Strong solutions for the steady incompressible MHD equations of non-Newtonian fluids

Strong solutions for the steady incompressible MHD equations of non-Newtonian fluids

Weiwei Shi and Changjia Wang

School of Science, Changchun University of Science and Technology, Changchun, 130022, P. R. China Received 10 February 2020, appeared 7 April 2020

Communicated by Maria Alessandra Ragusa

Abstract. In this paper we deal with a system of partial differential equations de- scribing a steady motion of an incompressible magnetohydrodynamic fluid, where the extra stress tensor is induced by a potential with p-structure (p=2 corresponds to the Newtonian case). By using a fixed point argument in an appropriate functional setting, we proved the existence and uniqueness of strong solutions for the problem in a smooth domain Ω ⊂ _Rⁿ (n = 2, 3) under the conditions that the external force is small in a suitable norm.

Keywords: strong solutions, existence and uniqueness, incompressible magnetohydro- dynamics, non-Newtonian fluids.

2020 Mathematics Subject Classification: 35M33, 35A01, 35D30.

1 Introduction and main result

Magnetohydrodynamics (MHD) concerns the interaction of electrically conductive fluids and electromagnetic fields. The system of partial differential equations in MHD are basically obtained through the coupling of the dynamical equations of the fluids with the Maxwell’s equations which is used to take into account the effect of the Lorentz force due to the mag- netic field, it has spanned a very large range of applications [21,24,25]. By neglecting the displacement current term, a commonly used simplified MHD system could be described by















u_t+ (u· ∇)u−divτ(Du) +∇p= ¹

µ(∇ ×b)×b+ f, in Q_T, b_t+ ¹

µcurl 1

σcurlb

=curl(u×b), in QT,

divu=0, divb=0, in Q_T,

(1.1)

where QT = _Ω×(_0,T), the unknown functions u = (_u1(_x,t)_,u2(_x,t)_{, . . . ,un}(_x,t)) _denotes the velocity of the fluid, b = (b₁(x,t),b₂(x,t), . . . ,bn(x,t)) the magnetic field, p = p(x,t) the pressure and f = (f₁(x,t),f₂(x,t), . . . ,f_n(x,t)) the external force applied to the fluid.

BCorresponding author. Email: wangchangjia@gmail.com

Also, τ = (τ_ij)is the stress tensor depending on the strain rate tensor Du = ¹₂(∇u+∇u^T), µ>0 and σ >0 denotes the permeability coefficient and the electric conductivity coefficient respectively. For the sake of simplicity, in this work, we takeµ=1 andσ=1.

Due to the conventional belief that the Navier–Stokes equations are an accurate model for the motion of incompressible fluids in many practical situations, the majority of the known work have assumed that the stress tensor τ(Du) is a linear function of the strain rate Du.

In this way we obtain the conventional system for MHD, and this classical model has been extensively studied. For instance, Duvaut and Lions [7] established the local existence and uniqueness of a solution in the Sobolev space H^s(R^N)(s ≥ N). They also proved the global existence of a solutions to this system with small initial data. Sermange and Temam [28]

proved the existence of a unique global solution in the two space dimensions. For the zero magnetic diffusion case, Lin, Xu and Zhang [22] and Xu and Zhang [29] established the global well-posedness in two and three dimensional space, respectively, under the assumption that the initial data are sufficiently close to the equilibrium state. The global existence of smooth solutions was proved by Lei [18] for the ideal MHD with axially symmetric initial datum in H^s(R³)withs≥2. For more details, one can also refer [3–5,8,9,11,13–16,23] and the reference cited therein.

In recent years, the flow of non-Newtonian fluids (i.e. the stress tensorτ(Du)being a non- linear function ofDu) has gained much importance in numerous technological applications.

