1 Introduction

Most of the fluid conceptions used to model flows in typical media like air, water or oil, are based on a Newtonian description. In an important number of cases, the supposition of Newtonian behavior does not seem to be accurate enough, leading to expand the theory to introduce additional rheological properties that end in non-Newtonian descriptions. As a set of examples that preclude its ubiquity, the non-Newtonian flow is experienced in industries of mines, where sludges and muds are frequently dealt, in lubrication and biomedical flows.

The non-Newtonian fluids are characterized in different types depending on their rheological characteristics. One of such types is known as Carreau fluid (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] for few relevant studies). More particularly, an interesting research on Carreau fluids is given in [4] where the fluid is analyzed over a shrinking surface in presence of an infinite shear rate viscosity. In addition, the study of Carreau flows in spheres are widely analyzed in [5, 6].

Currently, there is not a wide literature focused on developing the regularity criteria for equations obtained upon a Carreau fluid. Nonetheless, there exists an extensive literature developing the regularity criteria for Navier–Stokes equations (see [18,19,20,21,22,23]). Motivated by such facts, our intention in this study is to develop global regularity criteria for a magnetohydrodynamic (MHD) flow of Carreau fluid equations. The fluid is considered to flow between two stationary plates. Flow between the plates is induced by an initial velocity profile and a stationary linear law at \(y=0\) to the x-axis velocity component. In addition, a uniform magnetic field is applied in the transverse direction to the flow.

The paper layout is as follows: Firstly, the Carreau fluid model is discussed based on the general theory of fluid dynamics of non-Newtonian flows. In addition, some preliminary required results are introduced. Afterwards the theorems on existence of regular solutions are presented. The paper introduces progressively, in different sections, each of the required proofs for each of the regularity Theorems together with the supporting information required.

2 Model Proposal

We consider the two dimensional incompressible fluid flow equations of a Carreau fluid. The fluid is electrically conducting in the presence of an applied magnetic field \(B_{0}\). The MHD flow is governed by the following set of equations:

$$\begin{aligned}&{\mathbf {V}}=\left( u_{1},u_{2},0\right) , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \nabla \cdot {\mathbf {V}}=0, \end{aligned}$$
(2.1)
$$\begin{aligned}&\rho \frac{\partial {\mathbf {V}}}{\partial t}=\nabla \cdot \, {\tau }+\mathbf { J}\times \mathbf {B,} \end{aligned}$$
(2.2)

where \({\mathbf {V}}\) is the velocity field, t the time, \(\rho \) is the fluid density, \({\tau }\) is the Cauchy stress tensor, \({\mathbf {J}}\) is the current density and \(\mathbf {B=B_{0}+B_{1}}\) is the magnetic field for a Carreau fluid, which is given by

$$\begin{aligned} {\tau }=-p{\mathbf {I}}+\eta {\mathbf {A}}_{1}, \end{aligned}$$
(2.3)

with

$$\begin{aligned} \eta =\eta _{\infty }+\left( \eta _{0}-\eta _{\infty }\right) \left[ 1+\left( \varvec{\Gamma }\overset{{\cdot }}{\gamma }\right) ^{2} \right] ^{\frac{n-1}{2}}, \end{aligned}$$
(2.4)

where p is the pressure field, \({\mathbf {I}}\) the identity tensor, \(\eta _{0}\) the zero-shear-rate viscosity, \(\eta _{\infty }\) the infinite-shear-rate viscosity, \(\varvec{\Gamma }\) a material time constant, and n express the power law index (since it describes the slope of \(\frac{\eta _{0}-\eta _{\infty }}{\eta _{0}+\eta _{\infty }}\) in the power law region). The shear rate \(\overset{{\cdot }}{\gamma }\) is defined by

$$\begin{aligned} \overset{{\cdot }}{\gamma }=\sqrt{\underset{i}{\frac{1}{2}\sum } \underset{j}{\sum }\overset{{\cdot }}{\gamma }_{ij}\overset{{ \cdot }}{\gamma }_{ji}}= \sqrt{\frac{1}{2}\varvec{\Pi }}=\sqrt{\frac{1}{2} tr\left( {\mathbf {A}}_{1}^{2}\right) }. \end{aligned}$$
(2.5)

Here \(\varvec{\Pi }\) is the second invariant strain rate tensor and \(\mathbf { A}_{1}\) is given by

$$\begin{aligned} {\mathbf {A}}_{1} = \left( \nabla \,{\mathbf {V}}\right) +\left( \nabla \, {\mathbf {V}}\right) ^{T}. \end{aligned}$$
(2.6)

Note that \(\left( \cdot \right) ^{T}\) denotes the transpose of a matrix. From (2.1) to (2.6), the governing equations in the absence of pressure gradient are given by

$$\begin{aligned} \frac{\partial u_{1}}{\partial t}+u_1\frac{\partial u_{1}}{\partial x}+u_{2} \frac{\partial u_{1}}{\partial y}& = \nu \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right] ^{\frac{n-1}{2}}\nonumber \\&+\nu \left( n-1\right) \Gamma ^{2}\frac{\partial ^{2}u_{1}}{\partial y^{2}} \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right] ^{\frac{n-3}{ 2}}-\frac{\sigma B_{0}^{2}}{\rho }u_{1}, \end{aligned}$$
(2.7)

where \(\nu =\frac{\eta _{0}}{\rho }\) is the kinematic viscosity and \(\sigma \) is related with the electrical charges distribution.

