

$$\begin{aligned} \varvec{S}(D\varvec{u})=\tau (|D{\varvec{u}}|)D{\varvec{u}}, \qquad \tau (.) \text{ is } \text{ a } \text{ a } \text{ positive } \text{ real } \text{ function, } \end{aligned}$$

most of the mathematical literature devoted to the analysis of non-Newtonian fluids assume stress tensor to be of power type

$$\begin{aligned} \tau ( s)=(\tau _0 + \tau _1 s)^{p-2}, \qquad \tau _0\ge 0, ~~~\tau _1\ge 0. \end{aligned}$$

(2)

For the unsteady case, when there is no temperature T in (1), by using the Galerkin method and the monotone operator theory, Samokhin [11] proved the existence of weak solutions of the model with suitable boundary conditions in a bounded domain in the case of $\tau _0=0, ~~\tau _1>0$. For the case of bounded or periodic domains, Gunzburger et al. [12] established the unique solvability of the initial boundary value problem with $\tau _0>0, ~~\tau _1>0$. In the whole space $R^3$, by using Galerkin method, Kang and Kim [13] showed the existence of a weak and strong solution with a slitly more general type of the tensor structure. Kim [14] investigated the local-in-time existence results of solutions. Furthermore, if the existence of the global in time regular solution is guaranteed, the large time behavior result for solutions is obtained under the suitable assumption for the stress tensor part. For the steady case, by using a fixed point argument in an appropriate functional setting, Shi and Wang [15] proved the existence and uniqueness of strong solutions for the problem in a smooth domain under the conditions that the external force is small in a suitable sense. When the temperature equation is included, we have not found any relevant research to date. For more related results, one can also refer [16,17,18,19,20,21] and the reference cited therein.

In this work, we focus on system (1) with stress tensor ${\varvec{S}}$ being of the form (2). Instead of $2\nu {\varvec{S}}: {D(\varvec{u})}$, we consider the viscous heating term in a simplified version $2\nu {{\left| D(\varvec{u}) \right| }^{2}}$ ( Newtonian case) for technique reason. Also, for simplicity, we take the costant parameters $\nu =\rho _r=\mu =\sigma =\kappa =c_p=1$. More precisely, let $\Omega \subset R^3$ be an open bounded domain with smooth boundary, consider the system

$$\begin{aligned} \left\{ \begin{aligned}&-\textrm{div}\left[ {{\left( 1 +\left| D\varvec{u} \right| \right) }^{p-2}}D\varvec{u} \right] +\left( \varvec{u}\cdot \nabla \right) \varvec{u}+\nabla P=[1-\beta (T-{{T}_{r}})]\varvec{g}+\textrm{curl}\varvec{B}\times \varvec{B},&x\in \Omega \\&- \Delta \varvec{B}=\left( \varvec{B}\cdot \nabla \right) \varvec{u}-\left( \varvec{u}\cdot \nabla \right) \varvec{B},&\quad x\in \Omega \\&-\Delta T+\left( \varvec{u}\cdot \nabla \right) T={{\left| \textrm{curl}\varvec{B} \right| }^{2}}+2{{\left| D(\varvec{u}) \right| }^{2}}+\psi ,&\quad x\in \Omega \\&\textrm{div}~\varvec{u}=0,\quad ~ \textrm{div}~\varvec{B}=0,&\quad x\in \Omega \end{aligned} \right. \end{aligned}$$

(3)

completed by the boundary conditions

$$\begin{aligned} {{\left. \varvec{u} \right| }_{\partial \Omega }}=0,\quad {{\left. \varvec{B}\cdot \varvec{n} \right| }_{\partial \Omega }}=0,\quad {{\left. \textrm{curl}\varvec{B} \times \varvec{n} \right| }_{\partial \Omega }}=0,\quad {{\left. T \right| }_{\partial \Omega }}=0, \end{aligned}$$

(4)

where $\varvec{n}$ is the unit outward normal vector of $\partial \Omega$.

The aim of this paper is to prove the existence and uniqueness of regular solutions to system (3), (4). The proof is divided into two parts: firstly, we proved the existence and uniqueness of $C^{1,\gamma }({\overline{\Omega }})\times W^{2,r}(\Omega )\times W^{2,2}{(\Omega )}$ solution to the systems under suitable smallness assumptions on the force term by iteration methods; then, through the standard mollifier technique, we show that the solution possess a higher regularity of ${{W}^{2,2}}(\Omega )\cap {{C}^{1,\gamma }}({\overline{\Omega }})\times {{W}^{2,r}}(\Omega )\times {{W}^{2,2}}(\Omega ).$

Throughout the paper, as usual, by $\left( {{L}^{q}}\left( \Omega \right) ,{{\left\| \cdot \right\| }_{q}} \right)$ resp. $\left( {{W}^{k,p}}\left( \Omega \right) ,{{\left\| \cdot \right\| }_{k,p}} \right)$, we denote the classical Lebesgue and Sobolev spaces. Define $V=\left\{ v\in C_{0}^{\infty }\left( \Omega \right) ; \textrm{div}~v =0 \right\}$ and the spaces

$$\begin{aligned} {{V}_{q}}\left( \Omega \right) :=~the ~completion~ of ~V ~in ~the ~{{W}^{1,q}}-norm, \end{aligned}$$

for $q=2$ we simply write $V\left( \Omega \right)$. We also denote by $\left( {{C}^{m,\gamma }}\left( {\overline{\Omega }} \right) , {{\left\| \cdot \right\| }_{{{C}^{m,\gamma }}}} \right)$, m nonnegative integer and $\gamma \in \left( 0,1 \right)$, the Hölder space with order m. By ${{W}^{-1,{q}'}}\left( \Omega \right)$, ${q}'=\frac{q}{q-1}$, the strong dual of $W_{0}^{1,q}\left( \Omega \right)$ with norm ${{\left\| \cdot \right\| }_{-1,{q}'}}$.

