On the existence, uniqueness and regularity of solutions for a class of micropolar fluids

On the existence, uniqueness and regularity of solutions for a class of micropolar fluids

with shear dependent viscosities

Hui Yang and Changjia Wang

School of Science, Changchun University of Science and Technology, Changchun, 130022, P.R. China Received 26 April 2019, appeared 19 August 2019

Communicated by Maria Alessandra Ragusa

Abstract. In this paper we consider a model describing the motion of a class of microp- olar fluids with shear-dependent viscosities in a smooth domain Ω ⊂ R². Under the conditions that the external force and vortex viscosity µr are small in a suitable sense, we proved the existence and uniqueness of regularized solutions for the problem by using the iterative method.

Keywords: existence and uniqueness, regularity, shear dependent viscosity, micropolar fluids.

2010 Mathematics Subject Classification: 35M33, 35A01, 35D30.

1 Introduction and main result

The objective of the present work is to study the existence and uniqueness of strong solutions of a system associated to the steady equations for the motion of incompressible micropo- lar fluids with shear dependent viscosities in a bounded domain Ω ⊂ _R² having a smooth boundary. More precisely, we will study the following system











(u· ∇)u−div[(1+|Du|)^p−2Du] +∇η=µrrotw+ f, inΩ,

divu=0, inΩ,

(u· ∇)w−µ₁∆w−µ₂∇divw+µ_rw=µ_rrotu+g, inΩ,

(1.1)

together with the boundary conditions

u|_∂Ω=0, w|_∂Ω =0, (1.2)

where Du = ¹₂(∇u+∇u^T), p ∈ (1, 2) . The vector-valued functions u = (u₁,u₂,u₃),w = (w₁,w₂,w₃)and the scalar functionη denote respectively, the velocity, the angular velocity of rotation of particles and the pressure of the fluid. The vector-valued functions f andgdenote

BCorresponding author. Email: wangchangjia@gmail.com

respectively, the external sources of linear and angular momentum. The positive constantsµ₁ andµ₂ are the spin viscosities,µ_r is the vortex viscosity. For simplicity, in this paper, we take µ₁=µ₂=1.

The micropolar fluid model, firstly introduced by Eringen in [7], is a substantial general- ization of the classic Navier–Stokes equations in the sense that the microstructure of the fluid particles is taken into account. Physically, micropolar fluids represent fluids consisting of rigid randomly oriented (or spherical) particles suspended in a viscous medium, see e.g. [4,7,17].

We note that micropolar fluids enables us to consider some physical phenomena that can not be treated by the classical Navier–Stokes equations for the viscous incompressible fluids such as suspensions, lubricants, blood motion in animals and liquid crystals.

If the exponent index p=2, then (1.1)–(1.2) reduces the the classical micropolar fluid sys- tem and there are many results on the existence and uniqueness of solutions for it. For exam- ple, the existence of weak solutions in any connected open setS⊆_R^d(cf. [24], in any bounded domainΩ⊂ _R^d) and strong solutions in any bounded domain Ω⊂_R^d established by Galdi and Rionero [13], and Yamaguchi [27], respectively. For the same problem, Łukaszewicz [16]

proved the existence and uniqueness of strong solutions in 1989, and, in 1990, established the global existence of weak solutions for arbitrary initial data(u0,w0)∈ L²_σ×L²(see [17]). Using a spectral Galerkin method, Rojas-Medar [23] proved the local existence and uniqueness of strong solutions. Ortega-Torres and Rojas-Medar proved the global existence of a strong so- lution by assuming small initial data, (see [20]). Linearization and successive approximations have been considered in [3,21] to give sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations.

Recently, Loayza and Rojas-Medar [18] investigated regularity criteria for weak solutions of the micropolar fluid equations in a bounded three-dimensional domain. For more details, one can also refer [9–11,15,17,26] and the reference cited therein.