Further, the motion of the non-Newtonian fluids in the presence of a magnetic field in differ- ent contexts has been studied by several authors (see [2,6,26]). A typical form of the stress tensorτ(Du)is of somep– structure withDuwhich were firstly proposed by Ladyzhenskaya in [19,20]. For the MHD equations of non-Newtonian type (1.1), the known results are limited and here we only recall two results closely related to ours. In case thatτ(Du) =|Du|^p−2Du forp≥ ⁵₂, Samokhin proved in [27] the existence of weak solutions by using Galerkin method and the monotone theory, which solve the equations in the sense of distributions and satisfy the following energy inequality

sup

0≤t≤T

(ku(t)k²₂+kb(t)k²₂) +2 Z _T

0

(k∇u(t)k^p_p+k∇b(t)k²₂)dt≤(ku₀k²₂+kb₀k²₂).

Later on, Gunzburger and his collaborators considered (1.1) with τ(Du) = (1+|Du|^p−2)Du for the case of bounded or periodic domains, and they showed the existence and uniqueness of a weak solutions, see [12] for more details.

In this paper, in a smooth bounded domain Ω ⊂ _Rⁿ (n = 2 or 3), we consider a steady incompressible MHD equations of non-Newtonian fluids described by











−divh

2µ(1+|Du|²)^p−22Dui

+∇p= f −div(u⊗u) + (∇ ×b)×b, x∈_Ω,

−_∆b= (b· ∇)u−(u· ∇)b, x∈_Ω,

divu=_{0, div}b=_0, x∈_Ω,

(1.2)

supplemented by the boundary conditions

u|_∂Ω =0, b·n|_∂Ω=0, (∇ ×b)×n|_∂Ω=0, (1.3) where p>1,nis the unit outward normal vector of∂Ω.

Remark 1.1. Since u and b are divergence free (i.e. divu = 0, divb = 0), an elementary computations leads to the formulas

curl curlb=−_∆b, curl(u×b) = (b· ∇)u−(u· ∇)b. (1.4)

The aim of this paper is to prove the existence and uniqueness of strong solutions to system (1.2)–(1.3) under the assumption that theL^q-norm of the external force field f is small in a suitable sense. Our approach is based on regularity results for the Stokes problem and magnetic equation, and a fixed-point argument.

Throughout the paper, form∈N, the standard Lebesgue spaces are denoted byL^q(_Ω)_and their norms byk · k_q, the standard Sobolev spaces are denoted byW^m,q(_Ω)and their norms by k · k_m,q. We also denote byW^m,q₀ (_Ω)the closure inW^m,q(_Ω)of C^∞₀ (_Ω). W^−1,q(_Ω)denotes the dual ofW₀^1,q(_Ω) and their norms byk · k_−1,q;Ω. Forx,y ∈ _R we denote(x,y)⁺ = max{x,y}, x⁺ =max{x, 0}. We introduce the constants

Sp := (|p−2|, 2)⁺, rp := ¹+ (p−3)⁺−(p−4)⁺

2 , γp:= [(p, 3)⁺−2]^(p,3)+−2

[(p, 3)⁺−1]^{(p,3)+−1. (1.5)} We also introduce the space

V :={u∈ C₀^∞(_Ω), divu=0}; Vp :={u∈_W^1,p₀ (_Ω): divu=0};

V_m,p :={v∈W^1,p₀ (_Ω)∩W^m,p(_Ω): divv=0 }; W :={b∈W^1,2(_Ω)_{: div}b=_{0 ,}b·n|_∂Ω =₀}. Also, forq>r> nandδ >0, let us denote byB_δ the convex set defined by

B_δ :=ⁿ(ξ,η)∈V2,q×(W^2,r(_Ω)∩W):CEk∇ξk_1,q ≤δ, C_Eek∇ηk_1,r ≤δ o

, (1.6) where C_E is the norm of the embedding of W^1,q(_Ω) into L^∞(_Ω) and C_Ee is the norm of the embedding of W^1,r(_Ω) into L^∞(_Ω), also C_p denotes the Poincaré constant corresponding to the general Poincaré inequality k · k_s ≤ Cpk∇(·)k_s. We consider the space V2,q×W^2,r(_Ω) endowed with the norm

k(ξ,η)k_1,q,r:=max{k∇ξk_1,q, k∇ηk_1,r}. Now, we formulate the main theorem of this paper.