Before we establish the existence results, we first give a precise notion of a $C^{1,\gamma }\times W^{1,2}\times W_0^{1,2}$ solution.

Definition 1.1

Assume that $\varvec{g}\in {{L}^{q}}\left( \Omega \right)$, $\psi \in {{L}^{2}}\left( \Omega \right)$, $r>1$. We say that $\left( \varvec{u},\varvec{B},T \right)$ is a pair of ${{C}^{1,\gamma }}\left( {\overline{\Omega }} \right) \times {{W}^{1,2}}\left( \Omega \right) \times W_{0}^{1,2}\left( \Omega \right)$ $-$ solution of problem (3), (4). If $\varvec{u}\in {{C}^{1,\gamma }}\left( {\overline{\Omega }} \right)$ for some $\gamma \in \left( 0,1 \right)$, $\varvec{B}\in {{W}^{1,2}}\left( \Omega \right)$, $T\in {{W}^{1,2}}\left( \Omega \right)$, $\textrm{div}~\varvec{u}=0$, ${{\left. \varvec{u} \right| }_{\partial \Omega }}=0$, ${{\left. \varvec{B}\cdot \varvec{n} \right| }_{\partial \Omega }}=0$, ${{\left. \textrm{curl}\varvec{B} \times \varvec{n} \right| }_{\partial \Omega }}=0$, ${{\left. T \right| }_{\partial \Omega }}=0$, and it satisfies the following integral identity for $\forall {\varvec{\varphi }} \in V\left( \Omega \right)$, $\forall {\varvec{\eta }} \in C_{0}^{\infty }\left( \Omega \right)$ and $\forall \phi \in C_{0}^{\infty }\left( \Omega \right)$:

$$\begin{aligned} \int _{\Omega }{{{\left( 1+\left| D\varvec{u} \right| \right) }^{p-2}}D\varvec{u}D{\varvec{\varphi }} }dx= & {} \int _{\Omega }{\left[ 1-\beta \left( T-{{T}_{r}} \right) \right] \varvec{g}\cdot {\varvec{\varphi }}}dx \nonumber \\{} & {} +\int _{\Omega }{\textrm{curl}\varvec{B}\times \varvec{B}\cdot {\varvec{\varphi }}}dx -\int _{\Omega }{\left( \varvec{u}\cdot \nabla \right) \varvec{u}\cdot {\varvec{\varphi }} }dx, \end{aligned}$$

(5)

$$\begin{aligned} \int _{\Omega }{\nabla \varvec{B}\cdot \nabla {\varvec{\eta }} }dx= & {} \int _{\Omega } {\left( \varvec{B}\cdot \nabla \right) \varvec{u}\cdot {\varvec{\eta }} }dx-\int _{\Omega }{\left( \varvec{u}\cdot \nabla \right) \varvec{B}\cdot {\varvec{\eta }} }dx, \end{aligned}$$

(6)

$$\begin{aligned} \int _{\Omega }{\nabla T\cdot \nabla \phi }dx= & {} \int _{\Omega }{{{\left| \textrm{curl}\varvec{B} \right| }^{2}}\cdot \phi }dx +2\int _{\Omega }{{{\left| D\left( \varvec{u} \right) \right| }^{2}}\cdot \phi }dx\nonumber \\{} & {} +\int _{\Omega }{\psi \phi }dx -\int _{\Omega }{\left( \varvec{u}\cdot \nabla \right) T\cdot \phi }dx. \end{aligned}$$

(7)

Remark 1.1

We observe that if $\varvec{u}$ satisfies (5) then we can apply the theorem of deRham [22] to find a pressure P at least in ${{L}^{2}}\left( \Omega \right)$ such that the pair $\left( \varvec{u},P,\varvec{B},T \right)$ satisfies the following integral identity for $\forall {\varvec{\varphi }} \in C_{0}^{\infty }\left( \Omega \right)$:

$$\begin{aligned} \begin{aligned}&\int _{\Omega }{{{\left( 1+\left| D\varvec{u} \right| \right) }^{p-2}}D\varvec{u}D{\varvec{\varphi }} }dx-\int _{\Omega }{P\nabla \cdot {\varvec{\varphi }} }dx \\&\hspace{2cm} =\int _{\Omega }{\varvec{g}\left[ 1-\beta \left( T-{{T}_{r}} \right) \right] {\varvec{\varphi }}}dx +\int _{\Omega }{\textrm{curl}\varvec{B}\times \varvec{B}{\varvec{\varphi }}}dx-\int _{\Omega }{\left( \varvec{u}\cdot \nabla \right) \varvec{u}\cdot {\varvec{\varphi }} }dx. \end{aligned} \end{aligned}$$

The validity of the reverse implication is obvious. In the sequel, we shall refer to $\left( \varvec{u},\varvec{B},T \right)$ or $\left( \varvec{u},P,\varvec{B},T \right)$ as solution of systems (3), (4) without distinction.