Note that the subject boundary conditions are

$$\begin{aligned} u_{1}\left( x,y,t\right) =Ux, \, \, u_{2}\left( x,y,t\right) =0 \ \ \ \text { at} \ \ y=0, \end{aligned}$$
(2.8)
$$\begin{aligned} u_{1}\left( x,y,t\right) =0, \, u_{2}\left( x,y,t\right) =0 \ \ \ \text {at} \ \ y\rightarrow \infty , \end{aligned}$$
(2.9)

together with

$$\begin{aligned} u_{1}\left( x,y,t\right) =0, \, \, \, \, \, \, u_{2}\left( x,y,t\right) =0 \ \ \ \text {at} \ \ x=0, \, \, x=L. \end{aligned}$$
(2.10)

Where U is a constant. In addition, the following initial conditions hold

$$\begin{aligned} u_{1}\left( x,y,0\right) =u_{10}\left( x,y\right) { \ \ \ \ \ }u_{2}\left( x,y,t\right) =u_{20}\left( x,y\right) , \end{aligned}$$
(2.11)

such that \(\left( u_{10}\left( x,y\right) ,u_{20}\left( x,y\right) \right) \) corresponds to the vector of initial velocities. Note that the domain is given as \(\Omega = \lbrace (x,y) \in [0,L] \times [0, \infty ) \rbrace \).

In addition, consider that the shear stress is zero at \(y=0.\) As a consequence, the following holds

$$\begin{aligned} \tau _{xy}= \nu \frac{\partial u_{1}}{\partial y}\left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right] ^{\frac{n-1}{2}}=0, \end{aligned}$$

leading to \(\frac{ \partial u_{1}}{\partial y}=0\) at \(y=0.\)

3 Preliminaries and Statement of Results

3.1 Previous Results

Consider the well-known norm \(\left\| \cdot \right\| _{L^{p}}\) in the Lebesgue functional space \(L_{p}\left( \Omega \right) \) together with the usual Sobolev order m functional space defined by

$$\begin{aligned} H^{m}\left( \Omega \right) =\left\{ u\in L^{2}\left( \Omega \right) :\text { } \nabla ^{m}\left( u\right) \in L^{2}\left( \Omega \right) \right\} \end{aligned}$$

with the norm

$$\begin{aligned} \left\| u\right\| _{H^{m}}=\left( \left\| u\right\| _{L^{2}}^{2}+\left\| \nabla ^{m}u\right\| _{L^{2}}^{2}\right) ^{\frac{1 }{2}}. \end{aligned}$$

In addition, the following lemma is also needed (refer to Lemma 1 in [23])

Lemma 1

The following anisotropic Sobolev inequality holds

$$\begin{aligned} \iint \limits _{\Omega }\left| fgh\right| dxdy \le C_{0}\left\| f\right\| _{L^{2}}\left\| g\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial g}{\partial x}\right\| _{L^{2}}^{\frac{1}{2}}\left\| h\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial h}{ \partial y}\right\| _{L^{2}}^{\frac{1}{2}}. \end{aligned}$$

3.2 Statement of Results

The main results are stated as follows:

Theorem 3.1

Assuming \(\left( u_{10},u_{20}\right) \in H^{1}(\Omega )\), where \(\left| \nabla u_{0}\right| =\left| \frac{\partial u_{0}}{\partial y}\right| ,\) the equation (2.7) has solutions on \(\left( 0,T\right] \) in the defined strip \(\Omega =\left[ 0,L\right] \times \) \(\left[ 0,\infty \right) .\)

Theorem 3.2

Assuming \(\left( u_{10},u_{20}\right) \in L^{4}(\Omega )\), then the system (2.7) has solutions on \(\left( 0,T\right] \) in the defined strip \(\Omega = \left[ 0,L\right] \times \) \(\left[ 0,\infty \right) .\)

Theorem 3.3

Assuming \(\left( u_{10},u_{20}\right) \in H^{2}(\Omega )\), then the system (2.7) has solutions on \(\left( 0,T\right] \) when \(\frac{\partial u_{2} }{\partial x}\in L^{4}\left( \Omega \right) \) and \(\left( \frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) \), in the defined strip \(\Omega =\left[ 0,L\right] \times \) \( \left[ 0,\infty \right) .\)

4 Proof of Theorem 3.1

Firstly, the following proposition is required to be shown

Proposition 4.1

Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). Neglecting the higher powers of \(\Gamma ^{4}\), then \(u_{1}(x,y,t)\) satisfies

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| u_{1}\right\| _{L^{2}}^{2}+2\nu \int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{2}dt+\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{4}}^{4}dt \\&\quad +\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{6}}^{6}dt\le \overset{\sim }{C}_{1}\left\| u_{10}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{C}_{1}\) depends on a suitable constant M (to be defined in the proof) and T.

Proof

Multiplying the Eq. (2.7) by \(u_{1}\), operating and neglecting higher power of \(\Gamma ^{4},\) the following holds

$$\begin{aligned} \iint \limits _{\Omega }u_{1}\frac{\partial u_{1}}{\partial t}+I_{1}& = \nu \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}u_{1}dxdy+ \frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\iint \limits _{\Omega }\frac{ \partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y} \right) ^{2}u_{1}dxdy \\&+\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{ \partial u_{1}}{\partial y}\right) ^{4}u_{1}dxdy\\&-M^{2}\iint \limits _{\Omega }u_{1}^{2}dxdy, \end{aligned}$$

After using integration by parts

$$\begin{aligned} \frac{d}{dt}\left\| u_{1}\right\| _{L^{2}}^{2}& = 2I_{2}-2\nu \left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}-\left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{4}}^{4}\nonumber \\&-\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{6}}^{6}-2M^{2}\left\| u_{1}\right\| _{L^{2}}^{2}, \end{aligned}$$
(4.1)

where

$$\begin{aligned} I_{1}=\text { -}\iint \limits _{\Omega }u_{1}\left( u_{1}\frac{\partial u_{1} }{\partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right) dxdy. \end{aligned}$$

Integrating by parts

$$\begin{aligned} I_{1}=\frac{1}{2}\iint \limits _{\Omega }u_{1}^{2}\frac{\partial u_{2}}{ \partial y}dxdy. \end{aligned}$$