Theorem 1.2. Assume that q > r > n, p > 1, µ > 0, and let f ∈ L^q(_Ω). There exist positive constant C =C(C₀,C_p,C_E,C

Ee,C₋₁,c₂)such that if C

"

1+ ¹ µ

Ckfk_q µ

+S_pCkfk_q µ

2rp

1+ ^Ckfk_q µ

(p−4)⁺#

< ¹

4^(p−2,1)+, (1.7) then, problem(1.2)–(1.3)has a unique strong solution(u,b)∈V2,q×W^2,r(_Ω).

Remark 1.3. As usual, the pressureπ has disappeared from the notion of solution. Actually, the pressure may be recovered by de Rham Theorem at least in L²(_Ω), such that the triple (u,π,b)satisfies equations (1.2)–(1.3) almost everywhere (see [11]).

The rest of our paper is organized as follows: in Section 2, we review some known results and Section 3 is devoted to proving the main theorem to problem (1.2)–(1.3).

2 Preliminary lemmas

In this section, we recall some basic facts which will be used later.

Lemma 2.1([10, Theorem 6.1, pp. 225]). Let m≥ −1be an integer and letΩbe a bounded domain inRⁿ(n = 2, 3)with boundary ∂Ωof classC^k with k = (m+2, 2)⁺. Then for anyψ ∈ _W^m,ρ(_Ω), the following system







−_∆u+∇π =ψ, x ∈_Ω, divu=0, x ∈_Ω, u|_∂Ω=0,

admits a unique solution[u,π]∈W^m+2,ρ(_Ω)×W^m+1,ρ(_Ω). Moreover, the following estimate holds k∇uk_m+1,ρ+kπk_m+1,ρ/R≤ Cmkψk_m,ρ,

where C_m =C_m(n,ρ,Ω)is a positive constant.

Lemma 2.2([1]). Let r_p,γ_p are given by(1.5)and let G:R⁺ →_Rbe defined by G(δ) = Aδ²−δ+EδH(δ) +D,

where A,E,D are positive constants and H(x) = _x^2rp(₁+_x)^(p−4)+. Thus, if the following assertion holds

AD+ED^2rp(1+D)^(p−4)+ ≤ γp,

then G possesses at least one rootδ₀. Moreover,δ₀ >D and for every β∈ [1, 2]the following estimate holds

β−1

β δ₀+²−β

β Aδ₀²+ ^2rp+1−β

β Eδ₀H(δ₀) + ^E(p−4)⁺

β δ^2r₀ ^p+2(1+δ₀)^(p−4)+−1 ≤D.

Lemma 2.3([17]). Let X and Y be Banach spaces such that X is reflexive and X ,→ Y. Let B be a non-empty, closed, convex and bounded subset of X and let T: B→B be a mapping such that

kT(u)−T(v)k_Y ≤Kku−vk_Y, ∀ u,v∈ B (0<K<1), then T has a unique fixed point in B.

3 Proof of Theorem 1.2

Our proof relies on a Banach fixed point theorem. Toward this aim, we first reformulate the problem as follows















−_µ∆u+∇p= f −div(u⊗u) + (∇ ×b)×b+div[2µσ(|Du|²)Du], x ∈_Ω,

−_∆b= (b· ∇)u−(u· ∇)b, x ∈_Ω,

divu=0, divb=0, x ∈_Ω,

u|_∂Ω =0, b·n|_∂Ω=0, (∇ ×b)×n|_∂Ω =0,

(3.1)

whereσ(x) = (₁+x)^p−22 −_1.