The case of the exponent p 6= 2 (i.e. the non-Newtonian micropolar fluid or called the micropolar fluid with shear dependent viscosities) is less studied. Araújo et al. [1] proved the existence of weak solutions by using Galerkin and compactness arguments. Uniqueness and periodicity of solutions are also considered. In [2], the author studied the long time behavior of the two-dimensional flow for non-Newtonian micropolar fluids in bounded smooth domains, in the sense of pullback attractors. They proved the existence and upper semicontinuity of the pullback attractors with respect to the viscosity coefficient of the model.

In the present work, as we said previously, we are interested in the flow of micropolar fluids with shear-dependent viscosities in a smooth domainΩ ⊆ R². Under the conditions that the external sources f,g and the vortex viscosity µr are small in a suitable sense, we proved the existence and uniqueness of regularized solutions for the problem by using the iterative method.

Throughout the paper, as usual, we denote byV ={v∈C₀^∞(_Ω); divv=0}and the spaces V_q(_Ω)_:=the completion ofV _{in the}W^1,q-norm,

forq=2 we simply writeV(_Ω). We also denote by(C^m,γ(_Ω^¯), k · k_Cm,γ),mnonnegative integer andγ ∈ (0, 1), the Hölder space with order m. By W^−1,q0(_Ω), q⁰ = _q−^q₁, the strong dual of W₀^1,q(_Ω)with normk · k_−1,q.

Next, we introduce the notions of solutions to (1.1)–(1.2).

Definition 1.1. Assume that f ∈ L²(_Ω), g ∈ L²(_Ω). We say that(u,w)is a pair ofC^1,γ(_Ω^¯)× W^1,2(_Ω)-solution of problem (1.1)–(1.2). If u ∈ C^1,γ(_Ω^¯)_{, for some} γ ∈ (_{0, 1})_{, div}u = _0,

u|_∂Ω = 0, w ∈ W₀^1,2(_Ω) and it satisfies the following integral identity for ∀ϕ ∈ V_q(_Ω) and

∀ψ∈C₀^∞(_Ω)

Z

Ω(u· ∇)uϕdx+

Z

Ω(1+|Du|)^p−2DuDϕdx=

Z

Ωµrrotwϕdx+

Z

Ω fϕdx, (1.3) Z

Ω(u· ∇)wψdx+

Z

Ω∇w∇ψdx+

Z

Ωdivwdivψdx+

Z

Ωµ_rwψdx

=

Z

Ωµrrotuψdx+

Z

Ωgψdx. (1.4) Remark 1.2. We observe that if u satisfies (1.3) then we can apply the theorem of de Rham (see [25, Lemma 2.2.1]) to find a pressureηat least in L²(_Ω)such that the pair(u,η)satisfies the following integral identity for∀ϕ∈C₀^∞(_Ω)

Z

Ω(u· ∇)uϕdx+

Z

Ω(₁+|Du|)^p−2DuDϕdx−

Z

Ωη∇ ·ϕdx =

Z

Ωµ_rrotwϕdx+

Z

Ω fϕdx.

The validity of the reverse implication is obvious.In the sequel we shall refer to (u,w) or (u,η,w)as solution of system (1.1)–(1.2) without distinction.

Our aim is to prove the following theorems.

Theorem 1.3. Assume that p∈ (1, 2),q> 2,and letγ₀ =1− ²_q. LetΩbe a domain of class C², and let be f ∈ L^q(_Ω),g ∈ L²(_Ω). Ifkfk_q≤δ₁,kgk₂≤ δ2, µr< δ3whereδ₁, δ2, δ3are positive constants small in a suitable sense (see (3.2), (3.13), (3.16)), then there exist a unique C^1,γ(_Ω^¯)×W^1,2(_Ω)- solution(u,η,w)of problem(1.1)–(1.2)such that

u∈ C^1,γ(_Ω^¯), η∈ C^0,γ(_Ω^¯), w∈W^2,2(_Ω), ∀γ<γ₀, and

kuk_C1,γ +kηk_C0,γ+kwk_2,2≤2(c˜₀kfk_q+c₀kgk₂), wherec˜₀,c₀ are positive constants.