After using Eq. (2.1), the following reads

$$\begin{aligned} I_{1}=-\frac{1}{2}\iint \limits _{\Omega }u_{1}^{2}\frac{\partial u_{1}}{ \partial x}dxdy. \end{aligned}$$

Integrating again we get \(\left( I_{1}=0\right) \). Introducing the value of \( I_{1}\) into Eq. (4.1) and after Young’s inequality, the following applies

$$\begin{aligned}&\frac{d}{dt}\left\| u_{1}\right\| _{L^{2}}^{2}+2\nu \left\| \frac{ \partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}+\left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{4}}^{4} \\&\qquad +\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{6}}^{6} =-2M^{2}\left\| u_{1}\right\| _{L^{2}}^{2} \\&\quad \le \left| 2M^{2}\right| \left\| u_{1}\right\| _{L^{2}}^{2}. \end{aligned}$$

The Grownwall’s inequality yields

$$\begin{aligned}&\underset{0\le t \le T}{\sup }\left\| u_{1}\right\| _{L^{2}}^{2}+2\nu \int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{2}dt+\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{4}}^{4}dt \\&\quad +\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{6}}^{6}dt\le \overset{\sim }{C}_{1}\left\| u_{10}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{C}_{1}\) depends on M and T. \(\square \)

In addition, the following Proposition is also required.

Proposition 4.2

Assume a solution \(u_{1}\) to set of the equation and conditions (2.7)–(2.11) departing from \(u_{10}\). Then, neglecting higher powers of \(\Gamma ^{4}\), \(\frac{\partial u_{1}}{\partial y}\) satisfies

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{2}+2\nu \int _{0}^{T}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right\| _{L^{2}}^{2}dt+3\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial ^{2}u_{1}}{ \partial y^{2}}\frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right\| _{L^{2}}^{2}dt\le \overset{\sim }{\text { }C}_{2}\left\| \frac{\partial u_{10}}{\partial y} \right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{\text { }C}_{2}\) depends on M and T.

Proof

Multiplying (2.7) by \(-\frac{\partial ^{2}u_{1}}{\partial y^{2}}\) and integrating by parts:

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}& = \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2} }\left( V \cdot \nabla u_{1}\right) dxdy-\nu \iint \limits _{\Omega }\left( \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right) ^{2} \left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2} \right] ^{\frac{n-1}{2}}dxdy \\&-\nu \left( n-1\right) \Gamma ^{2}\iint \limits _{\Omega }\left( \frac{ \partial ^{2}u_{1}}{\partial y^{2}}\right) ^{2}\left( \frac{\partial u_{1}}{ \partial y}\right) ^{2}\left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{ \partial y}\right) ^{2}\right] ^{\frac{n-3}{2}}dxdy\\&-M^{2}\iint \limits _{\Omega }u_{1}\frac{\partial ^{2}u_{1}}{\partial y^{2}}dxdy. \end{aligned}$$

Expanding the term on the right side and neglecting the higher powers of \(\Gamma ^{4}\)

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}& = I_{2}-\nu \iint \limits _{\Omega }\left( \frac{\partial ^{2}u_{1} }{\partial y^{2}}\right) ^{2}dxdy-\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\iint \limits _{\Omega }\left( \frac{\partial ^{2}u_{1}}{\partial y^{2}} \right) ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}dxdy\nonumber \\&-\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\left( \frac{\partial ^{2}u_{1}}{\partial y^{2}} \right) ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{4}dxdy\nonumber \\&-M^{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y} \right) ^{2}dxdy, \end{aligned}$$
(4.2)

where

$$\begin{aligned} I_{2}& = \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}} \left( V \cdot \nabla u_{1}\right) dxdy \\& = \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}u_{1} \frac{\partial u_{1}}{\partial x}dxdy+\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}u_{2}\frac{\partial u_{1}}{\partial y}dxdy. \end{aligned}$$

Integrating \(I_{2},\)

$$\begin{aligned} I_{2}& = -\iint \limits _{\Omega }u_{1}\frac{\partial u_{1}}{\partial y}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}dxdy-\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial u_{1}}{\partial x }dxdy-\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{ \partial y}\right) ^{2}\frac{\partial u_{2}}{\partial y}dxdy \\& = -\iint \limits _{\Omega }u_{1}\frac{\partial u_{1}}{\partial y}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}dxdy-\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial u_{1}}{\partial x }dxdy+\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{ \partial y}\right) ^{2}\frac{\partial u_{1}}{\partial x}dxdy \\& = -\iint \limits _{\Omega }u_{1}\frac{\partial u_{1}}{\partial y}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}dxdy-\frac{1}{2}\iint \limits _{ \Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial u_{1}}{\partial x}dxdy, \end{aligned}$$

where (2.1) has been employed. Now, integrating again we get \(I_{2}=0.\) Introducing \(I_{2}\) in Eq. (4.2) and applying Young’s inequality, the following holds

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}\le & -\nu \left\| \frac{\partial ^{2}u_{1}}{\partial y^{2} }\right\| _{L^{2}}^{2}-\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{2}}^{2} \\&-\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right\| _{L^{2}}^{2}-M^{2}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2} , \end{aligned}$$

which implies that

$$\begin{aligned}&\frac{d}{dt}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}+2\nu \left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right\| _{L^{2}}^{2}+3\left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{2} \\&\qquad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right\| _{L^{2}}^{2}\le -2M^{2}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2} \\&\quad \le 2M^{2}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}. \end{aligned}$$

Now, applying Gronwall’s inequality

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{2}+2\nu \int _{0}^{T}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right\| _{L^{2}}^{2}dt+3\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial ^{2}u_{1}}{ \partial y^{2}}\frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right\| _{L^{2}}^{2}dt\le \overset{\sim }{\text { }C}_{2}\left\| \frac{\partial u_{10}}{\partial y} \right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{\text { }C}_{2}\) depends on M and T. \(\square \)

Finally, note that the Theorem 3.1 is proved by using Propositions 4.1 and 4.2.