Given(ξ,η)∈V_2,q×W^2,r(_Ω), we consider the following problem















−_µ∆u+∇p= f −div(ξ⊗ξ) + (∇ ×η)×η+div[2µσ(|Dξ|²)Dξ], x ∈_Ω,

−_∆b= (η· ∇)ξ−(ξ· ∇)η, x ∈_Ω,

divu =0, divb=0, x ∈_Ω,

u|_∂Ω =_0, b·n|_∂Ω=_0, (∇ ×b)×n|_∂Ω=_0.

(3.2)

From Lemma 2.1 and Proposition 2.30 in [11], there exists a unique solution (u,b) ∈ V_2,q× W^2,r(_Ω)to (3.2). We define the mapping

T :(ξ,η)→(u,b). Our purpose now is to prove that T_Bδ

0 is a contraction from B_δ0 to itself for someδ₀ > 0.

HereB_δ0 is the closed ball defined in (1.6).

Proposition 3.1. Let q> r > n, p>1,µ> 0, and let f ∈L^q(_Ω). There exists a positive constant M₁= M₁(C₀,C_p,C_E,C_Ee)such that if

M²₁kfk_q

µ² +M₁S_p

M₁kfk_q µ

2rp

1+ ^M1kfk_q µ

(p−4)⁺

≤γ_p, (3.3)

then T(Bδ₀)⊆ Bδ₀ for someδ₀>0.

Proof. Let(ξ,η)∈ B_δ. From Lemma 2.1,u∈V2,qand it satisfies k∇uk_1,q≤ ^C0

µ kfk_q+kξ· ∇ξk_q+k(∇ ×η)×ηk_q+kdiv[2µσ(|Dξ|²)Dξ]k_q. (3.4) Notice that

k(∇ ×η)×ηk_q≤ kηk_∞k∇ηk_q≤C_Eekηk_1,rk∇ηk_1,r

≤δ(Cp+1)k∇ηk_r≤δ(Cp+1)k∇ηk_1,r

≤ (C_p+1) CEe

δ², (3.5)

reasoning as in [1], we could obtain

kξ· ∇ξk_q+kdiv[2µσ(|Dξ|²)Dξ]k_q≤ ^Cp CE

δ²+^4µSp CE

δH(δ). (3.6) Combining (3.4), (3.5) and (3.6), we get

k∇uk_1,q≤ ^M1

µ kfk_q+δ²+µS_pδH(δ), where M₁=C₀max

1,_C^Cp

E + ^(C_C^p+1)

Ee ,_C⁴

E .

On the other hand, by Proposition 2.30 in [11], there exists a constantc₁>0 such that k∇bk_1,r≤c₁[kη· ∇ξk_r+kξ· ∇ηk_r]

≤c₁

C_Eekηk_1,rk∇ξk_1,q+CEkξk_1,qk∇ηk_1,r

≤c₁

CEe(C_p+₁)k∇_ηk_rk∇_ξk_1,q+C_E(C_p+₁)k∇_ξk_qk∇_ηk_1,r

≤c₁

C_Ee(Cp+1)k∇ηk_1,r ^δ

C_E +CE(Cp+1)k∇ξk_1,q ^δ C_Ee

≤c₁

(C_p+1) CE

δ²+ (C_p+1) C_Ee δ²

≤2M₂δ²,

(3.7)

where M₂ = c₁max(Cp+1) C_E ,^(C_C^p+1)

Ee . In order to ensure that T(B_δ)⊆ B_δ, it is enough to show that

k∇uk_1,q≤ ^M1

µ kfk_q+δ²+µSpδH(δ) ≤δ, k∇bk_1,r ≤2M2δ²≤ δ.

(3.8)

Using Lemma2.2 with A = ^M1

µ , E= M₁S_p andD = ^M1kfkq

µ , there existsδ₁ > ^M1kfkq

µ such

that M₁

µ kfk_q+δ²₁+µS_pδ₁H(δ₁) ≤δ₁, provided that

AD+ED^2rp(1+D)^(p−4)+ ≤ γ_p,

which holds from the hypothesis (3.3). Also, it holds (β=2 in Lemma2.2) that δ₁ ≤ ^2M1kfk_q

µ .

