# The Existence of Strong Solution for Generalized Navier-Stokes Equations withpx-Power Law under Dirichlet Boundary Conditions

Authors: Cholmin Sin
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling bypx-power law with Dirichlet boundary condition under the restriction 3n/n+2n+2<px<2n+1/n−1. In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction 2n+1/n+3<px<2n+1/n−1, which arises from the nonflatness of domain.


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## Body

## 1. Introduction

In this note, we consider the steady flows of non-Newtonian fluids inℝn, n=2,3, which is modeled by the following system:
(1)−div1+Du2px−2/2Du+u·∇u+∇π=f,inΩ,divu=0,inΩ,u=0,on∂Ω,where u is the velocity, π the pressure, f the external force, Du≔∇u+∇uT/2, Ω a bounded domain, and px>1 a prescribed function.This system arises from flows of electrorheological [1], thermorheological [2], chemically reacting non-Newtonian fluids [3].For the existence of weak solutions to the problem (1), we refer to [1, 4].Local higher differentiability for weak solutions to the problem (1) with px=const has first shown in [5] for 1.8<p<2 in 3D. This result was improved to p=1.8 in [6] by Wolf. Forp>2in 3D,Wloc2,2-regularity is proved (see [7]).Global higher differentiability for weak solutions to the problem (1) with px=const have been studied by several authors; for example, see [8–19] under the condition f∈Lp∧′Ω with p∧′=p^/p^−1 and p^≔minp,2. It was first established in [8] by Beirao da Veiga. He developed a crucial device which was to denote the second-order derivatives of the velocity in the normal direction through ones (and the first-order derivatives of the pressure) in the tangential directions by using the very explicit form of the main equations. But in contrast to interior regularity, the interaction between pressure and nonlinearity of leading term results in the lower regularity for the second-order derivatives of the velocity (and for the first-order derivatives of the pressure) in the normal direction, in comparison to the tangential directions. His idea reveals to be quite fruitful in many subsequent papers. He [10, 11] studied global higher differentiability of weak solution to the problem with the boundary condition (3) for p>15/8 in 3D cubic domain.With the help of the anisotropic Sobolev embedding theorem, Berselli [15] obtained an improved integrability of velocity gradient than in [11] in 3D cubic domain. His idea is that it is possible to apply the anisotropic Sobolev embedding theorem because of the difference in the regularity levels between the second-order derivatives of the velocity in the normal direction and the tangential ones. Beirao da Veiga [9] showed global W2,4p−2/p+1∩W1,4p−2-regularity for 20/11<p<2 by combining the idea from [15] with a delicate estimate on the convective term in 3D cubic domain and then in [12] extended it to nonflat boundaries. In [19], Crispo proved the same type of results in cylindrical domains. In [14], the authors showed global W2,q-regularity for shear-thickening flows, i.e., p>2 in n-D bounded smooth domain.Recently, global higher differentiability of weak solution to the problem (1) in 3D smooth domain is studied by us in [20] by using a global higher integrability condition, which holds under the condition f∈Lαp∧′xΩ, where p∧′x=p^x/p^x−1, p^x≔minpx,2, and α>1. This is slightly stronger rather than the standard condition f∈Lp∧′Ω for the case p=const. On the other hand, local higher differentiability of local weak solution to the problem (1) in 3D has been obtained in [21, 22] by relying on the local higher integrability result from [23].In [24], the existence and uniqueness of C1,γΩ¯∩W2,2Ω-solution corresponding to small data are proved, without further restrictions on the bounds on px.For interior or boundary partial regularity, we refer to [23, 25–27].If one assumes the conditionf∈Lp∧′xΩ for the problem (1), then when applying difference quotient, due to the px-dependence of leading term, the additional term will appear:
(2)∫1+Du2px−2/2Du2log21+Du2dx,which cannot be estimated in terms of a priori estimate on weak solutions. So for the system (1) with p≠const, the existence of strong solutions has been studied. In 3D, the existence of local strong solutions to the system (1) is first shown for 1.8<px<6 in [1] (chapter 3) by Ruzicka. In [28], Ettwein and Růžička showed the existence of Wloc2,3p^x/p^x+1-solutions without the artificial upper bound px<6. For 2D bounded domains, we refer to [29, 30].In [31, 32], we gain the existence of strong solution for the system (1) under the standard assumption f∈Lp∧′xΩ. But in that case, we consider the following the boundary condition: for Ω=0,1n, n=2,3, and Γ≔x∈∂Ω:∣x1∣,∣xn−1∣<1,xn=0orxn=1,
(3)uΓ=0,uisx′−periodic,where by x′-periodic, we mean periodic of period 1 both in x1 and xn−1. This allows us to consider a bounded domain and simultaneously a flat boundary. Thus, it is natural to ask whether the sharp results proved in [31, 32] are valid for smooth domain. This is the aim of this note.It seems to be possible to obtain the existence ofC1,αΩ¯-strong solution to the problem (1) in 2D by the result of this note and the same argument as in [32]. Very recently, we [32] show the result in the case of the boundary condition (3). For C1,αΩ-regularity in 2D, we refer to [18, 29, 30, 33–36].Set(4)Ω1≔x∈Ω∣px<2,Ω2≔Ω\Ω1.Forn=3, we define
(5)p¯x≔3px−px−2−μ,r¯x≔2p¯x22−px+p¯x,ifx∈Ω1,2p¯xpx+p¯x−2,ifx∈Ω2,where μ is arbitrarily close to 0 for p≠const, and for n=2,
(6)p¯x≡q,r¯x≔2−μ0,ifpx≠2,2,ifpx=2,where q is arbitrary real number such that 1<q<∞ and μ0>0 arbitrary close to 0.The main results are as follows.Theorem 1.
Letn=2,3. Assume that ∂Ω∈C2,1, px∈C0,1Ω¯, p≠const, and
(7)3nn+2<px<2n+1n−1,and f∈Lp∧′xΩ for
(8)p^x≔minpx,2.Then, forp¯x,r¯x from (5) and (6), there is a strong solution u,π to the problem (1) satisfying
(9)πp¯x,Ω+u1,p¯x,Ω≤C,(10)∇πr¯x,Ω+∇2ur¯x,Ω≤C,where the constants C depend on p−,p+,fp^′x.Moreover, for the problem (1) without the convective term, there is a unique strong solution satisfying (9) and (10) provided that
(11)2n+1n+3<px<2n+1n−1.Remark 2.
We note that ifp=const, then the condition (11) will be no longer needed (see [12–14, 16, 20]).But due topx-dependence of the leading term, we cannot obtain W2,q-regularity of weak solution to (1) provided f∈Lp∧′xΩ. So it is customary to consider an approximate problem. Fortunately, in [31, 32], the condition (11) does not appear but we consider the boundary condition (3). The condition (11) arises from the nonflatness of Ω and Dirichlet boundary condition. More precisely, due to them, there appears the term
(12)1+Duλ21+λDuλ2px−2/2Duλp^′x,in deriving of W2,q-regularity of the approximate solutions independent of parameter (see (41) and (45)). The term disappears in the case (3). For the estimate of the term, we need an additional condition (11). It is open whether the condition can be removed when one considers the approximate problem with Dirichlet boundary condition over nonflat domain. In fact, Kaplicky ([37], Lemma 4.2) showed (9) and (10) for the approximate problem to (1) with p=const∈2,4 in 2D.Due topx-dependence of the extra stress tensor, we consider an approximate problem with 1+Duλ2/1+λDuλ2px−2/2Duλ for λ∈0,1 instead of 1+Du2px−2/2Du. It is easy to see that the approximate solution belongs to V2Ω∩W2,2Ω owing to the trivial inequality 1≤1+a2/1+λa2≤1/λ. The main point is to derive the estimates about all derivatives of the approximation solutions in suitable Sobolev spaces, which are independent of parameter. This allows us to show convergence of approximation solutions to the one to problem (1).The paper is organized as follows. In Section2, we give preliminaries. Section 3 is devoted to prove the main results.