Theorem 1.4. In addition to the assumptions of Theorem1.3, if q > 4andkfk_q,kgk₂, µ_r are suffi- ciently small (see(4.4)), then there exists a solution(u,η,w)of problem(1.1)–(1.2)such that

u∈W^2,2(_Ω)∩C^1,γ(_Ω^¯), η∈W^1,2∩C^0,γ(_Ω^¯), w∈W^2,2(_Ω), ∀γ<γ₀.

The present work is organized as follows: in Section2 we state preliminaries results that will be used later in the paper. Section 3is dedicated to give the proof of Theorem1.3. More precisely, in Section3.1we construct approximate solutions to the original nonlinear problem by iterate scheme, then derive the uniform estimate for such approximate solutions. The results are used in Section 3.2 to prove the convergence of the solutions. The existence and uniqueness results are proved in Section3.3and Section3.4respectively. Finally, in Section4, we prove the regularity result (Theorem1.4).

2 Preliminary lemmas

In this section, we recall the following useful results.

Lemma 2.1([22]). For any q≥1, there exists a constant c₁such that kvk_q+k∇vk_q ≤c₁kDvk_q, for each v∈ V_q(_Ω). Hence the two quantities above are equivalent norms in Vq(_Ω).

Lemma 2.2([19]). If a distribution g is such that∇g∈W^−1,q(_Ω), then g∈ L^q(_Ω)and kgk_L^q

] ≤Ck∇gk_−1,q, where L^q_] = L^q/R.

Lemma 2.3([5]). For any given real numbersξ,η ≥0and1< p <2the following inequality holds true:

1

(1+ξ)^2−p − ¹ (1+η)^2−p

≤ (2−p)|ξ−η|.

Lemma 2.4([8]). For an arbitrary tensor D, define S(D)≡ (1+|D|)^p−2D, 1< p <2. Then there exist a constant C such that, for any pair of tensors D₁and D₂,

(S(D₁)−S(D₂))·(D₁−D₂)≥C |D₁−D₂|² (₁+|D1|+|D2|)^2−p.

3 The proof of Theorem 1.3

As already stated, in order to prove Theorem1.3we use the method of successive approxima- tions.

3.1 Approximating linear problems

We construct approximate solutions, inductively, as follows:

(i) first defineu⁻¹ =w⁻¹=0, and

(ii) assuming that(u^m−1,w^m−1)was defined form≥1, letu^m,w^m be the unique solution to the following boundary problems:















−div[(1+|Du^m−1|)^p−2Du^m] +∇η^m =µ_rrotw^m−1+ f −(u^m−1· ∇)u^m−1, in Ω,

divu^m =0, in Ω,

−_∆w^m− ∇divw^m =µ_rrotu^m−1−µ_rw^m−1+g−(u^m−1· ∇)w^m−1, in Ω, u^m|_∂Ω=0, w^m|_∂Ω =0.

(3.1)

The following result holds true.

Proposition 3.1. Assume that p ∈(_{1, 2})_, q> ₂and letγ₀ = ₁−²_q. LetΩbe a domain of class C², and let be f ∈L^q(_Ω), g ∈ L²(_Ω). Then, for any m∈_Nthere exists a weak solution(u^m,η^m,w^m)of problem(3.1)such that

u^m ∈C^1,γ0(_Ω^¯), η^m ∈C^0,γ0(_Ω^¯), w^m ∈W^2,2(_Ω).

Moreover, if f andgsatisfy the assumption

˜

c₀kfk_q+c₀kgk₂<

Cc˜₀µ_r+ (C+₁)c₀µ_r−₁²

4(c˜₀+c₀) ^{, (3.2)}

whereµ_ris properly small satisfying(3.13), then

ku^mk_C1,γ0 +kη^mk_C0,γ0 +kw^mk_2,2 ≤2(c˜₀kfk_q+c₀kgk₂), uniformly in m∈_N. (3.3) Proof. Setting I_m = ku^mk_C1,γ0 +kη^mk_C0,γ0 +kw^mk_{2,2. Let be}m= 0, first of all, we consider the following boundary-value problem

(−_∆w⁰− ∇divw⁰ =g, in Ω, w⁰|_∂Ω =0,

where g ∈ L²(_Ω). According to the theory of elliptic equation, we can find a solutionw⁰ ∈ W^2,2(_Ω)and get

kw⁰k_2,2≤c₀kgk₂. (3.4)