On the other hand, we reformulate the inequality(3.8)₂as

2M₂δ²−δ ≤0. (3.9)

Due to

∆=1>0, we deduce that for someδ, the inequality (3.9) is valid.

Take the constantDto satisfyδ⁻<D<2D<δ⁺, where δ^± = ¹

4M2

±√

1= ¹±4M₂ 4M2

.

Moreover, given that for every δ ∈ [δ⁻,δ⁺], the inequality (3.9) is valid, we can choose δ₂ ∈ (δ⁻,D)such that

2M₂δ₂²≤δ₂. In conclusion, we obtain

δ2< ^M1kfk_q

µ <δ₁ ≤ ^2M1kfk_q

µ .

Thus, takingδ₀= δ₁we obtain that T(B_δ0)⊆B_δ0.

Proposition 3.2. There is a positive constant m=m(C−1,Cp,c2,CE,C_Ee)such that if m

"

1+ ¹

µ

M₁kfk_q µ +S_p

M₁kfk_q µ

2rp

1+ ^M1kfk_q µ

(p−4)⁺#

< ¹

4^(p−2,1)+, (3.10) then T:B_δ0 → B_δ0 is a contraction inW^1,q₀ (_Ω)×_W^1,r(_Ω).

Proof. Let (ξ,η), (ξ^ˆ, ˆη) ∈ B_δ0 and let (u,b), (u, ˆˆ b) be their respective images under T. Then, from (3.2) we obtain















−_µ∆(u−uˆ) +∇(p−pˆ) =F, x∈_Ω,

−_∆(b−b^ˆ) =G, x∈_Ω,

div(u−uˆ) =0, div(b−b^ˆ) =0, x∈_Ω,

(u−u_ˆ)|_∂Ω =_0, (b−b^ˆ)·n|_∂Ω =_0, (∇ ×(b−b^ˆ))×n|_∂Ω =_0, where

F :=div(ξ^ˆ⊗ξ^ˆ−ξ⊗ξ) + (∇ ×η)×η−(∇ ×ηˆ)×ηˆ+2µdiv[σ(|Dξ|²)Dξ−σ(|Dξˆ|²)Dξˆ], G:= (η· ∇)ξ−(ηˆ· ∇)ξ^ˆ+ (ξ^ˆ· ∇)ηˆ−(ξ· ∇)η.

Applying Lemma2.1withψ =F we obtain k∇(u−uˆ)k_q≤ ^C−1

µ (kdiv(ξ^ˆ⊗ξ^ˆ−ξ⊗ξ)k_−1,q+k(∇ ×η)×η−(∇ ×ηˆ)×ηˆk_−1,q +2µkdiv[σ(|Dξ|²)Dξ−σ(|Dξˆ|²)Dξˆ]k_−1,q).

(3.11)

Notice that

k(∇ ×η)×η−(∇ ×ηˆ)×ηˆk_−1,q

≤ k(∇ ×η)×η−(∇ ×ηˆ)×ηˆk_r

= k(∇ ×η)×η−(∇ ×ηˆ)×η+ (∇ ×ηˆ)×η−(∇ ×ηˆ)×ηˆk_r

≤ k∇(η−ηˆ)k_rkηk_∞+k∇ηˆk_rkη−ηˆk_∞

≤C_Eekηk_1,rk∇(η−ηˆ)k_r+k∇ηˆk_1,rC_Eekη−ηˆk_1,r

≤C

Ee(C_p+1)k∇ηk_rk∇(η−ηˆ)k_r+δ₀(C_p+1)k∇(η−ηˆ)k_r

≤C_Ee(C_p+1)k∇ηk_1,rk∇(η−ηˆ)k_r+δ₀(C_p+1)k∇(η−ηˆ)k_r

≤ δ₀(C_p+1)k∇(η−ηˆ)k_r+δ₀(C_p+1)k∇(η−ηˆ)k_r,

=2δ₀(Cp+1)k∇(η−ηˆ)k_r,

(3.12)

