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

Authors: Jae-Myoung Kim
Journal: Advances in Mathematical Physics
 (2021)

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## 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 solutionu,b depending on a number q in ℝ2. Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1+t−1/2.


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## Body

## 1. Introduction

In this paper, we study the weak solutions to the incompressible 2D-MHD with power law-type nonlinear viscous fluid:(1)ut−∇·S+u·∇u+∇π=b·∇b,bt−△b+u·∇b−b·∇u=0,divu=0,divb=0,inQT≔ℝ2×0,T.Here,u:ℝ2×0,T→ℝ2 is the flow velocity vector, b:ℝ2×0,T→ℝ2 is the magnetic vector, and π:ℝ2×0,T→ℝ is the total pressure. We consider the initial value problem of (1), which requires initial conditions:
(2)ux,0=u0x,bx,0=b0xx∈ℝ2.We assume that the initial datau0x,b0x∈L2ℝ2 hold the incompressibility, i.e., ∇·u0x=0 and ∇·b0x=0, respectively. In this paper, we deal with S given as
(3)S≔SDu=μ0+μ1Du2q−2/2Du,Du=∇u+∇u⊥2,where μ0≥0 and μ1>0 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 S=μ∣D∣D (D is the rate of the strain tensor, μ·>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 μs=μ0+μ1sq−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 3n+2/n+2≤q was first obtained in [7, 8], which is unique for n+2/2≤q for any dimension n (cf. [9]). Later, the existence of weak solutions was investigated for 2n/n+2<q in [10, 11]. On the other hand, in the case of q=2, that is, SDu=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+2n/n+2 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∈1,2n+6/n+2 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.DenoteMsym2 by the vector space of all symmetric 2×2 matrices ζ=ζij1≤i,j≤2. Let S≔Duq−2Du and 1≤q<∞. The deviatoric stress tensor S=Sij, i,j=1,2, satisfies the following conditions:
(i)
S:QT×Msym2→Msym3 is a Carathéodory function(ii)
Symmetry:Sij=Sji(iii)
Polynomial growth:(4)Sijξ≤μ0+μ1ξq−2ξ.(iv)
Coercivity condition: there existsc1>1 such that(5)μ0+μ1ξq−2η2≤∂Sij∂ξklηklηij≤c1μ0+μ1ξq−2η2.(v)
Strict monotonicity: for allζ,η∈Msym2ζ≠η,Sζ−Sη:ζ−η>0By 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).
Letμ0≥0, μ1>0, and q∈1,∞. Suppose that u0,b0∈L2ℝ2. We say that u,b is a weak solution of the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) if u and b satisfy the following:
(6)u∈L∞0,T;L2ℝ2∩L20,T;W1,qℝ2,b∈L∞0,T;L2ℝ2∩L20,T;H1ℝ2.(i)
u,b satisfies (1) in the sense of distribution; that is,(7)∫0T∫ℝ2∂ϕ∂t+u·∇ϕudxdt+∫0T∫ℝ2SDu:∇ϕdxdt=∫ℝ2u0ϕx,0dx+∫0T∫ℝ2b·∇ϕbdxdt,∫0T∫ℝ2∂ϕ∂t+Δϕ+u·∇ϕbdxdt=∫ℝ2b0ϕx,0dx+∫0T∫ℝ2b·∇ϕudxdt,for all ϕ∈C0∞ℝ2×0,T with divϕ=0, and
(8)∫ℝ2u·∇ψdx=0,∫ℝ2b·∇ψdx=0,for every ψ∈C0∞ℝ2.Theorem 2.
Let2<q<∞ and μ0≥0 and μ1>0. Assume that u0,b0∈L2ℝ2. A weak solution u,b of the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) exists. In particular, in the case μ0>0 and μ1>0, the weak solution u,b is unique. Moreover, we obtain the following decay rate of the weak solution:
(9)utL2+btL2≤C1+t−1/2.