5 Proof of Theorem 3.2

To this end, the following Proposition is required

Proposition 5.1

Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). Neglecting the higher powers of \(\Gamma ^{4}\), then \(u_{1}(x,y,t)\) satisfies

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| u_{1}\right\| _{L^{4}}^{4}+\nu \int _{0}^{T}\left\| u_{1}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{2}}^{2}dt+\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| u_{1}^{\frac{1}{2}}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{4}}^{4}dt\qquad \\&\quad +\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| u_{1}^{\frac{1}{3}}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{6}}^{6}dt\le \overset{\sim }{C}_{3}\left\| u_{10}\right\| _{L^{4}}^{4}, \end{aligned}$$

where \(\overset{\sim }{C}_{3}\) depends on M and T.

Proof

Multiplying the Eq. (2.7) by \(u_{1}^{3}\), operating, neglecting higher power of \(\Gamma ^{4},\) and after using integration by parts

$$\begin{aligned} \iint \limits _{\Omega }u_{1}^{3}\frac{\partial u_{1}}{\partial t}+I_{3}& = \nu \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}} u_{1}^{3}dxdy+\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{ \partial u_{1}}{\partial y}\right) ^{2}u_{1}^{3}dxdy \\&+\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{ \partial u_{1}}{\partial y}\right) ^{4}u_{1}^{3}dxdy\\&-M^{2}\iint \limits _{\Omega }u_{1}^{4}dxdy, \end{aligned}$$

Afer using integration by parts

$$\begin{aligned} \frac{d}{dt}\left\| u_{1}\right\| _{L^{2}}^{4}& = 4I_{3}-12\nu \left\| u_{1}\frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}-6\left( n-1\right) \nu \Gamma ^{2}\left\| u_{1}^{\frac{1}{2} }\frac{\partial u_{1}}{\partial y}\right\| _{L^{4}}^{4}\nonumber \\&-\frac{3}{5}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| u_{1}^{\frac{1}{3}}\frac{\partial u_{1}}{\partial y}\right\| _{L^{6}}^{6}-4M^{2}\left\| u_{1}\right\| _{L^{2}}^{2}, \end{aligned}$$
(5.1)

where

$$\begin{aligned} I_{3}=\text { }\iint \limits _{\Omega }u_{1}^{4}\frac{\partial u_{1}}{\partial x}dxdy+\iint \limits _{\Omega }u_{1}^{3}u_{2}\frac{\partial u_{1}}{\partial y} dxdy. \end{aligned}$$

Applying integration by parts on the second term in right side and making use of Eq. (2.1)

$$\begin{aligned} I_{3}=\frac{1}{2}\iint \limits _{\Omega }u_{1}^{4}\frac{\partial u_{1}}{ \partial x}dxdy. \end{aligned}$$

Integrating again we get \(\left( I_{3}=0\right) \). Introducing the value of \( I_{3}\) in Eq. (5.1) and after Young’s inequality

$$\begin{aligned}&\frac{d}{dt}\left\| u_{1}\right\| _{L^{2}}^{4}+\nu \left\| u_{1} \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{2}+\left( n-1\right) \nu \Gamma ^{2}\left\| u_{1}^{\frac{1}{2}}\frac{\partial u_{1}}{\partial y}\right\| _{L^{4}}^{4} \\&\qquad +\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| u_{1}^{\frac{1}{3}}\frac{\partial u_{1}}{\partial y}\right\| _{L^{6}}^{6} =-2M^{2}\left\| u_{1}\right\| _{L4}^{4} \\&\quad \le \left| 2M^{2}\right| \left\| u_{1}\right\| _{L^{4}}^{4}. \end{aligned}$$

The Grownwall’s inequality yields

$$\begin{aligned}&\underset{0\le t \le T}{\sup }\left\| u_{1}\right\| _{L^{4}}^{4}+\nu \int _{0}^{T}\left\| u_{1}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{2}}^{2}dt+\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| u_{1}^{\frac{1}{2}}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{4}}^{4}dt\qquad \\&\quad +\frac{1}{10}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| u_{1}^{\frac{1}{3}}\frac{\partial u_{1}}{ \partial y}\right\| _{L^{6}}^{6}dt\le \overset{\sim }{C}_{3}\left\| u_{10}\right\| _{L^{4}}^{4}, \end{aligned}$$

where \(\overset{\sim }{C}_{3}\) depends on M and T. \(\square \)

Note that the Theorem 3.2 is shown as per the Proposition 5.1 introduced and proved.

6 Proof of Theorem 3.3

The Theorem is shown based on the coming Propositions.

Proposition 6.1

Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). In addition, assume the existence of \(\nabla u_{10}\) in \(L^{2}\left( \Omega \right) \) and that \(\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) \). Neglecting the higher power of \(\Gamma ^{4}\) then

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| \nabla u_{1}\right\| _{L^{2}}^{2}+\left( 2 \nu -\epsilon \right) \int _{0}^{T}\left\| \frac{ \partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt+3 \left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial u_{1} }{\partial y}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt\le \overset{\sim }{\text { }C}_{4}\left\| \nabla u_{10}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{\text { }C}_{4}\) depends on M and T.