Then we consider the following boundary-value problem











−¹₂4u⁰+∇η⁰ = f, in Ω,

divu⁰ =0, in Ω,

u⁰|_∂Ω=_0,

(3.5)

where f ∈ L²(_Ω). We can find a solution(u⁰,η⁰)∈C^1,γ0(_Ω^¯)×C^0,γ0(_Ω^¯)(see [6, Theorem 3.2]) and

ku⁰k_C1,γ0 +kη⁰k_C0,γ0 ≤c˜(ku⁰k_1,2+kfk_q) =c(ku⁰k_1,2+kfk_q), (3.6) where c > 1. By writing the definition of weak solution of (3.5)₁ with the test function ϕ replaced byu⁰ we get

Z

Ω|Du⁰|²dx=

Z

Ω f u⁰dx.

By Lemma2.1and the Hölder inequality, we have Z

Ω|Du⁰|²dx=kDu⁰k²₂ ≥ ¹

c²₁ku⁰k²_1,2, Z

Ω f u⁰dx≤ kfk₂ku⁰k₂≤ kfk_qku⁰k_1,2,

which obviously impliesku⁰k_1,2≤ c²₁kfk_q. So we can get from (3.4) and (3.6) that

I₀ =ku⁰k_C1,γ0 +kη⁰k_C0,γ0 +kw⁰k_2,2 ≤c(1+c²₁)kfk_q+c₀kgk₂. (3.7) Let bem≥1, assuming that(u^m,η^m,w^m)∈C^1,γ0(_Ω^¯)×C^0,γ0(_Ω^¯)×W^2,2(_Ω^¯)is a solution of (3.1). Firstly, we consider the following boundary-value problem

(−_∆w^m+1− ∇_divw^m+1 =µ_rrotu^m−µ_rw^m+g−(u^m· ∇)w^m, in Ω, w^m+1|_∂Ω =0,

whereg ∈ L²(_Ω). Since the assumption implies that

k(u^m· ∇)w^mk₂ ≤ ku^mk₂k∇w^mk₂≤ ku^mk_C1,γ0kw^mk_2,2,

thenµrrotu^m−µrw^m+g−(u^m· ∇)w^m belongs to L²(_Ω). According to the theory of elliptic equation, we can find a solutionw^m+1∈W^2,2(_Ω)and

kw^m+1k_2,2 ≤c₀(kµ_rrotu^mk₂+kµ_rw^mk₂+kgk₂+k(u^m· ∇)w^mk₂

≤c₀(Cµ_rku^mk_C1,γ0 +µ_rkw^mk_2,2+kgk₂+ku^mk_C1,γ0kw^mk_2,2). (3.8) Secondly, we consider the boundary-value problem











−div[(1+|Du^m|)^p−2Du^m+1] +∇η^m+1 =µrrotw^m+ f −(u^m· ∇)u^m, inΩ,

divu^m+1 =0, inΩ,

u^m+1|_∂Ω=0,

(3.9)

where f ∈ L^q(_Ω). Since the assumption implies that

kµ_rrotw^mk_q≤Cµ_rk∇w^mk_q≤ Cµ_rkw^mk_2,2, k(u^m· ∇)u^mk_q≤ ku^mk_∞k∇u^mk_q≤ ku^mk²

C^1,γ0,

thenµ_rrotw^m+ f −(u^m· ∇)u^m belongs toL^q(_Ω). Then we can get a solution(u^m+1,η^m+1)∈ C^1,γ0(_Ω^¯)×C^0,γ0(_Ω^¯)(see [6, Theorem 3.2]) such that

ku^m+1k_C1,γ0 +kη^m+1k_C0,γ0

≤c˜ ku^m+1k_1,2+kµrrotw^mk_q+kfk_q+k(u^m· ∇)u^mk_q

≤c(1+ku^mk_C1,γ0)^r·(ku^m+1k_1,2+Cµ_rkw^mk_2,2+kfk_q+ku^mk²

C^1,γ0),

(3.10)

where the exponentr is a real number greater than 2.