## 2. Preliminaries

In this section, we introduce the notation. LetI be a finite time interval. For 1≤q≤∞, we denote by Wk,qℝ2 the usual Sobolev spaces, namely, Wk,qℝ2=f∈Lqℝ2:Dαf∈Lqℝ2,0≤α≤k. The set of the q-th power Lebesgue integrable functions on ℝ2 is denoted by Lqℝ2, and Llocqℝ2 indicates the set of the locally q-th power Lebesgue integrable functions defined on ℝ2. For a function fx,t, O⊂ℝ2, and J⊂I, we denote fLx,tp,qO×J=fLpOLqJ. For vector fields u,v, we write uivji,j=1,2,3 as u⊗v. We denote A:B=aijbij for 3×3 matrices A=aij,B=bij. The letter C 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.
Letu,b be a solution to the initial value problem of (1)–(2) with the initial data u0,b0∈L2ℝ2∩L1ℝ2. Then, we have for 2≤p(10)u^ξ,t+b^ξ,t≤Cu^0ξ+b^0ξ+ξ∫0tusL2ℝ22+bsL2ℝ22ds+Cξ∫0tusL2ℝ2ds1/p−1,where C depends only on the L2∩L1ℝ2-norm of u0 and b0.Proof.
The proof is easily checked. Indeed, it is almost the same as that in [17] replacing (2.5) in [17] by
(11)∫0t∇usLp−1ℝ2p−1ds≤∫0tusL2ℝ21/p−1∇usLpℝ2pp−2/p−1ds≤C∫0tusL2ℝ2ds1/p−1∫0t∇usLpℝ2pdtp−2/p−1≤C∫0tusL2ℝ2ds1/p−1,p>2.