reasoning as in [1], we obtain

kdiv(_ξ^ˆ⊗ξ^ˆ−ξ⊗ξ)k_−1,q≤Ck(ξ^ˆ⊗ξ^ˆ−ξ⊗ξ)k_q

≤CC_p(C^q_p+1)^1qδ₀k∇(ξ−ξ^ˆ)k_q, (3.13) 2µkdiv[σ(|Dξ|²)Dξ−σ(|Dξˆ|²)Dξˆ]k_−1,q≤Cµk[σ(|Dξ|²)Dξ−σ(|Dξˆ|²)Dξˆ]k_q

≤CµS_pH(2δ₀)k∇(ξ−ξ^ˆ)k_q. (3.14) From (3.11)–(3.14) we obtain

k∇(u−uˆ)k_q≤ M3

2δ0

µ +SpH(2δ0)

max{k∇(ξ−ξ^ˆ)k_q,k∇(η−ηˆ)k_r}, (3.15)

where M₃=C₋₁max

CC_p(C^q_p+₁)^1q_{, 2}(C_p+₁)_,C .

On the other hand, again by Proposition 2.30 in [11], there exists a constant c₂ > 0 such that

k∇(b−b^ˆ)k_r≤ k∇(b−b^ˆ)k_1,r

≤ c₂

k(η· ∇)ξ−(ηˆ· ∇)ξ^ˆk_r+k(ξ^ˆ· ∇)ηˆ−(ξ· ∇)ηk_r

= c₂h

k(η· ∇)ξ−(ηˆ· ∇)ξ+ (ηˆ· ∇)ξ−(ηˆ· ∇)ξ^ˆk_r +k(ξ^ˆ· ∇)ηˆ−(_ξ^ˆ· ∇)η+ (_ξ^ˆ· ∇)η−(_ξ· ∇)ηk_rⁱ

≤ c₂h

kη−ηˆk_∞k∇ξk_r+kηˆk_∞k∇(ξ−ξ^ˆ)k_r +kξ^ˆk_∞k∇(ηˆ−η)k_r+kξ^ˆ−ξk_∞k∇ηk_rⁱ

≤ c2

h

C_Eekη−ηˆk_1,rk∇ξk_r+C_Eekηˆk_1,rk∇(ξ−ξ^ˆ)k_r +C_Ekξ^ˆk_1,qk∇(ηˆ−η)k_r+C_Ekξ^ˆ−ξk_1,qk∇ηk_rⁱ

≤ c₂h

C_Ee(C_p+1)k∇(η−ηˆ)k_rk∇ξk_1,q+C_Ee(C_p+1)k∇ηˆk_rk∇(ξ−ξ^ˆ)k_q +C_E(Cp+1)k∇ξ^ˆk_qk∇(ηˆ−η)k_r+C_E(Cp+1)k∇(ξ^ˆ−ξ)k_qk∇ηk_rⁱ

≤ c₂h

CEe(C_p+₁)k∇_ξk_1,qk∇(_η−_ηˆ)k_r+C

Ee(C_p+₁)k∇_ηˆk_1,rk∇(_ξ−_ξ^ˆ)k_q +C_E(C_p+1)k∇ξ^ˆk_1,qk∇(ηˆ−η)k_r+C_E(C_p+1)k∇ηk_1,rk∇(ξ^ˆ−ξ)k_qⁱ

≤ c₂

C_Ee(C_p+1)

C_E δ₀k∇(η−ηˆ)k_r+ (C_p+1)δ₀k∇(ξ−ξ^ˆ)k_q +(Cp+1)δ₀k∇(η−ηˆ)k_r+ ^CE(Cp+1)