## 2. Preliminaries

### 2.1. Notations

Byp′x, we denote the conjugate function of px. For p∈L∞Ω, p≥1, define p−≔essinfpx, p+≔esssuppx. Let p∗ be a Sobolev conjugate exponent, i.e., p∗=np/n−p when p<n and p∗=q for all q∈1,∞ when p=n and p∗=∞ otherwise.Forn×n-matrices F,H denote F:H=∑i,j=1nFijHij, F≡F:F1/2. For two vectors a and b, a⊗b=aibjij and aeb≔1/2a⊗b+a⊗bT.Forp∈L∞Ω, p≥1, the variable exponent Lebesgue space LpxΩ is defined by
(13)LpxΩ≔u:Ω⟶ℝ∣uismeasurableandρpxu≔∫Ωupxdx<∞,endowed with the norm upx,Ω≔infλ>0∣ρpxu/λ≤1. Then, we define the variable exponent Sobolev space by
(14)Wk,pxΩ≔u∣∇αu∈LpxΩ,∀α≤k,with the norm uk,px,Ω≔∑α≤k∇αupx,Ω. We define W0k,pxΩ as the closure of C0∞Ω in Wk,pxΩ. Let W−1,p′xΩ be dual of W01,pxΩ.We do not distinguish between scalar, vector-valued, and tensor-valued function spaces in the notations. Define(15)Vpx≔u∈W01,pxΩ∣divu=0.Definition 3.
We say that functionu is a weak solution to the problem (1) if u∈Vpx and it satisfies
(16)∫Ω1+Du2px−2/2Du:Dϕdx+∫Ωu·∇u·ϕdx=∫Ωf·ϕdx,∀ϕ∈Vpx.
We refer to the term strong solution as a weak solution which additionally satisfiesu∈W2,qΩ for some 1≤q<∞.


### 2.2. Some Problems Related to Flattening of the Boundary

As before, our problem is reduced to a problem involving a flat boundary by a suitable change of variables. Here, we follow the arguments and notations in [14]. Since ∂Ω∈C2,1, for each point P∈∂Ω, there are local coordinates such that in these coordinates, we have P=0 and ∂Ω is locally described by a C2,1-function θP:Qdn−10⟶Qd10, where Qdk0 is the k-dimensional cubic with center 0 and length 2d (which is small enough and will be fixed later), with the following properties:
(17)x∈ΩP≔Ω∩Qdn−10×Qd10⟺xn>θPx1,⋯xn−1,∇θP0=0,∇θPy<cd,y∈Qdn−10.As∂Ω is a compact, there exist a finite set of points Γ⊂∂Ω and an open set Ω0⊂⊂Ω such that Ω⊂Ω0∪⋃P∈ΓΩP. We construct a partition of unity ζ0,ζP,P∈Γ, corresponding to this covering, such that distsuppζP,∂ΩP\∂Ω≥h0 for all P∈Γ and some suitable small h0>0. Let us fix some P∈Γ.Setx′≔x1,⋯xn−1. For h<h0,i∈1,2, and a function φ with suppφ⊂suppζ, we define tangential translation through
(18)φτx′,xn≔φx′+hei,xn+θPx′+hei−θPx′and tangential derivative through
(19)∂τφ≔limh⟶0φτ−φh.Now, we give the two propositions below related to the tangential derivatives.Proposition 4 (see [12, 14]).
Letsuppφ⊂suppζP and φ∈W01,1ΩP. Then,
(20)∇∂τφ=∂τ∇φ+∂nφτ⊗∂τ∇θP,D∂τφ=∂τDφ+∂nφτ⨀∂τ∇θP,div∂τφ=∂τdivφ+∂nφτ·∂τ∇θP.Proposition 5 (see [20]).
Let1<px<∞, px∈C0,1ΩP. Then, there holds
(21)ζP∂τ∇upx≤cζP∂τDupx+c∇ζP∇upx,∀u∈W01,pxΩ.


## 2.1. Notations

Byp′x, we denote the conjugate function of px. For p∈L∞Ω, p≥1, define p−≔essinfpx, p+≔esssuppx. Let p∗ be a Sobolev conjugate exponent, i.e., p∗=np/n−p when p<n and p∗=q for all q∈1,∞ when p=n and p∗=∞ otherwise.Forn×n-matrices F,H denote F:H=∑i,j=1nFijHij, F≡F:F1/2. For two vectors a and b, a⊗b=aibjij and aeb≔1/2a⊗b+a⊗bT.Forp∈L∞Ω, p≥1, the variable exponent Lebesgue space LpxΩ is defined by
(13)LpxΩ≔u:Ω⟶ℝ∣uismeasurableandρpxu≔∫Ωupxdx<∞,endowed with the norm upx,Ω≔infλ>0∣ρpxu/λ≤1. Then, we define the variable exponent Sobolev space by
(14)Wk,pxΩ≔u∣∇αu∈LpxΩ,∀α≤k,with the norm uk,px,Ω≔∑α≤k∇αupx,Ω. We define W0k,pxΩ as the closure of C0∞Ω in Wk,pxΩ. Let W−1,p′xΩ be dual of W01,pxΩ.We do not distinguish between scalar, vector-valued, and tensor-valued function spaces in the notations. Define(15)Vpx≔u∈W01,pxΩ∣divu=0.Definition 3.
We say that functionu is a weak solution to the problem (1) if u∈Vpx and it satisfies
(16)∫Ω1+Du2px−2/2Du:Dϕdx+∫Ωu·∇u·ϕdx=∫Ωf·ϕdx,∀ϕ∈Vpx.
We refer to the term strong solution as a weak solution which additionally satisfiesu∈W2,qΩ for some 1≤q<∞.