Proof

Considering the inner product in Eq. (2.7) with \(\Delta u_{1}\) and integrating

$$\begin{aligned}&\iint \limits _{\Omega }\frac{\partial u_{1}}{\partial t}\Delta u_{1}dxdy +\iint \limits _{\Omega }\Delta u_{1}\left( u_{1}\frac{\partial u_{1}}{\partial x}+u_{2}\frac{\partial u_{2}}{\partial y}\right) dxdy \\&\quad =\nu \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right] ^{ \frac{n-1}{2}}\Delta u_{1}dxdy \\&\qquad +\nu \left( n-1\right) \Gamma ^{2}\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2} \right] ^{\frac{n-3}{2}}\Delta u_{1}dxdy\\&\qquad -M^{2}\iint \limits _{\Omega }u_{1}\Delta u_{1}dxdy. \end{aligned}$$

Expanding the term on the right side in the last equation and neglectng the higher powers of \(\Gamma ^{4},\)

$$\begin{aligned} \iint \limits _{\Omega }\frac{\partial u_{1}}{\partial t}\Delta u_{1}dxdy+I_{4}& = \nu \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{ \partial y^{2}}\Delta u_{1}dxdy\\&+\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\Delta u_{1}dxdy \\&+\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{4}\Delta u_{1}dxdy \\&-M^{2}\iint \limits _{\Omega }u_{1}\Delta u_{1}dxdy. \end{aligned}$$

Aplying interation by parts

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\left\| \nabla u_{1}\right\| _{L^{2}}^{2}-I_{4} \\&\quad =-\nu \iint \limits _{\Omega }\left( \frac{\partial \nabla u_{1}}{\partial y} \right) ^{2}dxdy-\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\left( \frac{\partial \nabla u_{1}}{\partial y}\right) ^{2}dxdy \\&\qquad -\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{4}\left( \frac{\partial \nabla u_{1}}{\partial y}\right) ^{2}dxdy-M^{2}\left\| \nabla u_{1}\right\| _{L^{2}}^{2}, \end{aligned}$$

which implies that

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\left\| \nabla u_{1}\right\| _{L^{2}}^{2}+\nu \left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}+\frac{3}{2} \left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial u_{1}}{\partial y} \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}\nonumber \\&\quad +\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}=I_{4}-M^{2}\left\| \nabla u_{1}\right\| _{L^{2}}^{2}, \end{aligned}$$
(6.1)

where

$$\begin{aligned} I_{4}=\iint \limits _{\Omega }\Delta u_{1}\left( u_{1}\frac{\partial u_{1}}{ \partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right) dxdy. \end{aligned}$$

Now, the intention is to further develop the integral \(I_{4},\) to this end

$$\begin{aligned} I_{4}& = -\iint \limits _{\Omega }\nabla u_{1}\nabla \left( u_{1}\frac{ \partial u_{1}}{\partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right) dxdy \\& = -\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial x}\right) ^{3}dxdy-\iint \limits _{\Omega }u_{1}\frac{\partial u_{1}}{\partial x}\frac{ \partial ^{2}u_{1}}{\partial x^{2}}dxdy-\iint \limits _{\Omega }\frac{ \partial u_{1}}{\partial x}\frac{\partial u_{1}}{\partial y}\frac{\partial u_{2}}{\partial x}dxdy \\&-\iint \limits _{\Omega }u_{2}\frac{\partial ^{2}u_{1}}{\partial x\partial y}\frac{\partial u_{1}}{\partial x}dxdy-\iint \limits _{\Omega }\left( \frac{ \partial u_{1}}{\partial y}\right) ^{2}\frac{\partial u_{1}}{\partial x} dxdy-\iint \limits _{\Omega }u_{1}\frac{\partial u_{1}}{\partial y}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}dxdy \\&-\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial u_{2}}{\partial y}dxdy-\iint \limits _{\Omega }u_{2}\frac{ \partial u_{1}}{\partial y}\frac{\partial ^{2}u_{1}}{\partial y^{2}}dxdy, \\& = -\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial x}\right) ^{3}dxdy-\iint \limits _{\Omega }\frac{\partial u_{1}}{\partial x} \frac{\partial u_{1}}{\partial y}\frac{\partial u_{2}}{\partial x}dxdy \\&-\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y }\right) ^{2}\frac{\partial u_{1}}{\partial x}dxdy-\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{ \partial u_{2}}{\partial x}dxdy. \end{aligned}$$

From Eq. (2.1), the following holds

$$\begin{aligned} I_{4}=-\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{ \partial x}\right) ^{3}dxdy+\iint \limits _{\Omega }\frac{\partial u_{2}}{ \partial y}\frac{\partial u_{1}}{\partial y}\frac{\partial u_{2}}{\partial x} dxdy. \end{aligned}$$

Integrating again

$$\begin{aligned} I_{4}& = -\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{ \partial x}\right) ^{3}dxdy-\iint \limits _{\Omega }u_{1}\frac{\partial ^{2}u_{2}}{\partial y^{2}}\frac{\partial u_{2}}{\partial x} dxdy-\iint \limits _{\Omega }u_{1}\frac{\partial u_{2}}{\partial y}\frac{ \partial u_{2}}{\partial x\partial y}dxdy \\& = -\frac{1}{2}\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial x }\right) ^{3}dxdy-\iint \limits _{\Omega }u_{1}\frac{\partial ^{2}u_{2}}{ \partial y^{2}}\frac{\partial u_{2}}{\partial x}dxdy+\frac{1}{2} \iint \limits _{\Omega }\left( \frac{\partial u_{2}}{\partial y}\right) ^{2} \frac{\partial u_{1}}{\partial x}dxdy \\& = -\iint \limits _{\Omega }u_{1}\frac{\partial ^{2}u_{1}}{\partial x\partial y}\frac{\partial u_{2}}{\partial x}dxdy, \end{aligned}$$

where Eq. (2.1) has been used, therefore \(I_{4}\) becomes

$$\begin{aligned} I_{4}\le & \left\| u\right\| _{L^{4}}\left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}\left\| \frac{\partial u_{2} }{\partial x}\right\| _{L^{4}} \\\le & \frac{\epsilon }{2}\left\| \frac{\partial \nabla u_{1}}{\partial y} \right\| _{L^{2}}^{2}+\frac{1}{2\epsilon }\left\| u_{1}\right\| _{L^{4}}^{2}\left\| \frac{\partial u_{2}}{\partial x}\right\| _{L^{4}}^{2}. \end{aligned}$$