By writing the definition of weak solution of (3.9)₁ with the test function ϕ replaced by u^m+1we get

Z

Ω(₁+|Du^m|)^p−2|Du^m+1|²dx=

Z

Ωµ_rrotw^mu^m+1dx+

Z

Ωf u^m+1dx−

Z

Ω(u^m· ∇)u^mu^m+1dx.

Since 1< p <2, by Lemma2.1and the Hölder inequality, there follows Z

Ω(1+|Du^m|)^p−2|Du^m+1|²dx≥(1+kDu^mk_∞)^p−2kDu^m+1k²₂

≥ ¹

c²₁(1+ku^mk_C1,γ0)^p−2ku^m+1k²_1,2,

Z

Ωµ_rrotw^mu^m+1dx+

Z

Ω f u^m+1dx−

Z

Ω(u^m· ∇)u^mu^m+1dx

≤ kµ_rrotw^mk₂ku^m+1k₂+kfk₂ku^m+1k₂+k(u^m· ∇)u^mk₂ku^m+1k₂

≤Cµ_rkw^mk_2,2ku^m+1k_1,2+kfk_qku^m+1k_1,2+ku^mk²

C^1,γ0ku^m+1k_1,2, which implies

ku^m+1k_1,2≤c²₁(1+ku^mk_C1,γ0)^2−p(Cµrkw^mk_2,2+kfk_q+ku^mk²_C1,γ0). (3.11)

Combining (3.8), (3.10) and (3.11), we obtain I_m+1=ku^m+1k_C1,γ0 +kη^m+1k_C0,γ0 +kw^m+1k_2,2

≤c(1+c²₁)(1+ku^mk_C1,γ0)^r+2−p(Cµrkw^mk_2,2+kfk_q+ku^mk²

C^1,γ0) +c0(Cµrku^mk_C1,γ0 +µrkw^mk_2,2+kgk₂+ku^mk_C1,γ0kw^mk_2,2)

≤c(₁+c²₁)(₁+I_m)^r+2−p(Cµ_rI_m+kfk_q+I_m²) +c₀

(C+₁)µ_rI_m+kgk₂+I_m² .

(3.12)

We shall prove the boundedness of the sequence{Im}by a fixed point argument. Setting, for any t≥0

ψ(t) =c(1+c²₁)(1+t)^r+2−p(Cµ_rt+kfk_q+t²) +c₀[(C+1)µ_rt+kgk₂+t²]−t.

We look for a root ofψ(t). Let us observe that if 0≤t≤1, then

ψ(t)≤c(1+c²₁)2^r+2−p(Cµrt+kfk_q+t²) +c₀[(C+1)µrt+kgk₂+t²]−t

=c(1+c²₁)2^r+2−p+c0

t²+Cc(1+c²₁)2^r+2−pµr+ (C+1)c0µr−1 t +c(₁+c²₁)₂^r+2−pkfk_q+c₀kgk_2.

Define

h(t) =c(1+c²₁)2^r+2−p+c₀

t²+Cc(1+c²₁)2^r+2−pµ_r+ (C+1)c₀µ_r−1 t +c(1+c²₁)2^r+2−pkfk_q+c₀kgk₂,

we note that if

µ_r≤ ¹

Cc(1+c²₁)2^r+2−p+ (C+1)c₀, (3.13) then the function h(t) has two positive roots s₁ < s₂, if and only if the discriminant ∆ > 0, namely

c(1+c²₁)2^r+2−pkfk_q+c₀kgk₂< [Cc(1+c²₁)2^r+2−pµ_r+ (C+1)c₀µ_r−1]² 4[c(1+c²₁)2^r+2−p+c₀] ^, and we have that

0< s₁ = ¹−Cc(1+c²₁)2^r+2−pµ_r−(C+1)c₀µ_r−√

∆ 2[c(1+c²₁)2^r+2−p+c₀] <1.