## 2.2. Some Problems Related to Flattening of the Boundary

As before, our problem is reduced to a problem involving a flat boundary by a suitable change of variables. Here, we follow the arguments and notations in [14]. Since ∂Ω∈C2,1, for each point P∈∂Ω, there are local coordinates such that in these coordinates, we have P=0 and ∂Ω is locally described by a C2,1-function θP:Qdn−10⟶Qd10, where Qdk0 is the k-dimensional cubic with center 0 and length 2d (which is small enough and will be fixed later), with the following properties:
(17)x∈ΩP≔Ω∩Qdn−10×Qd10⟺xn>θPx1,⋯xn−1,∇θP0=0,∇θPy<cd,y∈Qdn−10.As∂Ω is a compact, there exist a finite set of points Γ⊂∂Ω and an open set Ω0⊂⊂Ω such that Ω⊂Ω0∪⋃P∈ΓΩP. We construct a partition of unity ζ0,ζP,P∈Γ, corresponding to this covering, such that distsuppζP,∂ΩP\∂Ω≥h0 for all P∈Γ and some suitable small h0>0. Let us fix some P∈Γ.Setx′≔x1,⋯xn−1. For h<h0,i∈1,2, and a function φ with suppφ⊂suppζ, we define tangential translation through
(18)φτx′,xn≔φx′+hei,xn+θPx′+hei−θPx′and tangential derivative through
(19)∂τφ≔limh⟶0φτ−φh.Now, we give the two propositions below related to the tangential derivatives.Proposition 4 (see [12, 14]).
Letsuppφ⊂suppζP and φ∈W01,1ΩP. Then,
(20)∇∂τφ=∂τ∇φ+∂nφτ⊗∂τ∇θP,D∂τφ=∂τDφ+∂nφτ⨀∂τ∇θP,div∂τφ=∂τdivφ+∂nφτ·∂τ∇θP.Proposition 5 (see [20]).
Let1<px<∞, px∈C0,1ΩP. Then, there holds
(21)ζP∂τ∇upx≤cζP∂τDupx+c∇ζP∇upx,∀u∈W01,pxΩ.


## 3. The Proof of Main Results

We use universal constantsc,C>0, which may vary in different occurrences. In particular, C depends on p−,p+,Ω,n,∇px∞,fp^′x, while c on p−,p+,Ω,n,∇px∞.To begin with, let us define(22)Yλ≔1+Duλ21+λDuλ2,Mλ≔Yλpx−2/2,Sλx,Duλ≔MλDuλ,VpxλDuλ≔Mλ1/2Duλ.As before, in order to prove the main result, i.e., Theorem1, we will consider the following approximate problem:
(23)−divSλx,Duλ+uλ·∇uλ+∇πλ=f,inΩ,divuλ=0,inΩ,uλ=0,on∂Ω,where λ∈0,1. Let us denote
(24)L02Ω≔f∈L2Ω,∫Ωfdx=0.It is known that there exists a weak solutionuλ,πλ∈V2Ω×L02Ω to the problem (23) by compactness method if f∈L2Ω. Furthermore, the following facts are valid.Proposition 6.
Letuλ be a weak solution to the problem (23). Assume that f∈Lp∧′xΩ and 3/2≤px<∞. Then, the following hold:
(25)∫ΩMλDuλ2dx+∫ΩDuλp∧xdx≤c∫Ωfp∧′xdx+cΩ,(26)Mλ1/22px/px−2,Ω2+Mλ−1/22px/2−px,Ω1≤C,where p^x is from (8). For the problem (23) without convective term, these are valid for all 1<px<∞.Remark 7.
In fact, the proposition above was proved in ([31], Lemma 4.1) and [32] for the boundary condition (3). But it is easy to see that these inequalities hold also for Dirichlet boundary condition.Moreover, noting that(27)1≤1+a21+λa2≤1λ,forλ∈0,1,we can prove by the same line in [20] that if f∈L2Ω, px∈C0,1Ω¯, then a weak solution uλ,πλ of the problem (23) satisfies
(28)uλ∈V2Ω∩W2,2Ω,πλ∈L02Ω∩W1,2Ω.However, the norms ofuλ in V2Ω∩W2,2Ω and πλ in L02Ω∩W1,2Ω are dependent of λ.Thus, from now on, we focus on the estimates about the derivatives of the approximation solutions in suitable Sobolev spaces, which are independent of parameter. This allows us to show convergence of the approximation solutions to the one to problem (1) in the spaces.Hereafter, all constants are independent of parameterλ and introduce a shorthand notation Sλ≔Sλx,Duλ if it will be clear from the context.Let the assumptions of Theorem1 hold.To prove Theorem1, we will combine the methods in [20, 31, 32].The proof is divided into two cases: with and without the convective term.