Our aim is to prove the following theorems.

Theorem 1.1

Assume that $p\in \left( 1,2 \right)$, $r>3$, $q>3$, ${{\gamma }_{0}}=1-\frac{3}{q}$. Let $\Omega \in {{\mathbb {R}}^{3}}$ be a bounded smooth domain, and let $\varvec{g}\in {{L}^{q}}\left( \Omega \right)$, $\psi \in {{L}^{2}}\left( \Omega \right)$. If (9) is satisfied, then there exists at least one ${{C}^{1,\gamma }}\left( {\overline{\Omega }} \right) \times {{C}^{0,\gamma }}\left( {\overline{\Omega }} \right) \times {{W}^{2,r}}\left( \Omega \right) \times {{W}^{2,2}}\left( \Omega \right)$ $-$solution $\left( \varvec{u},P,\varvec{B},T \right)$ of problem (3), (4) such that

$$\begin{aligned} \varvec{u}\in {{C}^{1,\gamma }}\left( {\overline{\Omega }} \right) ,\quad P\in {{C}^{0,\gamma }}\left( {\overline{\Omega }} \right) , \quad \varvec{B}\in {{W}^{2,r}}\left( \Omega \right) ,\quad T\in {{W}^{2,2}}\left( \Omega \right) , \end{aligned}$$

and

$$\begin{aligned} {{\left\| {{\varvec{u}}} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}({\bar{\Omega }})}} +{{\left\| {{P}} \right\| }_{{{C}^{0,{{\gamma }_{0}}}}({\bar{\Omega }})}}+{{\left\| {{\varvec{B}}} \right\| }_{2,r}} +{{\left\| {{T}} \right\| }_{2,2}}\le 2{{K}_{2}}{{\left\| \varvec{g} \right\| }_{q}} +2{{c}_{2}}{{\left\| \psi \right\| }_{2}}+2CK\beta {{c}_{2}}{{\left\| \varvec{g} \right\| }_{q}}{{\left\| \psi \right\| }_{2}}, \end{aligned}$$

where K, ${{K}_{2}}$ is positive constant. Additionally, if $\frac{6}{5}<p<2$ the solution is unique.

Theorem 1.2

In addition to the assumptions of Theorem 1.1, if $q>6$ and ${{\left\| \varvec{g} \right\| }_{q}}+{{\left\| \psi \right\| }_{2}}$ is sufficiently small, then the ${{C}^{1,\gamma }}\left( {\overline{\Omega }} \right) \times {{C}^{0,\gamma }}\left( {\overline{\Omega }} \right) \times {{W}^{2,r}}\left( \Omega \right) \times {{W}^{2,2}}\left( \Omega \right) -$solutions $(\varvec{u},P,\varvec{B},T)$ of problems (3), (4) possesses a higher regularity of

$$\begin{aligned}{} & {} \varvec{u}\in ~{{W}^{2,2}}(\Omega )\cap {{C}^{1,\gamma }}({\overline{\Omega }}),~~P\in ~{{W}^{1,2}}(\Omega ) \cap {{C}^{0,\gamma }}({\overline{\Omega }}),\\{} & {} \varvec{B}\in ~{{W}^{2,r}}(\Omega ),~~T\in ~{{W}^{2,2}}(\Omega ),~~\forall \gamma <{{\gamma }_{0}}. \end{aligned}$$

The paper is organized as follows: In the next section, we collect some preliminaries results which will be needed in later analysis. In Sect. 3, the proof of Theorem 1.1 is presented. To be specific, in Sect. 3.1, we construct approximate solutions to the original nonlinear problem by iterate scheme, we also derive the uniform estimate, independent on the parameter k, for such approximate solutions. The results are used in Sect. 3.2 to prove the convergence of the solutions. The existence and uniqueness results are proved in Sects. 3.3 and 3.4, respectively. Then finally, the regularity result (Theorems 1.2) are proved in Sect. 4.

2 Preliminary Lemmas

We now recall some useful results that are needed for the treatment of the system (3), (4).

Lemma 2.1

[23] Let $q>n=3$, ${{\gamma }_{0}}=1-\frac{3}{q}$. Assume that $\Omega$ is a domain of class ${{C}^{1, {{\gamma }_{0}}}}$, ${\varvec{F}}\in {{L}^{q}}(\Omega )$. Then, there exists a weak solution $(\varvec{u}, P )$ such that

$$\begin{aligned} \left\{ \begin{aligned}&-\textrm{div} \left[ {{(\mu +\left| D\varvec{u} \right| )}^{p-2}}D\varvec{u}\right] +\nabla P={\varvec{F}}, \qquad in~\Omega , \\&\textrm{div}~\varvec{u}=0, \qquad in~\Omega , \\&\varvec{u}|_{\partial \Omega }=0. \end{aligned} \right. \end{aligned}$$

Moreover, $\varvec{u}\in {{C}^{1, {{\gamma }_{0}}}}({\bar{\Omega }})$, $P\in {{C}^{0, {{\gamma }_{0}}}}({\bar{\Omega }})$, the following estimate holds

$$\begin{aligned} {{\left\| \varvec{u} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}+{{\left\| P \right\| }_{{{C}^{0,{{\gamma }_{0}}}}}} \le {\tilde{c}}\left( {{\left\| \varvec{u} \right\| }_{1,2}}+{{\left\| {\varvec{F}} \right\| }_{q}}\right) , \end{aligned}$$

where ${\tilde{c}}={c}_{0}{{\left( 1+\frac{{{\left\| \varvec{u} \right\| }_{{{C}^{1,{{\gamma }_{0}}}}}}}{\mu }\right) }^{\alpha ({{\gamma }_{0}}, n, p)}}$, $\alpha >2$ is a positive constant.

