Abstract

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution depending on a number in . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate .

1. Introduction

In this paper, we study the weak solutions to the incompressible 2D-MHD with power law-type nonlinear viscous fluid:

Here, is the flow velocity vector, is the magnetic vector, and is the total pressure. We consider the initial value problem of (1), which requires initial conditions:

We assume that the initial data hold the incompressibility, i.e., and , respectively. In this paper, we deal with given as where and are constants (see, e.g., [1, 2]).

In modern industrial application, non-Newtonian fluids play an important role (see [3–5]). In particular, equation (1) is the simplest self-consistent model which describes the dynamics of electrically conducting liquid with involved rheological structure in a magnetic field.

Some examples of non-Newtonian fluids are coal-water, glues, soaps, etc. (see, e.g., [6]). One class of non-Newtonian fluids can be defined by ( is the rate of the strain tensor, a real function). That is, the relation between the shear stress and the strain rate is nonlinear. In this paper, we study the case which is called power law fluids. Commonly, the case of describes dilatant (or shear thickening) fluids whose viscosity increases with the rate of shear (see, e.g., [6]). On the other hand, pseudoplastic (or shear thinning) fluids correspond to the case of , where viscosity decreases with the increasing rate of shear (see, e.g., [1]).

In what follows, we review some known results related to our concerns. For incompressible Navier-Stokes equation for a non-Newtonian type, namely, in (1), the existence of weak solutions for was first obtained in [7, 8], which is unique for for any dimension (cf. [9]). Later, the existence of weak solutions was investigated for in [10, 11]. On the other hand, in the case of , that is, and , numerous results are known. Among them, we only mention that Ferreira and Villamizar-Roa [12] showed well-posedness, time decay, and stability for 3D magnetohydrodynamic equations.

In [13, 14], Samokhin first studied a nonstationary system of equations describing the motion of the Ostwald-de Waele media type and showed a unique existence of a generalized solution for to the problem based on the Faedo-Galerkin method and the monotone operator method. Later on, Gunzburger et al. in [15] proved the global unique solvability of the initial boundary value problem for the modified Navier-Stokes equations coupled with the Maxwell equations. Here, the authors use the strain tension containing the diffusion operator; that is, they do not deal with the degenerate power law fluids. Recently, Razafimandimby [16] proved the existence of weak solutions for to this model of bipolar type.

In this paper, we will prove the global-in-time existence and uniqueness of the weak solutions for the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) under a condition on the range of .

Our results are based on the standard Galerkin method and some uniform estimates.

Denote by the vector space of all symmetric matrices . Let and . The deviatoric stress tensor , , satisfies the following conditions: (i) is a Carathéodory function(ii)Symmetry: (iii)Polynomial growth:(iv)Coercivity condition: there exists such that(v)Strict monotonicity: for all

By the weak solution of the incompressible 2D-MHD with power law-type nonlinear viscous fluid, we mean solutions satisfying the following definitions:

Definition 1 (weak solution). Let , , and . Suppose that . We say that is a weak solution of the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) if and satisfy the following: (i) satisfies (1) in the sense of distribution; that is,for all with , and for every .

Theorem 2. Let and and . Assume that . A weak solution of the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) exists. In particular, in the case and , the weak solution is unique. Moreover, we obtain the following decay rate of the weak solution:

2. Preliminaries

In this section, we introduce the notation. Let be a finite time interval. For , we denote by the usual Sobolev spaces, namely, . The set of the -th power Lebesgue integrable functions on is denoted by , and indicates the set of the locally -th power Lebesgue integrable functions defined on . For a function , , and , we denote . For vector fields , we write as . We denote for matrices . The letter is used to represent a generic constant, which may change from line to line.

Before looking for a solution for the system (1), we give a lemma.

Lemma 3. Let be a solution to the initial value problem of (1)–(2) with the initial data . Then, we have for where depends only on the -norm of and .

Proof. The proof is easily checked. Indeed, it is almost the same as that in [17] replacing (2.5) in [17] by

3. Proof of Theorem 2

In this paper, we assume that and for convenience. Let with . Now, we will construct the existence of a weak solution to the system (1) via the standard Galerkin method. For this, first of all, we need to find a countable dense subset of the space in in Lemma 3.10 of [18].

Now, we consider Galerkin approximate solutions and , where the , are the eigenfunctions which are chosen by using Lemma 3.10 of [18]. for and . The initial conditions were where and . Indeed, the functions and satisfy the following ordinary differential equations as follows:

By the Carathéodory theorem (see [19], Theorem 3.4 in Appendix), there exist such that equation (16) has unique solutions on . Now set , .

Proof of Theorem 2. For a proof of existence for a weak solution, we assume that because it is easier for .
Part A: existence
Multiplying equation (13) by and equation (14) by and summing up the equations, we have where we use the divergence free condition, Korn’s inequality, and vector identity for the magnetic vector field . For the distributive time derivative , we have . Here, is the conjugate of , and is the dual space for . Indeed, for with , (i)Estimate of : using Hölder’s inequality and the energy estimate (18), we have(ii)Estimate of : since belongs to , we have(iii)Estimate of : using Hölder’s inequality, we haveWe combine (20), (21), and (22) to get To obtain the distributive time derivative , using the similar argument above, we have Indeed, for with , (iv)Estimate of : using Hölder’s inequality and the estimate (18), we have(v)Estimate of : using Hölder’s inequality, we haveDue to the energy estimate (18) and time derivative class for and , we can choose subsequences and such that when goes to . From the class of and in the convergence above and by the Aubin-Lions lemma (e.g., [20], Lemma 3.1), we have Thus, we have as . So then, due to the weak and strong convergence above, it is possible to pass to the limit in the nonlinear terms (see, e.g., [21]). Moreover, is uniformly bounded in , and so in this class. Hence, we will check which is shown by monotonicity trick (see [13], pp. 635-636). For this, we note that for , From the energy equality, we have for Define Here, . So, due to the property of the monotone operator and the semicontinuity of the norm, we obtain and also Then, due to the equality (32), we have Putting for and , we obtain As , we deduce which means that for a.e. . Hence, the proof of existence for weak solutions is completed.
Part B: uniqueness
For this part, we consider the equation for , , and : with and . Testing and to the equations above, we have that is, Applying Gronwall’s inequality, we obtain and in and hence and .
Part C: decay rate
A proof of this part is almost the same as that in [17]. For the convenience of the reader, it gives a proof. From the -energy inequality and Korn’s inequality, it follows that where is a Korn-type constant. Applying Plancherel’s theorem to (42) yields Put . Let be a smooth function of with and . Set . Then, Since we have Integrating in time, we get Set . From Lemma 3 with Young’s inequality and the energy estimate, we have Thus, we get Applying Gronwall’s inequality, we immediately deduce that thus, we finally obtain the desired result.