Introducing the assessed integral \(I_{4}\) into Eq. (6.1) and after using Proposition 3 and \(\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) \)

$$\begin{aligned}&\frac{d}{dt}\left\| \nabla u_{1}\right\| _{L^{2}}^{2}+\left( 2\nu - \epsilon \right) \left\| \frac{\partial \nabla u_{1}}{\partial y} \right\| _{L^{2}}^{2}+3\left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial u_{1}}{\partial y}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2} \\&\quad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}\le C_{16}\left\| \nabla u_{1}\right\| _{L^{2}}^{2} . \end{aligned}$$

From Grownwall’s inequality

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| \nabla u_{1}\right\| _{L^{2}}^{2}+\left( 2\nu -\epsilon \right) \int _{0}^{T}\left\| \frac{ \partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt+3 \left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial u_{1} }{\partial y}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt\le \overset{\sim }{\text { }C}_{4}\left\| \nabla u_{10}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{\text { }C}_{4}\) depends on M and T. \(\square \)

Proposition 6.2

Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). assume that in the strip \(\Omega \), there exists \(\left( \Delta u_{10},\frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) .\) Neglecting the higher power of \(\Gamma ^{4}\), then

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| \Delta u_{1}\right\| _{L^{2}}^{2}+3\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y}\frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +2\left( \nu -8\epsilon \right) \int _{0}^{T}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt\le \overset{\sim }{\text { }C} _{5}\left\| \Delta u_{10}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{\text { }C}_{5}\) depends on constants and T.

Proof

Apply the monotone operator \(\Delta \) to Eq. (2.7) and take inner product with \(\Delta u_{1}\). After integration by parts, the following reads

$$\begin{aligned}&\iint \limits _{\Omega }\Delta u_{1}\Delta \frac{\partial u_{1}}{\partial t }dxdy+\iint \limits _{\Omega }u_{1}\Delta \left( u_{1}\frac{\partial u_{1}}{ \partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right) \Delta dxdy \\&\quad =\nu \iint \limits _{\Omega }\Delta \left( \left( \frac{\partial ^{2}u_{1} }{\partial y^{2}}\right) \left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{ \partial y}\right) ^{2}\right] ^{\frac{n-1}{2}}\right) \Delta u_{1}dxdy \\&\qquad + \nu \left( n-1\right) \Gamma ^{2}\iint \limits _{\Omega }\Delta \left( \left( \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right) \left( \frac{ \partial u_{1}}{\partial y}\right) ^{2}\left[ 1+\Gamma ^{2}\left( \frac{ \partial u_{1}}{\partial y}\right) ^{2}\right] ^{\frac{n-3}{2}}\right) \Delta u_{1}dxdy \\&\qquad -\frac{\nu \phi }{k}\iint \limits _{\Omega }\Delta \left( \left[ 1+\Gamma ^{2}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right] ^{\frac{n-1}{ 2}}u_{1}\right) \Delta u_{1}dxdy-M^{2}\iint \limits _{\Omega }\Delta u_{1}\Delta u_{1}dxdy. \end{aligned}$$

Expanding the terms on the right side and neglecting the higher powers of \(\Gamma ^{4},\)

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\iint \limits _{\Omega }\left( \Delta u_{1}\right) ^{2}dxdy-I_{5}=\nu \iint \limits _{\Omega }\frac{\partial ^{2}u_{1}}{\partial y^{2}}\Delta u_{1}dxdy \\&\quad + \frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\iint \limits _{\Omega }\Delta \left( \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right) \Delta u_{1}dxdy \\&\quad +\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\iint \limits _{\Omega }\Delta \left( \frac{\partial ^{2}u_{1}}{\partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{4}\right) \Delta u_{1}dxdy-M^{2}\iint \limits _{\Omega }\left( \Delta u_{1}\right) ^{2}dxdy. \end{aligned} \end{aligned}$$

After Integration

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left\| \Delta u_{1}\right\| _{L^{2}}^{2}=&I_{5}-\nu \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}I_{6}\\&+\frac{1}{4} \left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}I_{7}-M^{2}\left\| \Delta u_{1}\right\| _{L^{2}}^{2}, \end{aligned}$$
(2.13)

where

$$\begin{aligned} I_{5}& = -\iint \limits _{\Omega }\Delta \left( u_{1}\frac{\partial u_{1}}{ \partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right) \Delta u_{1}dxdy , \\ I_{6}& = \iint \limits _{\Omega }\Delta \left( \frac{\partial ^{2}u_{1}}{ \partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\right) \Delta u_{1}dxdy , \\ I_{7}& = \iint \limits _{\Omega }\Delta \left( \frac{\partial ^{2}u_{1}}{ \partial y^{2}}\left( \frac{\partial u_{1}}{\partial y}\right) ^{4}\right) \Delta u_{1}dxdy. \end{aligned}$$