### 3.1. The Proof of Theorem1 without the Convective Term

Step 1.
Estimates of the approximate solutions independent of parameter in tangential directions.
In this step, our aim is to prove that(29)∫Mλ∂τDuλ2ζ2dx≤C+cfp∧′x2+cTλ+cSλp^′x2,(30)∂τπλζp˘′x≤C+cfp∧′x+CAλ1/2+Tλ1/2+cSλp∧′x,where p^x is from (8), p˘x≔max2,px, and
(31)Aλ≔∫Mλ∂τDuλ2ζ2dx,Tλ≔∫MλDuλ2log2Yλdx.FixP∈Γ, and let ζ=ζP, Ω=ΩP, and θ=θP be as in Subsection 2.2. For simplicity we will omit the symbol “ΩP” in integration ones and denote the symbols “ΩP∩Ωi,” i=1,2, by “Ωi.”By ([20], Proposition 14.3.15) and Propositions 4 and 5, there exists a solution ψ∈W01,p^xΩ to the problem
(32)divψ=−div∂τζ2∂τuλ,inΩ;ψ=0,on∂Ω,satisfying
(33)Dψp^x≤cζ∂τDuλp^x+c∇uλp^x,where the constant c depends on p−,p+,Ω,n.Multiplying the first equations in (23) by ϕ=∂τζ2∂τuλ+ψ, integrating by parts and using Proposition 4, we get
(34)∫ζ∂τSλ:Dζ∂τuλdx=−∫∂τSλ:∇ζ⨀ζ∂τuλdx−∫Sλ:∂nζ2∂τuλ⨀∂τ∇θdx+∫f·∂τζ2∂τuλ+ψdx−∫ΩSλ:Dψdx≕∑i=14Ii.We apply Proposition4 to the left hand side of (34) to get
(35)I0≔∫ζ∂τSλ:Dζ∂τuλdx=∫ζ2∂τSλ:∂τDuλdx+∫ζ2∂τSλ:∂nuλτ⨀∂τ∇θdx+∫∂τSλ:∂τuλ⨀∇ζζdx≕J1+J2+J3.It is clear that(36)∂τSλ=∑i,j=1n∂Sλ∂Dij∂τDijuλ+∂Sλ∂px∂τpx≕J1,1+J1,2.From ([31], (3.6)), we have
(37)∫J1,1:∂τDuλζ2dx≥cp−Aλ.where cp−=p^−−1. It is shown in ([31], (3.7)) that
(38)∫J1,2:∂τDuλζ2dx≤cp−16Aλ+cTλ.Combining (37) with (38) yields that
(39)15cp−16Aλ≤J1+cTλ.The termsJ2,J3 can be rewritten as
(40)J2+J3=−∫Sλ:∂τ∂nuλτ⨀∂τ∇θζ2dx−∫Sλ:∂τ∂τuλ⨀∇ζζdx.Hence these terms can be estimated as follows: by Korn’s inequality(41)J2+J3≤cSλp^′x∇uλp^x+∂τ∇uλζp^x≤13CSλp^′x1+∂τDuλζp^x≤CSλp^′x1+Mλ−1/22px/2−px,Ω1Aλ1/2+Aλ1/2≤14cSλp^′x2+cp−16Aλ+C.Combining (39), (41) with (35), we arrive at
(42)7cp−8Aλ≤cI0+cTλ+cSλp^′x2+C.SinceI1=−J3, the term I1 can be estimated as J3 by
(43)I1≤cSλp^′x2+cp−8Aλ+C.Note that by Proposition4(44)∂nζ2∂τuλ=ζ2∂τ∂nuλ+2ζ∂nζ∂τuλ.Hence the termI2 also can be estimated as J2+J3:
(45)I2=−∫Sλ:ζ2∂τ∂nuλ+2ζ∂nζ∂τuλdx≤cSλp^′x2+cp−8Aλ+C.We use Hölder’s inequality to get(46)I3≤c∫f∂τuλdx+c∫fζ∂τ∇uλ+ψdx≤13Cfp∧′x+fp∧′xψp^x+ζ∂τ∇uλp^x≤13,17Cfp∧′x+1+cfp∧′xζ∂τDuλp^x≤Cfp∧′x+1+cfp∧′xMλ−1/22px/2−px,Ω1+1Aλ1/2≤14C1+fp∧′x2+cp−8Aλ.Using (33) and (25) yields that
(47)I4≤∫SλDψdx≤cDψp^xSλp^′x≤cp−8Aλ+C+cSλp^′x2.Equation (34) together with (42)-(47) yields the desired estimate (29).Next, let us prove (30).In [32] (4.20), it is proved that
(48)πλp˘′x,Ω≤C.Indeed, though the formula (4.20) from [32] is proved in 2D for px>3/2, it also holds for px>3n/n+2, n=2,3. Moreover if neglecting the convective term, this holds for px>1.Letϕ∈C02Ω. Multiplying the first equations of (23) by ∂τζϕ and integrating by parts, we have
(49)∫∂τπλdivζϕdx=∫∂τSλ:Dζϕdx+∫πλ∂nζϕτ·∂τ∇θdx+∫f·∂τζϕdx−∫Sλ:∂nζϕτ⨀∂τ∇θdx.Sincep˘′x≤p∧′x, we have
(50)∫Sλ:∂nζϕτ⨀∂τ∇θdx≤cSλp˘′x∇ζϕp˘x≤cSλp^′xϕ1,p˘x,(51)∫πλ∂nζϕτ·∂τ∇θdx≤cπλp˘′x∇ζϕp˘x≤30Cϕ1,p˘x,(52)∫f·∂τζϕdx≤cfp^′x∇ζϕp^x≤cfp^′xϕ1,p˘x.It follows that(53)∫∂τSλ:Dζϕdx≤∫J1,1+J1,2Dζϕdx≤c∫Mλ∂τDuλ+MλDuλlogYλDζϕdx≤cAλ1/2Dϕ2,Ω1+Mλ1/22px/px−2,Ω2Dϕpx,Ω1+cTλ1/2Dϕp˘x≤14CAλ1/2+Tλ1/2ϕ1,p˘x.The left hand of (49) can be rewritten as
(54)∫∂τπλdivϕdx=−∫∇∂τπλ·ϕdx.Equation (49) together with (50)-(54) implies that
(55)∫∇∂τπλζ·ϕdx≤C+cfp∧′x+CAλ1/2+Tλ1/2ϕ1,p˘x+cSλp^′x.Thus, by the negative norm theorem ([38], Theorem 14.3.18) and (55), we have
(56)∂τπλζp˘′x≤c∇∂τπλζ−1,p˘′x+c∂τπλζ−1,p˘′x=csupϕ∈W01,p˘xΩ∇∂τπλζ,ϕϕ1,p˘x+csupϕ∈W01,p˘xΩ∂τπλζ,ϕϕ1,p˘x≤C+cfp∧′x+CAλ1/2+Tλ1/2+cSλp1′+csupϕ∈W01,p˘xΩπλp˘′x∂τϕp˘xϕ1,p˘x≤C+cfp∧′x+CAλ1/2+Tλ1/2+cSλp^′x,which is the just (30).Step 2.
Estimates of the approximate solutions independent of parameter in all directions.Our aim in this step is to show (79) and (80).Here, we follow the notations from [20]. By mimicking the derivation of [20], (4.51), we can get
(57)Mλ∇Duλ≤cMλ∂τ∇uλ+DuλlogYλ+c∂τπλ+f.Now, we want to derive some estimates on the first derivatives ofVpxλDuλ from (22), independent of parameter λ. This allows us to prove boundedness of Tλ and Sλp^′x and to improve regularity of solutions to problem (1).Since1/3∇2uλ≤D∇uλ≤3∇2uλ (see, [1]), it follows from (57) that
(58)∇VpxλDuλ≤cMλ1/2∇2uλ+cMλ1/2logYλDuλ≤cMλ1/2∂τ∇uλ+DuλlogYλ+cMλ−1/2∂τπλ+f.Similarly, we have(59)∂τVpxλDuλ≤cMλ1/2∂τDuλ+cMλ1/2logYλDuλ.Now, define(60)r1x≔2pxpx+px−2=minpx,p′x.It follows from (58) that
(61)∇VpxλDuλζr1x≤cMλ1/2∂τ∇uλζr1x+cMλ1/2logYλDuλζr1x+cMλ−1/2∂τπλ+fζr1x.Using Hölder’s inequality and the fact thatr1x≤2, we have by Korn’s inequality
(62)Mλ1/2∂τ∇uλζr1x≤∂τ∇uλζpx,Ω1+Mλ1/2∂τ∇uλζp′x,Ω2≤c∂τDuλζpx,Ω1+Mλ1/22px/px−2,Ω22,Ω2+C≤cMλ−1/22px/2−px,Ω1+Mλ1/22px/px−2,Ω2Aλ1/2+C≤14CAλ1/2+C.Taking into account the fact thatr1x≤2, we have
(63)cMλ1/2logYλDuλr1x≤cTλ1/2.From (30) and the fact that Mλ−1/2≤1 and r1x=p˘′x in Ω2, it follows that
(64)Mλ−1/2∣∂τπλ∣+∣f∣ζr1x≤Mλ−1/22px/2−px,Ω1∂τπλζ+f2,Ω1+∂τπλζ+fp′x,Ω2≤14C1+fp∧′x+Aλ1/2+Tλ1/2+Sλp∧′x.Gathering (61)-(64) and using (29), we obtain
(65)∇VpxλDuλζr1x≤C1+fp∧′x+Tλ1/2+cSλp∧′x.On the other hand, it follows from (59) and (29) that
(66)∂τVpxλDuλζ2≤cMλ1/2∂τDuλζ2+cMλ1/2logYλDuλζ2≤c1+fp∧′x+Tλ1/2+Sλp∧′x.Define(67)m1x≔1npx+px−22px+n−12−1−1−μ=2npxpxn−2+px−2−μ,where μ is an arbitrary small positive real number. Applying the anisotropic embedding theorem (see [39], Theorem 2.5), we obtain that by (65) and (66),
(68)VpxλDuλm1x≤c1+fp∧′x+Tλ1/2+Sλp∧′x.It is shown in [31, 32] that
(69)Tλ≤ε0VpxλDuλm1x2+C.Now, let us estimate the termSλp^′x. Note that
(70)pxpx−1<m1x,provided that
(71)px<2,px>2n+1n+3.This implies immediately that(72)Sλp′x,Ω1≤VpxλDuλp′x,Ω1≤ε0VpxλDuλm1x+C.Before we begin with the estimate onSλ2,Ω2, we claim that for p≥2,
(73)1+a21+λa2p−2/2a≤c1+a21+λa2p−2/4a2p−1/p+c.Indeed, settingY≔1+a2/1+λa2 and using Young’s inequality with a pair p/p−2,p/2, we get
(74)Yp−2/4=Yp−2p−2/4p−2p−2/pYp−2/41−p−2p−2/p=Yp−2/4Y1/2p−2/p≤cYp−2/41+ap−2/p≤cYp−2/4ap−2/p+cYp−2p−2/4p−2p−2/p≤cYp−2/4ap−2/p+12Yp−2/4+c,which implies (73).Now, let us estimate the termSλ2,Ω2. Using the interpolation and Young’s inequalities yields that
(75)Sλ2,Ω2≤52c+cVpxλDuλ4p2−1/p2,Ω22p2−1/p2≤c+cVpxλDuλ2,Ω21−βVpxλDuλm−1,Ω2β2p2−1/p2,where
(76)p2≔supΩP∩Ω2px,m−1≔infΩP∩Ω2m1x=2np2p2n−2+p2−2−μ,β≔np2p2−22p2−1p2+2.Hence, noting thatm1x is decreased in p over Ω2, we have
(77)Sλ2,Ω2≤52c+cVpxλDuλm−1,Ω2np2−2/p2+2≤ε0VpxλDuλm1x+C,provided that np2−2/p2+2<1, i.e.,
(78)px>2,px<2n+1n−1.Inserting (69), (72), and (77) into (68) and taking (71) and (78) into account, we conclude that
(79)VpxλDuλm1x≤c1+fp∧′xand furthermore from (65) that
(80)∇VpxλDuλζr1x≤C1+fp∧′x,provided that
(81)2n+1n+3<px<2n+1n−1,which is the just (11).Step 3.
Sincem1x>2, the estimate (79) gives us the better information on VpxλDuλ than an a priori estimate (25). By the same argument, it is possible to gradually improve the integrability of VpxλDuλ with the estimates independent of parameter and to get (7) and (8) with uλ instead of u. This bootstrap argument is the same as in [20, 31, 32], and so we omit them. In particular, since the estimates (9) and (10) with uλ are independent of parameter λ, it is possible to apply limiting process λ⟶0 and as a result, we have that there exists a strong solution to (1) without the convective term satisfying (9) and (10).Thus, Theorem1 is proved without the convective term.