Lemma 2.2

[24] Let m be a non-negative integer and $1<p<\infty$. Let $\varvec{g}\in {{W}^{m, p}}(\Omega )$, with $\textrm{div}~{\varvec{g}}=0$ and $\varvec{g}\cdot \varvec{n}=0$ on $\partial \Omega$, $\varvec{k}\in {{W}^{m+1-\frac{1}{p}, p}}(\partial \Omega )$, $q \in {{W}^{m+2-\frac{1}{p}, p}}(\partial \Omega )$. Then there exists a unique $\varvec{B}\in {{W}^{m+2, p}}(\Omega )$ such that

$$\begin{aligned} \left\{ \begin{aligned}&\textrm{curl}~(\textrm{curl}\varvec{B})=\varvec{g}, \qquad in~\Omega , \\&\textrm{div}~\varvec{B}=0, \qquad in~\Omega , \\&\varvec{B}\cdot \varvec{n}{{|}_{\partial \Omega }}=q, \\&\textrm{curl}\varvec{B}\times \varvec{n}{{|}_{\partial \Omega }}=\varvec{k}\times \varvec{n}, \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} {{\left\| \varvec{B} \right\| }_{{{W}^{m+2, p}}}}\le {{c}_{1}}\left( {{\left\| \varvec{g} \right\| }_{{{W}^{m, p}}}} +{{\left\| \varvec{k} \right\| }_{{{W}^{m+1-\frac{1}{p}, p}}}}+{{\left\| q \right\| }_{{{W}^{m+2-\frac{1}{p}, p}}}}\right) . \end{aligned}$$

Lemma 2.3

[25] Let ${\varvec{f}}\in {{L}^{2}}(\Omega )$, $\varvec{u}\in H_{0}^{1}(\Omega )$ is the weak solution of

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta \varvec{u}={\varvec{f}}, \qquad in~\Omega , \\&\varvec{u}{{|}_{\partial \Omega }}=0. \end{aligned} \right. \end{aligned}$$

If $\partial \Omega \in {{C}^{2}}$, then $\varvec{u}\in {{H}^{2}}({\Omega })$, such that

$$\begin{aligned} \begin{aligned} {{\left\| \varvec{u} \right\| }_{{{H}^{2}}}}\le {{c}_{2}}{{\left\| {\varvec{f}} \right\| }_{{{L}^{2}}}}. \end{aligned} \end{aligned}$$

Where ${c}_{2}$ is a constant that depends only on the space dimensions n and $\Omega$.

Lemma 2.4

[26] For any $q\ge 1$, there exists a constant ${{c}_{3}}$, such that

$$\begin{aligned} {{\left\| {\varvec{v}} \right\| }_{q}}+{{\left\| \nabla {\varvec{v}} \right\| }_{q}}\le {{c}_{3}}{{\left\| D{\varvec{v}} \right\| }_{q}} , \qquad {for~each~} {\varvec{v}}\in {{V}_{q}}(\Omega ). \end{aligned}$$

Hence, the two quantities above are equivalent norms in ${{V}_{q}}(\Omega )$.

Lemma 2.5

[27] If a distribution $\varvec{g}$ is such that $\nabla \varvec{g}\in {{W}^{-1, q}}(\Omega )$, then $\varvec{g}\in {{L}^{q}}(\Omega )$ and ${{\left\| \varvec{g} \right\| }_{L_{\#}^{q}}}\le c{{\left\| \nabla \varvec{g} \right\| }_{-1, q}}$, where $L_{\#}^{q}={{L}^{q}}/\mathbb {R}$.

Lemma 2.6

[28] For an arbitrary tensor ${\varvec{\xi }}$, define $S({\varvec{\xi }})\equiv {{(1+\left| {\varvec{\xi }} \right| )}^{p-2}}{\varvec{\xi }}$, $1< p < 2$. Then there exists a constant C, such that, for any pair of tensors ${{\varvec{\xi }}_{1}}$ and ${{\varvec{\xi }}_{2}}$,

$$\begin{aligned} (S({{\varvec{\xi }}_{1}})-S({{\varvec{\xi }}_{2}}))\cdot ({{\varvec{\xi }}_{1}}-{{\varvec{\xi }}_{2}})\ge C\frac{{{\left| {{\varvec{\xi }}_{1}}-{{\varvec{\xi }}_{2}} \right| }^{2}}}{{{(1+\left| {{\varvec{\xi }}_{1}} \right| +\left| {{\varvec{\xi }}_{2}} \right| )}^{2-p}}}. \end{aligned}$$

Lemma 2.7

[23] For any given real numbers $s, t \ge 0$ and $1< p <2$, the following inequality holds true

$$\begin{aligned} \left| \frac{1}{{{(1+s )}^{2-p}}}-\frac{1}{{{(1+t )}^{2-p}}} \right| \le (2-p)\left| s -t \right| . \end{aligned}$$

3 The Proof of the Theorem 1.1

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.