C_Ee δ₀k∇(ξ^ˆ−ξ)k_q

≤4M₄δ₀max

k∇(ξ−ξ^ˆ)k_q,k∇(η−ηˆ)k_r ,

(3.16)

where M₄= c₂max{^CEe(C_C^p+1)

E ,(Cp+1),^CE(C_C^p+1)

eE }. Combining (3.15) and (3.16), we deduce that

max{k∇(u−uˆ)k_q,k∇(b−b^ˆ)k_r}

≤

2M3δ0

µ +4M₄δ₀+M₃SpH(2δ₀)

·max{k∇(ξ−ξ^ˆ)k_q,k∇(η−ηˆ)k_r}.

From here, and taking into account thatδ₀≤^2M1kfkq

µ ,His nondecreasing,H(4y)≤4^(p−2,1)+H(y) and definingm=max{2M₃, 4M₄}, we get

max{k∇(u−uˆ)k_q,k∇(b−b^ˆ)k_r}

≤ m δ₀

µ +δ₀+S_pH(2δ₀)

max{k∇(ξ−ξ^ˆ)k_q,k∇(η−ηˆ)k_r}

≤ m

2M₁kfk_q

µ² +^2M1kfk_q

µ +Sp4^(p−2,1)+H

M₁kfk_q µ

·_max{k∇(_ξ−_ξ^ˆ)k_q,k∇(_η−_ηˆ)k_r}

=m

"

1+ ¹

µ

2M₁kfk_q

µ +4^(p−2,1)+Sp

M₁kfk_q µ

2rp

1+ ^M1kfk_q µ

(p−4)⁺#

·_max{k∇(_ξ−_ξ^ˆ)k_q,k∇(_η−_ηˆ)k_r}

≤4^(p−2,1)+m

"

1+ ¹

µ

M₁kfk_q µ +S_p

M₁kfk_q µ

2rp

1+ ^M1kfk_q µ

(p−4)⁺#

·_max{k∇(_ξ−_ξ^ˆ)k_q,k∇(_η−_ηˆ)k_r}_{. (3.17)} Considering the space Y := _W^1,q₀ (_Ω)×_W^1,r(_Ω), with norm max{k∇ · k_q,k∇ · k_r}_{, the} inequality (3.17) implies that

kT ξˆ, ˆη

−T(ξ,η)k_Y≤4^(p−2,1)+m

1+ ¹ µ

M₁kfk_q µ

+S_p

M₁kfk_q µ

2rp

1+ ^M1kfk_q µ

⁽p−4)⁺#

ξˆ, ˆη

−(ξ,η) Y. From which and hypothesis (3.10), we obtain T : B_δ0 → B_δ0 is a contraction in W^1,q₀ (_Ω)× W^1,r(_Ω).

Proof of Theorem1.2. Notice that for p ≤ 3, γ_p = 1/4 = 1/4^(p−2,1)+ and for p > 3, γ_p >

1/4^(p−2,1)+. Thus, by taking C = (M₁,m)⁺ and because of (1.7) implies (3.3) and (3.10), Propositions3.1 and Propositions3.2yield that the mapping T: B_δ0 → B_δ0 is a contraction in W^1,q₀ (_Ω)×_W^1,r(_Ω)_.

Applying Lemma 2.3 with X = V2,q×W^2,r(_Ω), Y = W^1,q₀ (_Ω)×W^1,r(_Ω) and B = B_δ0, we could obtain that T has a unique fixed point inB_δ0 and this implies the original problem (1.2)–(1.3) has a unique strong solution(u,b)∈V_2,q×W^2,r(_Ω).

The proof of Theorem1.2is finished.

Acknowledgment

This work was supported by the fund of the “Thirteen Five” Scientific and Technological Re- search Planning Project of the Department of Education of Jilin Province (Grant No.

JJKH20200727KJ).

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