### 3.2. The Proof of Theorem1 with the Convective Term

Step 1.
Estimates of the approximate solutions independent of parameter in tangential directions.Our aim in this step is to show that the estimates (29) and (30) are valid under the additional assumption px>3n/n+2.To begin with, let us show the validity of (29).In this case, on (34), there will be added the term
(82)I5=∫∂τuλ·∇uλ+uλτ·∇∂τuλ−uτ·∂nuλτ⊗∂τ∇θ·ζ2∂τuλdx−∫uλ·∇uλ·ψdx=∫∂τuλ2∇ζ·uλζ+uλτ·∂nuλτ⊗∂τ∇θ·∂τuλζ2dx−∫ζ2∂τuλ·∇uλ·∂τuλdx−∫uλ·∇uλ·ψdx≕I51+I52+I53,see [20], (5.1).Letp1≔infΩPpx. Taking into account 2p^′1<p∧1∗ for p^1>3n/n+2 and using Hölder’s, Sobolev’s, and Korn’s inequalities, we have
(83)I52≤c∂τuλp^1uλ2p^′1ζ∂τuλ2p^′1≤c∇uλp^1uλp^1∗ζ∂τuλp^1∗≤c∇uλp^x2Dζ∂τuλp^x≤13C+CMλ−1/22px/2−px,Ω1+1Aλ1/2≤14C+cp−24Aλ.On the other hand, by Hölder’s inequality, we have(84)I51≤ζ∂τuλ2p^′12∇uλp^1.Note that3n/n+2<p^1≤2. Since p^1<2p^′1<p∧1∗, we can interpolate ζ∂τu2p^′1 between Lp∧1 and Lp∧1∗. Using this interpolation and Korn’s, Hölder’s, and Sobolev’s inequalities, it follows that for α=n+2p^1−3n/2p^1,
(85)I51≤13Cζ∂τuλ2p^1′2≤Cζ∂τuλp^12αζ∂τuλp∧1∗21−α≤Cζ∂τuλp^12+Cζ∂τuλp^12αζ∂τDuλp^x21−α≤C+Cζ∂τuλp^12αMλ−1/22px/2−px,Ω121−α+1Aλ1−α≤14C+cp−24Aλ,where, in the last estimate, we use Young’s inequality with a pair 1/1−α,1/α. From (33), (25), and (26), it follows that
(86)I53≤uλ2p^′12∇ψp^1≤C+CDζ∂τuλp^x≤C+CMλ−1/22px/2−px,Ω1+1Aλ1/2≤C+cp−24Aλ.Thus, identity (82) together with (83)-(86) yields
(87)I5≤C+cp−8Aλ.Thus, the estimate (29) continues to hold for the full problem (23).Next, let us prove that there also holds (30). Indeed, for the full problem (23), there will be additionally added the term
(88)−∫uλ·∇uλ·∂τζϕdx,on the right hand side of (49). It is clear that
(89)npxn+1px−2n≤2∗,x∈Ω1,4≤p∗x,x∈Ω2.Recalling thatW1,p˘xΩ°LqΩ for any q<∞ and using Hölder’s inequality, we obtain
(90)∫Ωuλ·∇uλ·∂τζϕdx=∫Ω∂τuλ·∇uλ·ζϕdx+∫Ωuλ·∇∂τuλ·ζϕdx≤c∂τuλζp∗x,Ω1∇uλpx,Ω1ϕnpx/n+1px−2n,Ω1+∂τuλζ4,Ω2∇uλ2,Ω2ϕ4,Ω2+cuλp∗x,Ω1∇∂τuλζpx,Ω1ϕnpx/n+1px−2n,Ω1+uλ4,Ω2∇∂τuλζ2,Ω2ϕ4,Ω2≤13,14CAλ1/2ϕ1,p˘x.Thus, there also holds (30) for the problem (23).Step 2.
Estimates of the approximate solutions independent of parameter in all directions.Here, we prove the validity of (79).For the problem (23), there will be additionally added the term
(91)cMλ−1/2uλ·∇uλζr1xon the right hand side of (61) and hence,
(92)VpxλDuλm1x≤c1+fp∧′x+Tλ1/2+Sλp∧′x+Mλ−1/2uλ·∇uλζr1x,where m1x is from (67).By (25) and (26) and Korn’s, Hölder’s, and Young’s inequalities, we have
(93)Mλ−1/2uλ·∇uλζr1x,Ω1≤Mλ−1/22px/2−px,Ω1uλnpx/n−px,Ω1∇uλζ2npx/n+2px−2n,Ω1≤CDuλζ2npx/n+2px−2n,Ω1+C≤CMλ−1/22px/2−px,Ω1VpxλDuλζnpx/n+1px−2n,Ω1+C≤ε0VpxλDuλm1x+C,where, to perform the last estimate, we use the fact
(94)npxn+1px−2n<2npxpxn−3+2=m1x,forn=2,3,x∈Ω1,which holds by the condition px>3n/n+2.On the other hand, by the same line above, we obtain that(95)Mλ−1/2uλ·∇uλζr1x,Ω2≤uλ2n/n−2,Ω2∇uλζ2npx/n+2px−2n,Ω2≤CDuλζ2npx/n+2px−2n,Ω2+C≤Mλ1/2≥1CVpxλDuλζ2npx/n+2px−2n,Ω2+C≤ε0VpxλDuλm1x+C,where, to perform the last estimate, we use the fact
(96)2npxn+2px−2n<2npxpxn−1−2=m1x,forn=2,3,x∈Ω2,which also holds by the condition px>3n/n+2.Inserting (93) and (95) into (92) and noting (69), (72), and (77), we conclude that
(97)VpxλDuλm1x≤C1+fp∧′x.This is just the estimate (79).Step 3.
The rest is the same as the previous subsection. Thus, Theorem1 is completely proved.