4 Proof of Theorem 1.2

Throughout the proof, the conditions of assumption (36) are satisfied, in this way, all the hypotheses of Theorem 1.1 are satisfied and we can find the sequences $\{ {\varvec{u}^m}\}$, $\{ {P^m}\}$, $\{ {\varvec{B}^m}\}$ and $\{ {T^m}\}$, as in Proposition 3.2, converging to the solution $(\varvec{u},P,\varvec{B},T)$. To get ${D^2}{\varvec{u}} \in {L^2}(\Omega )$ we proceed by induction on m. First, we have (see [27], Theorem 3.2)

$$\begin{aligned} {\left\| {{\varvec{u}^0}} \right\| _{2,2}} + {\left\| {{P^0}} \right\| _{1,2}} \le {{\widetilde{c}}}({\left\| {{\varvec{u}^0}} \right\| _{1,2}} + {\left\| {[1 - \beta ({T^0} - {T_r})]\varvec{g}} \right\| _q}). \end{aligned}$$

Since ${\left\| {{\varvec{u}^0}} \right\| _{1,2}} \le c_3^2{\left\| {[1 - \beta ({T^0} - {T_r})]\varvec{g}} \right\| _q}$, there follows

$$\begin{aligned} {\left\| {{\varvec{u}^0}} \right\| _{2,2}} + {\left\| {{P^0}} \right\| _{1,2}} \le {{\widetilde{c}}}(1 + c_3^2){\left\| {[1 - \beta ({T^0} - {T_r})]\varvec{g}} \right\| _q}, \end{aligned}$$

it implies that ${\left\| {{D^2}{\varvec{u}^0}} \right\| _2} \le {{\widetilde{c}}}(1 + c_3^2){\left\| {[1 - \beta ({T^0} - {T_r})]\varvec{g}} \right\| _q}$. We assume that ${D^2}{\varvec{u}^m} \in {L^2}(\Omega )$ and we go forward with the step $m+1$. Let us consider the following boundary value problem

$$\begin{aligned} \left\{ \begin{aligned}&-\textrm{div}[{{(1+{{J}_{\varepsilon }}(\left| D{{\varvec{u}}^{m}} \right| ))}^{p-2}}D\varvec{u}_{\varepsilon }^{m+1}]+\nabla P_{\varepsilon }^{m+1} \\&\qquad =[1-\beta ({{T}^{m+1}}-{{T}_{r}})]\varvec{g}-({{\varvec{u}}^{m}}\cdot \nabla ){{\varvec{u}}^{m}}+\textrm{curl}{{\varvec{B}}^{m}}\times {{\varvec{B}}^{m}}, \\&\textrm{div}~\varvec{u}_{\varepsilon }^{m+1}=0, \\&\varvec{u}_{\varepsilon }^{m+1}{{|}_{\partial \Omega }}=0. \end{aligned} \right. \end{aligned}$$

(43)

Where ${J_\varepsilon }$ denotes the Friedrichs mollifier. Since $[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g} - ({\varvec{u}^m} \cdot \nabla ){\varvec{u}^m} + \textrm{curl}{\varvec{B}^m} \times {\varvec{B}^m} \in {L^q}(\Omega )$, then, as in (19), there exists a solution $(\varvec{u}_\varepsilon ^{m + 1}, P_\varepsilon ^{m + 1}) \in {C^{1,{\gamma _0}}}({{\overline{\Omega }}} ) \times {C^{0,{\gamma _0}}}({{\overline{\Omega }}} )$ and satisfies

$$\begin{aligned} \begin{aligned}&{\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{{C^{1,{\gamma _0}}}}} + {\left\| {P_\varepsilon ^{m + 1}} \right\| _{{C^{0,{\gamma _0}}}}} \\&\le {c_0}{\left( 1 + {\left\| {{J_\varepsilon }\left( \left| {D{\varvec{u}^m}} \right| \right) } \right\| _{{C^{0,{\gamma _0}}}}}\right) ^\alpha } \cdot \bigg ({\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{1,2}} + {\left\| {\left[ 1 - \beta ({T^{m + 1}} - {T_r}\bigg )\right] \varvec{g}} \right\| _q} \\&\quad + \left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C\left\| {{\varvec{B}^m}} \right\| _{2,r}^2). \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} {\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{1,2}}\le & {} c_3^2{(1 + {\left\| {{J_\varepsilon }\left( \left| {D{\varvec{u}^m}} \right| \right) } \right\| _{{C^{0,{\gamma _0}}}}})^{2 - p}} \cdot ({\left\| {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}} \right\| _q} \\{} & {} + \left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C\left\| {{\varvec{B}^m}} \right\| _{2,r}^2), \\ {\left\| {{J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| )} \right\| _{{C^{0,{\gamma _0}}}}}\le & {} {\left\| {D{\varvec{u}^m}} \right\| _{{C^{0,{\gamma _0}}}}} \le {\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}}, \end{aligned}$$

there follows

$$\begin{aligned} \begin{aligned}&{\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{{C^{1,{\gamma _0}}}}} + {\left\| {P_\varepsilon ^{m + 1}} \right\| _{{C^{0,{\gamma _0}}}}} \\&\le {c_0}(1 + c_3^2){(1 + {\left\| {{J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| )} \right\| _{{C^{0,{\gamma _0}}}}})^{\alpha + 2 - p}} \cdot ({\left\| [1 - \beta ({T^{m + 1}} - {T_r})]{\varvec{g}} \right\| _q} \\&\quad + \left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C\left\| {{\varvec{B}^m}} \right\| _{2,r}^2). \end{aligned} \end{aligned}$$

(44)

Further, by Theorem 3.2 in [27], we have $(\varvec{u}_\varepsilon ^{m + 1},P_\varepsilon ^{m + 1}) \in {W^{2,2}}(\Omega ) \times {W^{1,2}}(\Omega )$. Next, we dedicate to derive the corresponding estimates which is uniformly in $\varepsilon$.