Since c > 1 and consequently 2[_c(₁+_c²₁)₂^r+2−p+_c0] > _{1. Since} ψ(₀) > _{0 and} ψ(_t) ≤ h(_t)_, whent∈ [0, 1], there existst₁, with 0<t₁<s₁such thatψ(t₁) =0, i.e.

c(1+c²₁)(1+t₁)^r+2−p(Cµrt₁+kfk_q+t²₁) +c₀[(C+1)µrt₁+kgk₂+t²₁]−t₁ =0.

Since t₁ > 0, it follows that c(1+c²₁)kfk_q+c0kgk₂−t₁ ≤ 0, recalling (3.7), we get t₁ ≥ c(1+c²₁)kfk_q+c₀kgk₂ ≥ I₀. If we suppose that I_m ≤ t₁, by inequality (3.12) and the fact that ψ(t₁) =0 we obtain

I_m+1 ≤c(1+c²₁)(1+I_m)^r+2−p[Cµ_rI_m+kfk_q+I_m²] +c₀

(C+1)µ_rI_m+kgk₂+I_m²

≤c(1+c²₁)(1+t₁)^r+2−p(Cµ_rt₁+kfk_q+t²₁) +c₀

(C+1)µ_rt₁+kgk₂+t²₁

=ψ(t₁) +t₁

=t₁,

which proves our claim. Therefore

Im ≤t₁ <s₁ <c(1+c²₁)2^r+3−pkfk_q+2c0kgk₂ ≤1, ∀m∈_N.

Let ˜c₀ =c(₁+c²₁)₂^r+2−p, we can get I_m ≤₂(c˜₀kfk_q+c₀kgk₂)_.

3.2 Convergence of approximate solutions

For any j∈ _{N, set} P^j+1 = u^j+1−u^j, Q^j+1 = η^j+1−η^j, R^j+1 = w^j+1−w^j. Taking m= jand j+1, respectively, in the weak formula of (3.1)₁ , then subtracting one from the other, we can get for∀ϕ∈C^∞₀ (_Ω)

Z

Ω(1+|Du^j|)^p−2Du^j+1Dϕdx=

Z

Ω(1+|Du^j−1|)^p−2Du^jDϕdx +

Z

Ωµ_rrotR^jϕdx−

Z

Ω(P^j· ∇)u^jϕdx

−

Z

Ω(u^j−1· ∇Pϕdx+

Z

ΩQ^j+1divϕdx.

Next, by subtractingR

Ω(₁+|Du^j|)^p−2Du^jDϕdx from both sides of the above equality, we could obtain

Z

Ω(1+|Du^j|)^p−2DP^j+1Dϕdx =

Z

Ω[(1+|Du^j−1|)^p−2−(1+|Du^j|)^p−2]Du^jDϕdx +

Z

Ωµ_rrotR^jϕdx−

Z

Ω(P^j· ∇)u^jϕdx

−

Z

Ω(u^j−1· ∇)P^jϕdx+

Z

ΩQ^j+1divϕdx.

(3.14)

This identity, by a continuity argument, still holds withϕ∈V(_Ω), in which case the last term of (3.14) vanishes. Here, we recall thatu^j−1 =w^j−1=0 for j=0 and thenP⁰= u⁰,R⁰= w⁰.

Similarly, by taking m = j and j+1, respectively, in the weak formula of (3.1)₃, then subtracting one from the other, we can get for∀ψ∈ H₀¹(_Ω)

Z

ΩdivR^j+1divψdx+

Z

Ω∇R^j+1∇ψdx =

Z

Ωµ_rrotP^jψdx−

Z

Ωµ_rR^jψdx

−

Z

Ω(P^j· ∇)w^jψdx−

Z

Ω(u^j−1· ∇)R^jψdx.

(3.15) Proposition 3.2. Assume that all the assumptions of Proposition3.1are satisfied and let{u^m},{η^m} and{w^m}be the corresponding sequence. Then, if

1+2 ˜c₀kfk_q+2c₀kgk₂^2−p

·^h(2−p+2c²₁+Cc₁+C)·(2 ˜c₀kfk_q+2c₀kgk₂) +C(c₁+1)µ_r i

<1, (3.16) withc˜₀ and c₀ given by Proposition 3.1, the series∑mP^m converges to a functionPin W^1,2(_Ω), the series ∑mQ^m converges to a function Q in L²(_Ω), the series ∑mR^m converges to a function R in W^1,2(_Ω).