## 3.1. The Proof of Theorem1 without the Convective Term

Step 1.
Estimates of the approximate solutions independent of parameter in tangential directions.
In this step, our aim is to prove that(29)∫Mλ∂τDuλ2ζ2dx≤C+cfp∧′x2+cTλ+cSλp^′x2,(30)∂τπλζp˘′x≤C+cfp∧′x+CAλ1/2+Tλ1/2+cSλp∧′x,where p^x is from (8), p˘x≔max2,px, and
(31)Aλ≔∫Mλ∂τDuλ2ζ2dx,Tλ≔∫MλDuλ2log2Yλdx.FixP∈Γ, and let ζ=ζP, Ω=ΩP, and θ=θP be as in Subsection 2.2. For simplicity we will omit the symbol “ΩP” in integration ones and denote the symbols “ΩP∩Ωi,” i=1,2, by “Ωi.”By ([20], Proposition 14.3.15) and Propositions 4 and 5, there exists a solution ψ∈W01,p^xΩ to the problem
(32)divψ=−div∂τζ2∂τuλ,inΩ;ψ=0,on∂Ω,satisfying
(33)Dψp^x≤cζ∂τDuλp^x+c∇uλp^x,where the constant c depends on p−,p+,Ω,n.Multiplying the first equations in (23) by ϕ=∂τζ2∂τuλ+ψ, integrating by parts and using Proposition 4, we get
(34)∫ζ∂τSλ:Dζ∂τuλdx=−∫∂τSλ:∇ζ⨀ζ∂τuλdx−∫Sλ:∂nζ2∂τuλ⨀∂τ∇θdx+∫f·∂τζ2∂τuλ+ψdx−∫ΩSλ:Dψdx≕∑i=14Ii.We apply Proposition4 to the left hand side of (34) to get
(35)I0≔∫ζ∂τSλ:Dζ∂τuλdx=∫ζ2∂τSλ:∂τDuλdx+∫ζ2∂τSλ:∂nuλτ⨀∂τ∇θdx+∫∂τSλ:∂τuλ⨀∇ζζdx≕J1+J2+J3.It is clear that(36)∂τSλ=∑i,j=1n∂Sλ∂Dij∂τDijuλ+∂Sλ∂px∂τpx≕J1,1+J1,2.From ([31], (3.6)), we have
(37)∫J1,1:∂τDuλζ2dx≥cp−Aλ.where cp−=p^−−1. It is shown in ([31], (3.7)) that
(38)∫J1,2:∂τDuλζ2dx≤cp−16Aλ+cTλ.Combining (37) with (38) yields that
(39)15cp−16Aλ≤J1+cTλ.The termsJ2,J3 can be rewritten as
(40)J2+J3=−∫Sλ:∂τ∂nuλτ⨀∂τ∇θζ2dx−∫Sλ:∂τ∂τuλ⨀∇ζζdx.Hence these terms can be estimated as follows: by Korn’s inequality(41)J2+J3≤cSλp^′x∇uλp^x+∂τ∇uλζp^x≤13CSλp^′x1+∂τDuλζp^x≤CSλp^′x1+Mλ−1/22px/2−px,Ω1Aλ1/2+Aλ1/2≤14cSλp^′x2+cp−16Aλ+C.Combining (39), (41) with (35), we arrive at
(42)7cp−8Aλ≤cI0+cTλ+cSλp^′x2+C.SinceI1=−J3, the term I1 can be estimated as J3 by
(43)I1≤cSλp^′x2+cp−8Aλ+C.Note that by Proposition4(44)∂nζ2∂τuλ=ζ2∂τ∂nuλ+2ζ∂nζ∂τuλ.Hence the termI2 also can be estimated as J2+J3:
(45)I2=−∫Sλ:ζ2∂τ∂nuλ+2ζ∂nζ∂τuλdx≤cSλp^′x2+cp−8Aλ+C.We use Hölder’s inequality to get(46)I3≤c∫f∂τuλdx+c∫fζ∂τ∇uλ+ψdx≤13Cfp∧′x+fp∧′xψp^x+ζ∂τ∇uλp^x≤13,17Cfp∧′x+1+cfp∧′xζ∂τDuλp^x≤Cfp∧′x+1+cfp∧′xMλ−1/22px/2−px,Ω1+1Aλ1/2≤14C1+fp∧′x2+cp−8Aλ.Using (33) and (25) yields that
(47)I4≤∫SλDψdx≤cDψp^xSλp^′x≤cp−8Aλ+C+cSλp^′x2.Equation (34) together with (42)-(47) yields the desired estimate (29).Next, let us prove (30).In [32] (4.20), it is proved that
(48)πλp˘′x,Ω≤C.Indeed, though the formula (4.20) from [32] is proved in 2D for px>3/2, it also holds for px>3n/n+2, n=2,3. Moreover if neglecting the convective term, this holds for px>1.Letϕ∈C02Ω. Multiplying the first equations of (23) by ∂τζϕ and integrating by parts, we have
(49)∫∂τπλdivζϕdx=∫∂τSλ:Dζϕdx+∫πλ∂nζϕτ·∂τ∇θdx+∫f·∂τζϕdx−∫Sλ:∂nζϕτ⨀∂τ∇θdx.Sincep˘′x≤p∧′x, we have
(50)∫Sλ:∂nζϕτ⨀∂τ∇θdx≤cSλp˘′x∇ζϕp˘x≤cSλp^′xϕ1,p˘x,(51)∫πλ∂nζϕτ·∂τ∇θdx≤cπλp˘′x∇ζϕp˘x≤30Cϕ1,p˘x,(52)∫f·∂τζϕdx≤cfp^′x∇ζϕp^x≤cfp^′xϕ1,p˘x.It follows that(53)∫∂τSλ:Dζϕdx≤∫J1,1+J1,2Dζϕdx≤c∫Mλ∂τDuλ+MλDuλlogYλDζϕdx≤cAλ1/2Dϕ2,Ω1+Mλ1/22px/px−2,Ω2Dϕpx,Ω1+cTλ1/2Dϕp˘x≤14CAλ1/2+Tλ1/2ϕ1,p˘x.The left hand of (49) can be rewritten as
(54)∫∂τπλdivϕdx=−∫∇∂τπλ·ϕdx.Equation (49) together with (50)-(54) implies that
(55)∫∇∂τπλζ·ϕdx≤C+cfp∧′x+CAλ1/2+Tλ1/2ϕ1,p˘x+cSλp^′x.Thus, by the negative norm theorem ([38], Theorem 14.3.18) and (55), we have
(56)∂τπλζp˘′x≤c∇∂τπλζ−1,p˘′x+c∂τπλζ−1,p˘′x=csupϕ∈W01,p˘xΩ∇∂τπλζ,ϕϕ1,p˘x+csupϕ∈W01,p˘xΩ∂τπλζ,ϕϕ1,p˘x≤C+cfp∧′x+CAλ1/2+Tλ1/2+cSλp1′+csupϕ∈W01,p˘xΩπλp˘′x∂τϕp˘xϕ1,p˘x≤C+cfp∧′x+CAλ1/2+Tλ1/2+cSλp^′x,which is the just (30).Step 2.