## 3. Proof of Theorem2

In this paper, we assume thatμ0=0 and μ1=1 for convenience. Let
(12)Vq≔φ∈D′ℝ22:∇·φ=0,with φVq≔DφLqℝ2. 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 φ∈Dℝ2:∇·φ=0 in W2,2ℝ2∩Vq in Lemma 3.10 of [18].Now, we consider Galerkin approximate solutionsumt=Σi=1mgjmtφj and bmt=Σi=1mhjmtψj, where the φj, ψj are the eigenfunctions which are chosen by using Lemma 3.10 of [18].
(13)utm−∇·SDu+um·∇um−bm·∇bm,φ=0,(14)btm−△bm+um·∇bm−bm·∇um,ψ=0,for φ∈spanφ1,φ2,⋯φm and ψ∈spanψ1,ψ2,⋯ψm. The initial conditions were
(15)umx,0=Σi=1maiφix,bmx,0=Σi=1mciψix,where ai=∫ℝ2umx,0·φix and ci=∫ℝ2bmx,0·ψix. Indeed, the functions gjmt and hjmt satisfy the following ordinary differential equations as follows:
(16)ġjmt+λugjmt+gkmtp−1∫ℝ2∇φkp−2∇φk·∇φj+gkmtglmt∫ℝ2φk·∇φlφj−hkmthlmt∫ℝ2ψk·∇ψlφj=0,(17)ḣjmt+λbhjmt+gkmthlmt∫ℝ2φk·∇ψlψj−hkmtglmt∫ℝ2ψk·∇φlψj=0.By the Carathéodory theorem (see [19], Theorem 3.4 in Appendix), there exist Tm such that equation (16) has unique solutions on 0,Tm. Now set Tm=T, T<∞.Proof of Theorem 2.
For a proof of existence for a weak solution, we assume thatμ0=0 because it is easier for μ0>0.
Part A: existence
Multiplying equation (13) by um and equation (14) by bm and summing up the equations, we have
(18)umTL2ℝ22+bmTL2ℝ22+∇umLqQTq+∇bmL2QT2≤um0L2ℝ22+bm0L2ℝ22,where we use the divergence free condition, Korn’s inequality, and vector identity for the magnetic vector field b. For the distributive time derivative dum/dt, we have dum/dt∈Lq′0,T;W1,q∗+Lq0,T;W1,q′∗+L20,T;W1,2∗. Here, q′ is the conjugate of p, and W1,q′ℝ2∗ is the dual space for W1,q′ℝ2. Indeed, for ϕ∈Lq0,T;W1,q∩Lq′0,T;W1,q′ℝ2∩L20,T;W1,2ℝ2 with ∇·ϕ=0,
(19)∫0T∫ℝ2dumdt·ϕdxdt=∫0T∫ℝ2∇·SDumdx−∇·um⊗um+∇·bm⊗bm·ϕdxdt=−∫0T∫ℝ2Dumq−2Dum:∇ϕdxdt+∫0T∫ℝ2um⊗um:∇ϕdxdt−∫0T∫ℝ2bm⊗bm:∇ϕdxd≔I1+I2+I3.(i)
Estimate ofI1: using Hölder’s inequality and the energy estimate (18), we have(20)I1≤Dumq−1Lq′0,T;Lq′∇ϕLq0,T;Lq≤DumLq0,T;Lqq−1∇ϕLq0,T;Lq≤Cu0L2ℝ22,T,q∇ϕLq0,T;Lq.(ii)
Estimate ofI2: since um belongs to L2q0,T;L2q, we have(21)I2≤um⊗umLq0,T;Lq∇ϕLq/q−10,T;Lq/q−1≤umL2q0,T;L2qumL2q0,T;L2q∇ϕLq/q−10,T;Lq/q−1≤C.(iii)
Estimate ofI3: using Hölder’s inequality, we have(22)I3≤bm⊗bmL20,T;L2∇ϕL20,T;L2≤bmL40,T;L4bmL40,T;L4∇ϕL20,T;L2≤C.
We combine (20), (21), and (22) to get
(23)dumdt∈Lq′0,T;W1,q∗+Lq0,T;W1,q′∗+L20,T;W1,2∗.
To obtain the distributive time derivativedbm/dt, using the similar argument above, we have
(24)dbmdt∈L20,T;W1,2∗+L20,T;W1,4∗.
Indeed, forϕ∈L20,T;W1,2∩L20,T;W1,4 with ∇·ϕ=0,
(25)∫0T∫ℝ2dumdt·ϕdxdt=−∫0T∫ℝ2∇bm:∇ϕdxdt+∫0T∫ℝ2um⊗bm:∇ϕdxdt≔J1+J2.(iv)
Estimate ofI1: using Hölder’s inequality and the estimate (18), we have(26)J1≤∇bmL20,T;L2∇ϕL20,T;L2≤C.(v)
Estimate ofI2: using Hölder’s inequality, we have(27)J2≤CumL2q0,T;L2qbmL40,T;L4∇ϕL4q3q−20,T;L4q3q−2≤C.
Due to the energy estimate (18) and time derivative class for um and bm, we can choose subsequences umk and bmk such that
(28)umk⇀uweaklyinL∞I;L2ℝ2∩LqI;W1,qℝ2,bmk⇀bweaklyinL∞I;L2ℝ2∩L2I;W1,2ℝ2,∂tumk⇀∂tuweaklyinLq′0,T;W1,q∗+Lq0,T;W1,q′∗+L20,T;W1,2∗,∂tbmk⇀∂tbweaklyinL20,T;W1,2∗+L20,T;W1,4∗,when k goes to ∞. From the class of umk and bmk in the convergence above and by the Aubin-Lions lemma (e.g., [20], Lemma 3.1), we have