Proof. First, let us verify that the following estimates hold:

(a)

kDP¹k₂+kR¹k_1,2 ≤(2 ˜c₀kfk_q+2c₀kgk₂)·(1+2 ˜c₀kfk_q+2c₀kgk₂)^2−p

·(2−p+c₁+C)(2 ˜c₀kfk_q+2c₀kgk₂) +Cµr(1+c₁); (b) if, forj≥1, it holds that

kDP^jk₂+kR^jk_1,2

≤(2 ˜c₀kfk_q+2c₀kgk₂)· (2−p+c₁+C)(2 ˜c₀kfk_q+2c₀kgk₂) +Cµ_r(1+c₁) (2−p+2c²₁+Cc₁+C)(2 ˜c₀kfk_q+2c₀kgk₂) +C(c₁+1)µ_r

·ⁿ(₁+_{2 ˜}c₀kfk_q+2c₀kgk₂)^2−p(₂−p+2c²₁+Cc₁+C)(_{2 ˜}c₀kfk_q+2c₀kgk₂)+C(c₁+₁)µ_roj

,

then

kDP^j+1k₂+kR^j+1k_1,2 (3.17)

≤ (2 ˜c0kfk_q+2c0kgk₂) (₂−p+c₁+C)(_{2 ˜}c₀kfk_q+2c₀kgk₂) +Cµ_r(₁+c₁) (2−p+2c²₁+Cc₁+C)(2 ˜c0kfk_q+2c0kgk₂) +C(c₁+1)µr

·ⁿ(1+2 ˜c₀kfk_q+2c₀kgk₂)^2−p(2−p+2c²₁+Cc₁+C)(2 ˜c₀kfk_q+2c₀kgk₂) +C(c₁+1)µroj+1

. By the above arguments, setting j=0 and testing withP¹in (3.14), we get

Z

Ω(1+|Du⁰|)^p−2|DP¹|²dx

=

Z

Ω[1−(1+|Du⁰|)^p−2]Du⁰DP¹dx+

Z

Ωµrrotw⁰P¹dx−

Z

Ω(u⁰· ∇)u⁰P¹dx.

Since p<2, then Z

Ω(1+|Du⁰|)^p−2|DP¹|²dx≥(1+kDu⁰k_∞)^p−2kDP¹k²₂≥ (1+ku⁰k_C1,γ0)^p−2kDP¹k²₂, by using the Hölder inequality, Lemma2.3 and Lemma2.1we get