Estimates of the approximate solutions independent of parameter in all directions.Our aim in this step is to show (79) and (80).Here, we follow the notations from [20]. By mimicking the derivation of [20], (4.51), we can get
(57)Mλ∇Duλ≤cMλ∂τ∇uλ+DuλlogYλ+c∂τπλ+f.Now, we want to derive some estimates on the first derivatives ofVpxλDuλ from (22), independent of parameter λ. This allows us to prove boundedness of Tλ and Sλp^′x and to improve regularity of solutions to problem (1).Since1/3∇2uλ≤D∇uλ≤3∇2uλ (see, [1]), it follows from (57) that
(58)∇VpxλDuλ≤cMλ1/2∇2uλ+cMλ1/2logYλDuλ≤cMλ1/2∂τ∇uλ+DuλlogYλ+cMλ−1/2∂τπλ+f.Similarly, we have(59)∂τVpxλDuλ≤cMλ1/2∂τDuλ+cMλ1/2logYλDuλ.Now, define(60)r1x≔2pxpx+px−2=minpx,p′x.It follows from (58) that
(61)∇VpxλDuλζr1x≤cMλ1/2∂τ∇uλζr1x+cMλ1/2logYλDuλζr1x+cMλ−1/2∂τπλ+fζr1x.Using Hölder’s inequality and the fact thatr1x≤2, we have by Korn’s inequality
(62)Mλ1/2∂τ∇uλζr1x≤∂τ∇uλζpx,Ω1+Mλ1/2∂τ∇uλζp′x,Ω2≤c∂τDuλζpx,Ω1+Mλ1/22px/px−2,Ω22,Ω2+C≤cMλ−1/22px/2−px,Ω1+Mλ1/22px/px−2,Ω2Aλ1/2+C≤14CAλ1/2+C.Taking into account the fact thatr1x≤2, we have
(63)cMλ1/2logYλDuλr1x≤cTλ1/2.From (30) and the fact that Mλ−1/2≤1 and r1x=p˘′x in Ω2, it follows that
(64)Mλ−1/2∣∂τπλ∣+∣f∣ζr1x≤Mλ−1/22px/2−px,Ω1∂τπλζ+f2,Ω1+∂τπλζ+fp′x,Ω2≤14C1+fp∧′x+Aλ1/2+Tλ1/2+Sλp∧′x.Gathering (61)-(64) and using (29), we obtain
(65)∇VpxλDuλζr1x≤C1+fp∧′x+Tλ1/2+cSλp∧′x.On the other hand, it follows from (59) and (29) that
(66)∂τVpxλDuλζ2≤cMλ1/2∂τDuλζ2+cMλ1/2logYλDuλζ2≤c1+fp∧′x+Tλ1/2+Sλp∧′x.Define(67)m1x≔1npx+px−22px+n−12−1−1−μ=2npxpxn−2+px−2−μ,where μ is an arbitrary small positive real number. Applying the anisotropic embedding theorem (see [39], Theorem 2.5), we obtain that by (65) and (66),
(68)VpxλDuλm1x≤c1+fp∧′x+Tλ1/2+Sλp∧′x.It is shown in [31, 32] that
(69)Tλ≤ε0VpxλDuλm1x2+C.Now, let us estimate the termSλp^′x. Note that
(70)pxpx−1<m1x,provided that
(71)px<2,px>2n+1n+3.This implies immediately that(72)Sλp′x,Ω1≤VpxλDuλp′x,Ω1≤ε0VpxλDuλm1x+C.Before we begin with the estimate onSλ2,Ω2, we claim that for p≥2,
(73)1+a21+λa2p−2/2a≤c1+a21+λa2p−2/4a2p−1/p+c.Indeed, settingY≔1+a2/1+λa2 and using Young’s inequality with a pair p/p−2,p/2, we get
(74)Yp−2/4=Yp−2p−2/4p−2p−2/pYp−2/41−p−2p−2/p=Yp−2/4Y1/2p−2/p≤cYp−2/41+ap−2/p≤cYp−2/4ap−2/p+cYp−2p−2/4p−2p−2/p≤cYp−2/4ap−2/p+12Yp−2/4+c,which implies (73).Now, let us estimate the termSλ2,Ω2. Using the interpolation and Young’s inequalities yields that
(75)Sλ2,Ω2≤52c+cVpxλDuλ4p2−1/p2,Ω22p2−1/p2≤c+cVpxλDuλ2,Ω21−βVpxλDuλm−1,Ω2β2p2−1/p2,where
(76)p2≔supΩP∩Ω2px,m−1≔infΩP∩Ω2m1x=2np2p2n−2+p2−2−μ,β≔np2p2−22p2−1p2+2.Hence, noting thatm1x is decreased in p over Ω2, we have
(77)Sλ2,Ω2≤52c+cVpxλDuλm−1,Ω2np2−2/p2+2≤ε0VpxλDuλm1x+C,provided that np2−2/p2+2<1, i.e.,
(78)px>2,px<2n+1n−1.Inserting (69), (72), and (77) into (68) and taking (71) and (78) into account, we conclude that
(79)VpxλDuλm1x≤c1+fp∧′xand furthermore from (65) that
(80)∇VpxλDuλζr1x≤C1+fp∧′x,provided that
(81)2n+1n+3<px<2n+1n−1,which is the just (11).Step 3.
Sincem1x>2, the estimate (79) gives us the better information on VpxλDuλ than an a priori estimate (25). By the same argument, it is possible to gradually improve the integrability of VpxλDuλ with the estimates independent of parameter and to get (7) and (8) with uλ instead of u. This bootstrap argument is the same as in [20, 31, 32], and so we omit them. In particular, since the estimates (9) and (10) with uλ are independent of parameter λ, it is possible to apply limiting process λ⟶0 and as a result, we have that there exists a strong solution to (1) without the convective term satisfying (9) and (10).Thus, Theorem1 is proved without the convective term.