Multiply (43) by $\Delta \varvec{u}_\varepsilon ^{m + 1}$ and integrate on ${\Omega _\eta } = \{ x \in \Omega :dist(x,\partial \Omega ) > \eta \}$, for some $\varepsilon < \eta$, we get

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\int _{{\Omega _\eta }} {{{(1 + {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))}^{p - 2}} {{\left| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right| }^2}dx} \\&= \int _{{\Omega _\eta }} {(2 - p){{(1 + {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))}^{p - 3}} D\varvec{u}_\varepsilon ^{m + 1}(\Delta \varvec{u}_\varepsilon ^{m + 1} \otimes \nabla {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))dx} \\&\quad + \int _{{\Omega _\eta }} {\nabla P_\varepsilon ^{m + 1}\Delta \varvec{u}_\varepsilon ^{m + 1}dx} - \int _{{\Omega _\eta }} {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}\Delta \varvec{u}_\varepsilon ^{m + 1}dx} \\&\quad + \int _{{\Omega _\eta }} {({\varvec{u}^m} \cdot \nabla ){\varvec{u}^m}\Delta \varvec{u}_\varepsilon ^{m + 1}dx} \\&\quad - \int _{{\Omega _\eta }} {\textrm{curl}{\varvec{B}^m} \times {\varvec{B}^m}\Delta \varvec{u}_\varepsilon ^{m + 1}dx} \buildrel \Delta \over = \sum \limits _{i = 1}^5 {{I_i}}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} {\left\| {\nabla {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| )} \right\| _{2,{\Omega _\eta }}} = {\left\| {{J_\varepsilon }(\nabla \left| {D{\varvec{u}^m}} \right| )} \right\| _{2,{\Omega _\eta }}} \le {\left\| {\nabla \left| {D{\varvec{u}^m}} \right| } \right\| _{2,{\Omega _\eta }}}, \end{aligned}$$

there follows

$$\begin{aligned} \begin{aligned} \left| {{I_1}} \right|&\le {\left\| {D\varvec{u}_\varepsilon ^{m + 1}} \right\| _{\infty ,{\Omega _\eta }}} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} {\left\| {\nabla {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| )} \right\| _{2,{\Omega _\eta }}} \\&\le {\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} {\left\| {\nabla \left| {D{\varvec{u}^m}} \right| } \right\| _{2,{\Omega _\eta }}}. \end{aligned} \end{aligned}$$

Using the divergence theorem

$$\begin{aligned} \left| {{I_2}} \right| = \left| {\int _{\partial {\Omega _\eta }} {P_\varepsilon ^{m + 1}\Delta \varvec{u}_\varepsilon ^{m + 1} \cdot \varvec{n}d\sigma } } \right| \le {\left\| {P_\varepsilon ^{m + 1}} \right\| _{{W^{\frac{1}{2},2}}(\partial {\Omega _\eta })}} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1} \cdot \varvec{n}} \right\| _{{W^{ - \frac{1}{2},2}}(\partial {\Omega _\eta })}}. \end{aligned}$$

Since ${\gamma _0} = 1 - \frac{3}{q}$, $q > 6$, there follows ${\gamma _0} > \frac{1}{2}$, then ${\left\| {P_\varepsilon ^{m + 1}} \right\| _{{W^{\frac{1}{2},2}}(\partial {\Omega _\eta })}} \le C{\left\| {P_\varepsilon ^{m + 1}} \right\| _{{C^{0,{\gamma _0}}}}}$ (see [30]). ${\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1} \cdot \varvec{n}} \right\| _{{W^{ - \frac{1}{2},2}}(\partial {\Omega _\eta })}} \le {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} + {\left\| {\nabla \cdot \Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} = {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}}$ (see [31]). Hence,

$$\begin{aligned} \left| {{I_2}} \right| \le C{\left\| {P_\varepsilon ^{m + 1}} \right\| _{{C^{0,{\gamma _0}}}}} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}}. \end{aligned}$$

Moreover,

$$\begin{aligned} \left| {{I_3}} \right|\le & {} {\left\| {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}} \right\| _2} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} \\\le & {} C {\left\| {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}} \right\| _q} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}}, \\ \left| {{I_4}} \right|\le & {} {\left\| {({\varvec{u}^m} \cdot \nabla ){\varvec{u}^m}} \right\| _2} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} \\\le & {} \left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}^2{\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}}, \\ \left| {{I_5}} \right|\le & {} {\left\| {\textrm{curl}{\varvec{B}^m} \times {\varvec{B}^m}} \right\| _{2,{\Omega _\eta }}} {\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} \\\le & {} C\left\| {{\varvec{B}^m}} \right\| _{2,r}^2{\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \left| {\int _{{\Omega _\eta }} {{{(1 + {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))}^{p - 2}}{{\left| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right| }^2}dx} } \right|&\ge {(1 + {\left\| {{J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| )} \right\| _{{C^{1,{\gamma _0}}}}})^{p - 2}}\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}^2 \\&\ge {(1 + {\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}})^{p - 2}}\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}^2. \end{aligned} \end{aligned}$$