(29)umk⇀ustronglyinLlocpℝ2×I,p∈1,2q,bmk⇀bstronglyinLlocp~ℝ2×I,p~∈1,4.
Thus, we have(30)umk→ustronglyinLloc2ℝ2×I,bmk→bstronglyinLloc2ℝ2×I,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, SDum is uniformly bounded in Lq′ℝ2×0,T, and so SDu⇀A in this class. Hence, we will check A=SDu which is shown by monotonicity trick (see [13], pp. 635-636). For this, we note that for q≥2,
(31)t→∫ℝ2u·∇u·udx∈L10,T,t→∫ℝ2u·∇b·bdx∈L10,Tt→∫ℝ2b·∇b·udx∈L10,T,t→∫ℝ2b·∇u·bdx∈L10,T.
From the energy equality, we have for0≤s≤T(32)12usL22+bsL22+∫0s∇bL2dt+∫0sA·Dudt=12u0L22+b0L22.
Define(33)Xsm=∫0sSDum−SDϕ,Dum−Dϕdt+12usL22,ϕ∈Lq0,T;Wσ1,q.
Here,Wσ1,q≔v∈W1,qℝ2:∇·v=0. So, due to the property of the monotone operator S and the semicontinuity of the norm, we obtain
(34)liminfm→∞Xsm≥12us2,and also
(35)limm→∞Xsm=∫0s12u0L22+∫0sb·∇b·udt−∫0sA,Dϕdt−∫0sSDϕ,Du−Dϕdt
Then, due to the equality (32), we have
(36)∫0sA−SDϕ·Du−Dϕdt≥0,a.e.s∈0,T.
Puttingϕ=u−λw for λ>0 and w∈Lq0,T;Wσ1,q, we obtain
(37)∫0sA−SDu−λw·wdt≥0.
Asλ→0, we deduce
(38)∫0sA−SDu·wdt≥0,which means that A=SDu for a.e. s∈0,T. Hence, the proof of existence for weak solutions is completed.
Part B: uniqueness
For this part, we consider the equation forv=u1−u2, h=b1−b2, and π~=π1−π2:
(39)∂tv−∇·1+Du1q−2Du1+∇·1+Du2q−2Du2+u1·∇v+v·∇u2−b1·∇h−h·∇b2+∇π~=0,∂th−Δh~+u1·∇h+v·∇b2−b1·∇v−h·∇u2=0,with divv=0 and divh=0. Testing v and h to the equations above, we have
(40)12ddtvL2ℝ22+hL2ℝ22+∇vL2ℝ22+∇hL2ℝ22≤∫ℝ2v·∇u2·v−∫ℝ2h·∇b2·v+∫ℝ2v·∇b2·h−∫ℝ2h·∇u2·h≤vL22u2L22∇u2L22+hL22b2L22∇b2L22+vL22b2L22∇b2L22+hL22u2L22∇u2L22+12∇vL22+∇hL22,that is,
(41)12ddtvL2ℝ22+hL2ℝ22+∇vL2ℝ22+∇hL2ℝ22≤vL22+hL22u2L22∇u2L22+b2L22∇b2L22.
Applying Gronwall’s inequality, we obtainv=0 and h=0 in ℝ2 and hence u1=u2 and b1=b2.
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 L2-energy inequality and Korn’s inequality, it follows that
(42)12ddt∫ℝ3u2+b2dx+minC,1∫ℝ3∇u2+∇b2≤0,where C>0 is a Korn-type constant. Applying Plancherel’s theorem to (42) yields
(43)12ddt∫ℝ3u∧ξ,t2+b∧ξ,t2dξ+minC,1∫ℝ3ξ2u∧ξ,t2+b∧ξ,t2dξ≤0.
PutΦξ,t≔u∧ξ,t2+b∧ξ,t2. Let ft be a smooth function of t with f0=1,ft>0 and f′t>0. Set St=ξ∈ℝn:2minC,1ftξ2≤f′t. Then,
(44)2minC,1ft∫ℝ3ξ2Φξ,t2dξ≥f′t∫ℝ3Φξ,t2dξ−f′t∫StΦξ,t2dξ.
Since(45)ddtft∫ℝ3Φξ,t2dξ+2minC,1ft∫ℝ3ξ2Φξ,t2dξ≤f′t∫ℝ3Φξ,t2dξ,we have
(46)ddtft∫ℝnu∧ξ,t2dξ≤f′t∫Stu∧ξ,t2dξ.
Integrating in time, we get(47)ft∫ℝnu∧ξ,t2+b∧ξ,t2dξ≤∫ℝnu∧0ξ2+b∧0ξ2dξ+C∫0tf′s∫Ssu∧ξ,s2+b∧ξ,s2dξds.
Setft=1+t2. From Lemma 3 with Young’s inequality and the energy estimate, we have
(48)1+t2∫ℝnu∧ξ,t2+b∧ξ,t2dξ≤∫ℝnu∧0ξ2+b∧0ξ2dξ+C∫0t1+s∫Ssu∧0ξ2+b∧0ξ2dξds+C∫0t1+s∫Ssξ2∫0tusL22+bsL22ds2dξds+C∫0t1+s∫Ssξ2∫0tusL2ℝ22ds1/p−1dξds≤∫ℝnu∧0ξ2+b∧0ξ2dξ+C∫0t1+s∫Ssu∧0ξ2+b∧0ξ2dξds+C∫0t1+s∫Sssξ2∫0tusL24+bsL24dsdξds+C∫0t1+s∫Sssξ2∫0tusL2ℝ22ds+Cdξd≤∫ℝnu∧0ξ2+b∧0ξ2dξ+C∫0t1+s∫Ssu∧0ξ2+b∧0ξ2dξd+C∫0t1+s2∫Ssξ2∫0tusL24+bsL24dsdξds+C∫0t1+s2∫Ssξ2∫0tusL2ℝ22dsdξds≤C+C1+t+∫0tusL22+bsL22ds.
Thus, we get(49)1+t∫ℝ3uξ,t2+bξ,t2dξ≤C+C∫0t1+sus2+bs21+s−1ds.
Applying Gronwall’s inequality, we immediately deduce that(50)utL2+btL2≤C1+t−1/2,thus, we finally obtain the desired result.



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