Z

Ω[1−(1+|Du⁰|)^p−2]Du⁰DP¹dx+

Z

Ωµ_rrotw⁰P¹dx−

Z

Ω(u⁰· ∇)u⁰P¹dx

≤ (2−p)kDu⁰k₂kDu⁰k_∞kDP¹k₂+kµrrotw⁰k₂kP¹k₂+k(u⁰· ∇)u⁰k₂kP¹k₂

≤ (2−p)ku⁰k²

C^1,γ0kDP¹k₂+Cc₁µ_rkw⁰k_1,2kDP¹k₂+c₁ku⁰k²

C^1,γ0kDP¹k₂, hence

kDP¹k₂≤(1+ku⁰k_C1,γ0)^2−p·^h(2−p)ku⁰k²

C^1,γ0 +Cc₁µ_rkw⁰k_1,2+c₁ku⁰k²

C^1,γ0

i

. (3.18) Nextly, settingj=0 and testing withR¹ in (3.15), we get

Z

Ω|divR¹|²dx+

Z

Ω|∇R¹|²dx=

Z

Ωµrrotu⁰R¹dx−

Z

Ωµrw⁰R¹dx−

Z

Ω(u⁰· ∇)w⁰R¹dx, by the Hölder inequality and Lemma2.1we get

Z

Ω|divR¹|²dx+

Z

Ω|∇R¹|²dx≥CkR¹k²_1,2, and

Z

Ωµ_rrotu⁰R¹dx−

Z

Ωµ_rw⁰R¹dx−

Z

Ω(u⁰· ∇)w⁰R¹dx

≤ kµ_rrotu⁰k₂kR¹k₂+kµ_rw⁰k₂kR¹k₂+k(u⁰· ∇)w⁰k₂kR¹k₂

≤Cµ_rku⁰k_C1,γ0kR¹k_1,2+µ_rkw⁰k_2,2kR¹k_1,2+ku⁰k_C1,γ0kw⁰k_2,2kR¹k_1,2, hence

kR¹k_1,2 ≤Cµ_rku⁰k_C1,γ0 +Cµ_rkw⁰k_2,2+Cku⁰k_C1,γ0kw⁰k_2,2. (3.19)

Combining (3.18) (3.19) and by using estimate (3.3), we obtain kDP¹k₂+kR¹k_1,2 ≤(₁+ku⁰k_C1,γ0)^2−p·(₂−p+c₁)ku⁰k²

C^1,γ0 +Cc₁µ_rkw⁰k_1,2 +Cµ_rku⁰k_C1,γ0 +Cµ_rkw⁰k_2,2+Cku⁰k_C1,γ0kw⁰k_2,2

≤(1+2 ˜c₀kfk_q+2c₀kgk₂)^2−p·(2−p+c₁+C)(2 ˜c₀kfk_q+2c₀kgk₂) +Cµr(1+c₁)(2 ˜c₀kfk_q+2c₀kgk₂).

We arrive at(a).

Let us pass to estimate(b). Assume that the hypothesis in(b)holds. As for(a), by setting j≥1 andϕ=P^j+1 ∈V(_Ω)in (3.14), we get

Z

Ω(1+|Du^j|)^p−2|DP^j+1|²dx

=

Z

Ω

(1+|Du^j−1|)^p−2−(1+|Du^j|)^p−2Du^jDP^j+1dx +

Z

Ωµ_rrotR^jP^j+1dx−

Z

Ω(P^j· ∇)u^jP^j+1dx−

Z

Ω(u^j−1· ∇)P^jP^j+1dx.

Sincep <2, we get Z

Ω(1+|Du^j|)^p−2|DP^j+1|²dx≥ (1+kDu^jk_∞)^p−2kDP^j+1k²₂

≥ (1+ku^jk_C1,γ0)^p−2kDP^j+1k²₂, then the Hölder inequality , Lemma2.3and Lemma2.1 yield that

Z

Ω

(1+|Du^j−1|)^p−2−(1+|Du^j|)^p−2Du^jDP^j+1dx

≤(2−p)kDP^jk₂kDu^jk_∞kDP^j+1k₂

≤(2−p)kDP^jk₂ku^jk_C1,γ0kDP^j+1k₂,

Z

ΩµrrotR^jP^j+1dx−

Z

Ω(P^j· ∇)u^jP^j+1dx−

Z

Ω(u^j−1· ∇)P^jP^j+1dx

≤ kµ_rrotR^jk₂kP^j+1k₂+k(P^j· ∇)u^jk₂kP^j+1k₂+k(u^j−1· ∇)P^jk₂kP^j+1k₂

≤Cc₁µrkR^jk_1,2kDP^j+1k₂+c²₁kDP^jk₂ku^jk_C1,γ0kDP^j+1k₂ +c²₁ku^j−1k_C1,γ0kDP^jk₂kDP^j+1k₂,

hence

kDP^j+1k₂ ≤(1+ku^jk_C1,γ0)^2−p·^h(2−p)kDP^jk₂ku^jk_C1,γ0

+Cc₁µrkR^jk_1,2+c²₁kDP^jk₂ku^jk_C1,γ0 +c²₁ku^j−1k_C1,γ0kDP^jk₂ⁱ.

(3.20)

Then settingj≥1 and testing withR^j+1 in (3.15), we get Z

Ω|divR^j+1|²dx+

Z

Ω|∇R^j+1|²dx =

Z

Ωµ_rrotP^jR^j+1dx−

Z

Ωµ_rR^jR^j+1dx

−

Z

Ω(P^j· ∇)w^jR^j+1dx−

Z

Ω(u^j−1· ∇)R^jR^j+1dx,