The above estimates, recalling (44), imply that $\Delta \varvec{u}_\varepsilon ^{m + 1} \in {L^2}({\Omega _\eta })$ and

$$\begin{aligned} \begin{aligned}&{\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}} \\&\le 2{(1 + {\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}})^{2 - p}} ({\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{{C^{1,{\gamma _0}}}}} {\left\| {\nabla \left| {D{\varvec{u}^m}} \right| } \right\| _{2,{\Omega _\eta }}} + C{\left\| {P_\varepsilon ^{m + 1}} \right\| _{{C^{0,{\gamma _0}}}}} \\&\quad + {\left\| {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}} \right\| _q} \\&\quad + \left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}^2 + C\left\| {{\varvec{B}^m}} \right\| _{2,r}^2) \\&\le C{c_0}(1 + c_3^2){(1 + 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})^{\alpha + 4 - 2p}}\\&\quad \left( {\left\| {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}} \right\| _q}\right. \\&\quad \left. + C{(2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})^2}\right) \cdot ({\left\| {\nabla \left| {D{\varvec{u}^m}} \right| } \right\| _{2,{\Omega _\eta }}} + C). \end{aligned} \end{aligned}$$

(45)

Since the previous estimate holds uniformly for $\eta > 0$, by taking $\eta \rightarrow 0$, we can replace ${\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,{\Omega _\eta }}}$ with ${\left\| {\Delta \varvec{u}_\varepsilon ^{m + 1}} \right\| _{2,\Omega }}$. By the boundedness of $\Delta \varvec{u}_\varepsilon ^{m + 1}$ in ${L^2}(\Omega )$, uniformly in $\varepsilon$, we deduce the existence of a subsequence weekly converging in ${L^2}(\Omega )$.

On the other hand, for any fixed $m \in \mathbb {N}$, $\varvec{u}_\varepsilon ^{m + 1}$ converges to ${\varvec{u}^{m + 1}}$ in ${W^{1,2}}(\Omega )$ as $\varepsilon \rightarrow 0$. Using the definition of weak solution for $\varvec{u}_\varepsilon ^{m + 1}$ and ${\varvec{u}^{m + 1}}$ and testing with ${\varvec{u}^{m + 1}} - \varvec{u}_\varepsilon ^{m + 1}$, we get

$$\begin{aligned} \begin{aligned} \int _\Omega {{(1 + {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))}^{p - 2}}&D\varvec{u}_\varepsilon ^{m + 1}(D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1})dx \\&= \int _\Omega {{{(1 + \left| {D{\varvec{u}^m}} \right| )}^{p - 2}} D{\varvec{u}^{m + 1}}(D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1})dx}, \end{aligned} \end{aligned}$$

hence,

$$\begin{aligned} \begin{aligned} \int _\Omega (1 +&\left| {D{\varvec{u}^m}} \right| )^{p - 2}{{\left| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right| }^2}dx \\&= \int _\Omega {\left| {{{(1 + {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))}^{p - 2}} - {{(1 + \left| {D{\varvec{u}^m}} \right| )}^{p - 2}}} \right| Du_\varepsilon ^{m + 1}(D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1})dx}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned}{} & {} \left| {\int _\Omega {{{(1 + \left| {D{\varvec{u}^m}} \right| )}^{p - 2}}{{\left| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right| }^2}dx} } \right| \\{} & {} \ge {(1 + {\left\| {D{\varvec{u}^m}} \right\| _\infty })^{p - 2}}\left\| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right\| _2^2 \\{} & {} \ge {(1 + {\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}})^{p - 2}}\left\| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right\| _2^2, \\{} & {} \Big | \int _\Omega \Big | {(1 + {J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ))}^{p - 2} - {{(1 + \left| {D{\varvec{u}^m}} \right| )}^{p - 2}} \Big |D\varvec{u}_\varepsilon ^{m + 1}(D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1})dx \Big | \\{} & {} \le (2 - p){\left\| {{J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ) - D{\varvec{u}^m}} \right\| _2}{\left\| {D\varvec{u}_\varepsilon ^{m + 1}} \right\| _\infty }{\left\| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right\| _2} \\{} & {} \le (2 - p){\left\| {{J_\varepsilon }(\left| {D{\varvec{u}^m}} \right| ) - D{\varvec{u}^m}} \right\| _2}{\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{{C^{1,{\gamma _0}}}}}{\left\| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right\| _2}, \end{aligned}$$

there follows

$$\begin{aligned} {\left\| {D{\varvec{u}^{m + 1}} - D\varvec{u}_\varepsilon ^{m + 1}} \right\| _2} \le (2 - p) \cdot {(1 + {\left\| {{\varvec{u}^m}} \right\| _{{C^{1,{\gamma _0}}}}})^{2 - p}}{\left\| {{J_\varepsilon } (\left| {D{\varvec{u}^m}} \right| ) - D{u^m}} \right\| _2}{\left\| {\varvec{u}_\varepsilon ^{m + 1}} \right\| _{{C^{1,{\gamma _0}}}}}, \end{aligned}$$

we can get $D{\varvec{u}^{m + 1}} \in {L^2}(\Omega )$ as $\varepsilon \rightarrow 0$.

