The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous ﬂ uid. In this paper, we establish the global existence and uniqueness of a weak solution ð u , b Þ depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the ﬂ uid ﬂ ows has decay rate ð 1 + t Þ − 1/2 .

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 S = μð|D | ÞD (D is the rate of the strain tensor, μð·Þ > 0 a real function). That is, the relation between the shear stress and the strain rate is nonlinear. In this paper, we study the case μðsÞ = μ 0 + μ 1 s q−2 which is called power law fluids. Commonly, the case of q > 2 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 1 < q < 2, 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, b = 0 in (1), the existence of weak solutions for ð3n + 2Þ/ðn + 2Þ ≤ q was first obtained in [7,8], which is unique for ðn + 2Þ/2 ≤ q for any dimension n (cf. [9]). Later, the existence of weak solutions was investigated for 2n/ðn + 2Þ < q in [10,11]. On the other hand, in the case of q = 2, that is, SðDuÞ = Du and n = 3, 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 q ≥ 1 + ð2n/ðn + 2ÞÞ 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 q ∈ ð1, ð2n + 6Þ/ðn + 2Þ 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 q.
Our results are based on the standard Galerkin method and some uniform estimates.

Preliminaries
In this section, we introduce the notation. Let I be a finite time interval. For 1 ≤ q ≤ ∞, we denote by W k,q ðℝ 2 Þ the usual Sobolev spaces, namely, The set of the q-th power Lebesgue integrable functions on ℝ 2 is denoted by L q ðℝ 2 Þ, and L q loc ð ℝ 2 Þ indicates the set of the locally q-th power Lebesgue integrable functions defined on ℝ 2 . For a function f ðx, tÞ, O ⊂ ℝ 2 , and J ⊂ I, we denote k f k L p,q x,t ðO×JÞ = kkf k L p ðOÞ k L q ðJÞ . For vector fields u, v, we write ðu i v j Þ i,j=1,2,3 as u ⊗ v. We denote A : B = a ij b ij for 3 × 3 matrices A = ða ij Þ, B = ðb ij Þ. The letter C is used to represent a generic constant, which may change from line to line.

Advances in Mathematical Physics
Before looking for a solution for the system (1), we give a lemma.

Lemma 3.
Let ðu, bÞ be a solution to the initial value problem of (1)-(2) with the initial data where C depends only on the ðL 2 ∩ L 1 Þðℝ 2 Þ-norm of u 0 and b 0 .
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 μ 0 = 0 and μ 1 = 1 for convenience. Let with kφk V q ≔ kDφk L q ðℝ 2 Þ . 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 fφ ∈ Dðℝ 2 Þ: ∇·φ = 0g in W 2,2 ðℝ 2 Þ ∩ V q in Lemma 3.10 of [18]. Now, we consider Galerkin approximate solutions u m ðt Þ = Σ m i=1 g m j ðtÞφ j and b m ðtÞ = Σ m i=1 h m j ðtÞψ j , where the φ j , ψ j are the eigenfunctions which are chosen by using Lemma 3.10 of [18].
Proof of Theorem 2. For a proof of existence for a weak solution, we assume that μ 0 = 0 because it is easier for μ 0 > 0.
Here, q′ is the conjugate of p, and ðW 1,q ′ ðℝ 2 ÞÞ * is the dual space for W 1,q′ ðℝ 2 Þ. Indeed, for ϕ ∈ L q ð0, T ; W 1,q Þ ∩ L q′ ð0, T ; ðW 1,q′ ðℝ 2 ÞÞÞ ∩ L 2 ð0, T ; ðW 1,2 ðℝ 2 ÞÞÞ with ∇·ϕ = 0, (i) Estimate of I 1 : using Hölder's inequality and the energy estimate (18), we have 3 Advances in Mathematical Physics (ii) Estimate of I 2 : since u m belongs to L 2q ð0, T ; L 2q Þ, we have (iii) Estimate of I 3 : using Hölder's inequality, we have We combine (20), (21), and (22) to get To obtain the distributive time derivative db m /dt, using the similar argument above, we have Indeed, for ϕ ∈ L 2 ð0, T ; W 1,2 Þ ∩ L 2 ð0, T ; W 1,4 Þ with ∇·ϕ = 0, (iv) Estimate of I 1 : using Hölder's inequality and the estimate (18), we have (v) Estimate of I 2 : using Hölder's inequality, we have Due to the energy estimate (18) and time derivative class for u m and b m , we can choose subsequences u m k and b m k such that when k goes to ∞. From the class of u m k and b m k in the convergence above and by the Aubin-Lions lemma (e.g., [20], Lemma 3.1), we have Thus, we have as k → ∞. 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, SðDu m Þ is uniformly bounded in L q′ ðℝ 2 × ð0, TÞÞ, and so SðDuÞ ⇀ A in this class. Hence, we will check A = SðDuÞ which is shown by monotonicity trick (see [13], pp. 635-636). For this, we note that for q ≥ 2, From the energy equality, we have for 0 ≤ s ≤ T Advances in Mathematical Physics Here, W 1,q σ ≔ fv ∈ W 1,q ðℝ 2 Þ: ∇·v = 0g. So, due to the property of the monotone operator S and the semicontinuity of the norm, we obtain and also Then, due to the equality (32), we have Putting ϕ = u − λw for λ > 0 and w ∈ L q ð0, T ; W 1,q σ Þ, we obtain As λ → 0, we deduce which means that A = SðDuÞ for a.e. s ∈ ½0, T. Hence, the proof of existence for weak solutions is completed. Part B: uniqueness For this part, we consider the equation with div v = 0 and div h = 0. Testing v and h to the equations above, we have that is, Applying Gronwall's inequality, we obtain v = 0 and h = 0 in ℝ 2 and hence u 1 = u 2 and b 1 = b 2 .
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 L 2 -energy inequality and Korn's inequality, it follows that Applying Gronwall's inequality, we immediately deduce that thus, we finally obtain the desired result.