Now for \(I_{5}\)

$$\begin{aligned} I_{5}& = -\iint \limits _{\Omega }\Delta \left( u_{1}\frac{\partial u_{1}}{ \partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right) \Delta u_{1}dxdy \\& = -\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{\partial x}\left( \frac{\partial ^{2}u_{1}}{\partial x^{2}}+\frac{\partial ^{2}u_{1}}{\partial y^{2}}\right) +\frac{\partial u_{1}}{\partial y}\left( \frac{\partial ^{2}u_{2}}{\partial x^{2}}+\frac{\partial ^{2}u_{2}}{\partial y^{2}}\right) \right) \Delta u_{1}dxdy \\&-\int \int \left( 2\frac{\partial u_{1}}{\partial x}\frac{\partial ^{2}u_{1}}{\partial x^{2}}+2\frac{\partial u_{1}}{\partial y}\frac{\partial ^{2}u_{1}}{\partial x\partial y}+2\frac{\partial u_{2}}{\partial x}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}+2\frac{\partial u_{2}}{\partial y} \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right) \Delta u_{1}dxdy \\&-\iint \limits _{\Omega }\left( u_{1}\frac{\partial ^{3}u_{1}}{\partial x^{3}}+u_{1}\frac{\partial ^{3}u_{1}}{\partial y^{2}\partial x}+u_{2}\frac{ \partial ^{3}u_{1}}{\partial x^{2}\partial y}+u_{2}\frac{\partial ^{3}u_{1}}{ \partial y^{3}}\right) \Delta u_{1}dxdy. \end{aligned}$$

Applying integration by parts and continuity

$$\begin{aligned} \iint \limits _{\Omega }\left( u_{1}\frac{\partial ^{3}u_{1}}{\partial x^{3}} +u_{1}\frac{\partial ^{3}u_{1}}{\partial y^{2}\partial x}+u_{2}\frac{ \partial ^{3}u_{1}}{\partial x^{2}\partial y}+u_{2}\frac{\partial ^{3}u_{1}}{ \partial y^{3}}\right) \Delta u_{1}dxdy=0. \end{aligned}$$

Then

$$\begin{aligned} I_{5}& = -\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{1}}{\partial x }\Delta u_{1}dxdy-\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{1}}{ \partial y}\Delta u_{2}dxdy-2\iint \limits _{\Omega }\Delta u_{1}\frac{ \partial u_{1}}{\partial x}\frac{\partial ^{2}u_{1}}{\partial x^{2}}dxdy \nonumber \\&-2\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{1}}{\partial y} \frac{\partial ^{2}u_{1}}{\partial x\partial y}dxdy-2\iint \limits _{\Omega }u_{1}\frac{\partial u_{2}}{\partial x}\frac{\partial ^{2}u_{1}}{\partial x\partial y}dxdy-2\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{2}}{ \partial y}\frac{\partial ^{2}u_{1}}{\partial y^{2}}dxdy \nonumber \\& = k_{1}+k_{2}+k_{3}+k_{4}+k_{5}+k_{6}\text {.} \end{aligned}$$
(6.2)

In order to solve the above six terms, the Lemma 1 is employed so that

$$\begin{aligned} k_{1}& = -\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{1}}{\partial x }\Delta u_{1}dxdy \nonumber \\\le & C_{0 }\left\| \Delta u_{1}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{2}u_{1}}{\partial x^{2}}\right\| _{L^{2}}^{\frac{1 }{2}} \nonumber \\& = C_{0 }\left\| \Delta u_{1}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{2}u_{2}}{\partial x \partial y}\right\| _{L^{2}}^{\frac{1 }{2}} \nonumber \\\le & C_{0 }\left\| \Delta u_{1}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \nabla u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}} \nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{1}}{\partial x} \right\| _{L^{2}}^{\frac{2}{3}}\left\| \Delta u_{1}\right\| _{L^{2}}^{2} \left\| \frac{\partial \nabla u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{2}{3}}. \end{aligned}$$
(6.3)
$$\begin{aligned} k_{2}& = -\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{1}}{\partial y }\Delta u_{2}dxdy \nonumber \\\le & C_{0 }\left\| \Delta u_{2}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{2}u_{1}}{\partial x\partial y}\right\| _{L^{2}}^{ \frac{1}{2}} \nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{2}\left\| \Delta u_{1}\right\| _{L^{2}}^{2} \left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \Delta u_{2}\right\| _{L^{2}}^{2}. \end{aligned}$$
(6.4)
$$\begin{aligned} k_{3}& = 2\iint \limits _{\Omega }\Delta u_{1}\frac{\partial u_{2}}{ \partial y}\frac{\partial ^{2}u_{1}}{\partial x^{2}}dxdy \nonumber \\\le & C_{0 }\left\| \frac{\partial ^{2}u_{1}}{\partial x^{2}}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2} }\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{ \frac{1}{2}}\left\| \frac{\partial u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{2}u_{2}}{\partial x\partial y}\right\| _{L^{2}}^{\frac{1}{2}} \nonumber \\\le & C_{0 }\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{3}{2} }\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{ \frac{1}{2}}\left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \nabla u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{1 }{2}} \nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{2}}{\partial y} \right\| _{L^{2}}^{\frac{2}{3}}\left\| \Delta u_{1}\right\| _{L^{2}}^{2}\left\| \frac{\partial \nabla u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{2}{3}}{ .\ } \end{aligned}$$
(6.5)
$$\begin{aligned} k_{4}& = -2\iint \limits _{\Omega }\frac{\partial u_{1}}{\partial y}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}\Delta u_{1}dxdy \nonumber \\\le & C_{0 }\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1 }{2}}\left\| \frac{\partial ^{2}u_{1}}{\partial x\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{3}u_{1}}{\partial x^{2}\partial y}\right\| _{L^{2}}^{\frac{1}{2}} \nonumber \\\le & C_{0 }\left\| \frac{\partial u_{1}}{\partial y}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{ \partial \Delta u_{1}}{\partial y}\right\| _{L^{2}} \left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1 }{2}}\nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{1}}{\partial y} \right\| _{L^{2}}^{4}\left\| \Delta u_{1}\right\| _{L^{2}}^{2} +C_{\epsilon }\left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}. \end{aligned}$$
(6.6)
$$\begin{aligned} k_{5}& = -2\iint \limits _{\Omega }\frac{\partial u_{1}}{\partial x}\frac{ \partial ^{2}u_{1}}{\partial x\partial y}\Delta u_{1}dxdy \nonumber \\\le & C_{0 }\left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1 }{2}}\left\| \frac{\partial ^{2}u_{1}}{\partial x\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{3}u_{1}}{\partial x^{2}\partial y}\right\| _{L^{2}}^{\frac{1}{2}} \nonumber \\\le & C_{0 }\left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{ \partial \Delta u_{1}}{\partial y}\right\| _{L^{2}} \left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1 }{2}}\nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{2}}{\partial x} \right\| _{L^{2}}^{4}\left\| \Delta u_{1}\right\| _{L^{2}}^{2} +C_{\epsilon }\left\| \frac{\partial \nabla u_{1}}{\partial y}\right\| _{L^{2}}^{2}. \end{aligned}$$
(6.7)
$$\begin{aligned} k_{6}& = -2\iint \limits _{\Omega }\frac{\partial u_{2}}{\partial y}\frac{ \partial ^{2}u_{1}}{\partial y^{2}}\Delta u_{1}dxdy \nonumber \\\le & C_{0 }\left\| \Delta u_{1}\right\| _{L^{2}} \left\| \frac{\partial u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}} \left\| \frac{\partial ^{2}u_{2}}{\partial x \partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{\partial ^{2}u_{1}}{\partial y^{2}}\right\| _{L^{2}}^{\frac{1}{2}} \left\| \frac{\partial ^{3}u_{1}}{\partial y^{3}}\right\| _{L^{2}}^{\frac{1}{2}} \nonumber \\\le & C_{0 }\left\| \Delta u_{1}\right\| _{L^{2}}^{\frac{3}{2}} \left\| \frac{\partial u_{1}}{\partial x}\right\| _{L^{2}}^{\frac{1}{2}} \left\| \frac{ \partial \nabla u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}}\left\| \frac{ \partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{\frac{1}{2}} \nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{1}}{\partial x} \right\| _{L^{2}}^{\frac{2}{3}}\left\| \Delta u_{1}\right\| _{L^{2}}^{2} \left\| \frac{ \partial \nabla u_{2}}{\partial y}\right\| _{L^{2}}^{\frac{2}{3}}. \end{aligned}$$
(6.8)