Using the strong convergence of $\varvec{u}_\varepsilon ^{m + 1}$ to ${\varvec{u}^{m + 1}}$ in ${W^{1,2}}(\Omega )$, we also deduce that the limit point of the subsequence of $\Delta \varvec{u}_\varepsilon ^{m + 1}$ in ${L^2}(\Omega )$ is $\Delta {\varvec{u}^{m + 1}}$.

Since ${\left\| {{D^2}{\varvec{u}^{m + 1}}} \right\| _2} \le C{\left\| {\Delta {\varvec{u}^{m + 1}}} \right\| _2}$, setting

$$\begin{aligned} {X_1}= & {} {(1 + 2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})^{\alpha + 4 - 2p}}, \\ {X_2}= & {} {\left\| {[1 - \beta ({T^{m + 1}} - {T_r})]\varvec{g}} \right\| _q} + (1 + C){(2{K_2}{\left\| \varvec{g} \right\| _q} + 2{c_2}{\left\| \psi \right\| _2} + 2CK\beta {c_2}{\left\| \psi \right\| _2}{\left\| \varvec{g} \right\| _q})^2}, \end{aligned}$$

estimate (45) yields that

$$\begin{aligned} {\left\| {{D^2}{\varvec{u}^{m + 1}}} \right\| _2} \le C{c_0}(1 + c_3^2){X_1}{X_2}{\left\| {{D^2}{\varvec{u}^m}} \right\| _2} + C{c_0}(1 + c_3^2){X_1}{X_2}. \end{aligned}$$

(46)

Set

$$\begin{aligned} \Phi (z) = C{c_0}(1 + c_3^2){X_1}{X_2}z + C{c_0}(1 + c_3^2){X_1}{X_2}. \end{aligned}$$

If

$$\begin{aligned} C{c_0}(1 + c_3^2){X_1}{X_2} < 1, \end{aligned}$$

(47)

then there exists ${z_0} > 0$ such that $\Phi ({z_0}) = {z_0}$. Let ${\overline{m}} = \min \{ m \in \mathbb {N}:{\left\| {{D^2}{\varvec{u}^m}} \right\| _2} \le {z_0}\}$. Assume that ${\overline{m}} = + \infty$. Since $\Phi (z) < z$ for any $z > {z_0}$, then using (46) we get

$$\begin{aligned} {\left\| {{D^2}{\varvec{u}^{m + 1}}} \right\| _2} \le \Phi ({\left\| {{D^2}{\varvec{u}^m}} \right\| _2}) \le {\left\| {{D^2}{\varvec{u}^m}} \right\| _2}. \end{aligned}$$

Therefore, ${\left\| {{D^2}{\varvec{u}^m}} \right\| _2} \le {\left\| {{D^2}{\varvec{u}^0}} \right\| _2} \le {{\widetilde{c}}}(1 + c_3^2){\left\| {[1 - \beta ({T^0} - {T_r})]\varvec{g}} \right\| _q}$, for every $m \in \mathbb {N}$. On the other hand, if ${\overline{m}} < + \infty$, since $\Phi (z)$ is increasing, for any $m \ge {\overline{m}}$, we have

$$\begin{aligned} {\left\| {{D^2}{\varvec{u}^{m + 1}}} \right\| _2} \le \Phi ({\left\| {{D^2}{\varvec{u}^m}} \right\| _2}) \le \Phi ({z_0}) = {z_0}. \end{aligned}$$

Hence, by induction, ${\left\| {{D^2}{\varvec{u}^m}} \right\| _2} \le {z_0}$ for any $m \ge {\overline{m}}$. Finally,

$$\begin{aligned} {\left\| {{D^2}{\varvec{u}^m}} \right\| _2} \le \max \{ {{\widetilde{c}}}(1 + c_3^2){\left\| {[1 - \beta ({T^0} - {T_r})]\varvec{g}} \right\| _q},{z_0}\}. \end{aligned}$$

By the uniform boundedness of the ${L^2}$ $-$norm of ${D^2}{\varvec{u}^m}$, using the strong convergence in ${W^{1,2}}(\Omega )$ of ${\varvec{u}^m}$ to the solution $\varvec{u}$ of problem (3), we deduce that if the condition (47) and (36) are satisfied, then $\varvec{u} \in {W^{2,2}}(\Omega )$, $\varvec{B} \in {W^{2,r}}(\Omega )$ and $T \in {W^{2,2}}(\Omega )$. By (3), we have that $\nabla P = [1 - \beta ({T} - {T_r})]\varvec{g} - (\varvec{u} \cdot \nabla )\varvec{u} + \textrm{curl}\varvec{B} \times \varvec{B} + \textrm{div}[{(1 + \left| {D\varvec{u}} \right| )^{p - 2}}D\varvec{u}]$ in the distribution sense. Observing that the right-hand side of the previous identity belongs to ${L^2}(\Omega )$, we obtain that $\nabla P \in {L^2}(\Omega )$.

5 Conclusions

In this paper, we proved the existence and uniqueness of regular solutions for a class of steady non-Newtonian incompressible MHD equations coupled to the heat equation. For the full system, up to our knowedge, the known results are all regarding the Newtonian case, and we have not found any relevant research related to the non-Newtonian type. The results obtained in this work are new and generalize many related problems in the literature.