Integrating \(I_{6}\), we have

$$\begin{aligned} I_{6}& = -2\iint \limits _{\Omega }\frac{\partial u_{1}}{\partial y}\left( \frac{\partial \nabla u_{1}}{\partial y}\right) ^{2}\frac{\partial \Delta u_{1}}{\partial y}dxdy-\iint \limits _{\Omega }\left( \frac{\partial u_{1}}{ \partial y}\right) ^{2}\left( \frac{\partial \Delta u_{1}}{\partial y} \right) ^{2}dxdy \nonumber \\\le & \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \frac{\partial u_{1}}{\partial y} \left( \frac{\partial \nabla u_{1}}{\partial y}\right) ^{2}\right\| _{L^{2}}^{2}- \left\| \frac{\partial u_{1}}{\partial y}\frac{ \partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}, \end{aligned}$$
(6.9)

where we made use of the Young’s inequality. Similarly to solve \(I_{7}\)

$$\begin{aligned} I_{7}\le \epsilon \left\| \frac{\partial \Delta u_{1}}{\partial y} \right\| _{L^{2}}^{2}+C_{\epsilon }\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\left( \frac{\partial \nabla u_{1}}{\partial y} \right) ^{2}\right\| _{L^{2}}^{2}- \left\| \left( \frac{ \partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}. \end{aligned}$$
(2.22)

Combining Eq. (2.13) to Eq. (2.22), After using Proposition 1,  Proposition 2,  Proposition 3 and \(\Delta u_{2},\frac{\partial \nabla u_{2}}{\partial y}\in L^{2}\left( \Omega \right) ,\) we get

$$\begin{aligned}&\frac{d}{dt}\left\| \Delta u_{1}\right\| _{L^{2}}^{2} +3\left( n-1\right) \nu \Gamma ^{2}\left\| \frac{\partial u_{1}}{ \partial y}\frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}\\&\quad + \frac{1}{2}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{ \partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2} \\&\quad +2\left( \nu -8\epsilon \right) \left\| \frac{\partial \Delta u_{1}}{ \partial y}\right\| _{L^{2}}^{2}\le C_{1}\left\| \Delta u_{1}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(C_{1}\) is suitable constant. Applying Grownwall’s inequality again

$$\begin{aligned}&\underset{0\le t\le T}{\sup }\left\| \Delta u_{1}\right\| _{L^{2}}^{2}+\frac{3}{2}\left( n-1\right) \nu \Gamma ^{2}\int _{0}^{T}\left\| \frac{\partial u_{1}}{\partial y}\frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\frac{1}{4}\left( n-3\right) \left( 2n-5\right) \nu \Gamma ^{4}\int _{0}^{T}\left\| \left( \frac{\partial u_{1}}{\partial y}\right) ^{2}\frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt \\&\quad +\left( \nu -8\epsilon \right) \int _{0}^{T}\left\| \frac{\partial \Delta u_{1}}{\partial y}\right\| _{L^{2}}^{2}dt\le \overset{\sim }{\text { }C} _{5}\left\| \Delta u_{10}\right\| _{L^{2}}^{2}, \end{aligned}$$

where \(\overset{\sim }{\text { }C}_{5}\) depends on suitable constants and T. \(\square \)

Finally, the Theorem 3.3 is shown by using Proposition 6.1 and Proposition 6.2.

7 Conclusion

The proposed Theorems 3.1, 3.2 and 3.3 have been shown in the different supporting propositions. Such Theorems lead to confirm on the existence of regular solutions departing from an initial data generalized under the defined functional spaces. The shown solutions are applicable to the two dimensional Carreau fluid on the defined strip \(\Omega = [0,L] \times [0, \infty )\).