## 3.2. The Proof of Theorem1 with the Convective Term

Step 1.
Estimates of the approximate solutions independent of parameter in tangential directions.Our aim in this step is to show that the estimates (29) and (30) are valid under the additional assumption px>3n/n+2.To begin with, let us show the validity of (29).In this case, on (34), there will be added the term
(82)I5=∫∂τuλ·∇uλ+uλτ·∇∂τuλ−uτ·∂nuλτ⊗∂τ∇θ·ζ2∂τuλdx−∫uλ·∇uλ·ψdx=∫∂τuλ2∇ζ·uλζ+uλτ·∂nuλτ⊗∂τ∇θ·∂τuλζ2dx−∫ζ2∂τuλ·∇uλ·∂τuλdx−∫uλ·∇uλ·ψdx≕I51+I52+I53,see [20], (5.1).Letp1≔infΩPpx. Taking into account 2p^′1<p∧1∗ for p^1>3n/n+2 and using Hölder’s, Sobolev’s, and Korn’s inequalities, we have
(83)I52≤c∂τuλp^1uλ2p^′1ζ∂τuλ2p^′1≤c∇uλp^1uλp^1∗ζ∂τuλp^1∗≤c∇uλp^x2Dζ∂τuλp^x≤13C+CMλ−1/22px/2−px,Ω1+1Aλ1/2≤14C+cp−24Aλ.On the other hand, by Hölder’s inequality, we have(84)I51≤ζ∂τuλ2p^′12∇uλp^1.Note that3n/n+2<p^1≤2. Since p^1<2p^′1<p∧1∗, we can interpolate ζ∂τu2p^′1 between Lp∧1 and Lp∧1∗. Using this interpolation and Korn’s, Hölder’s, and Sobolev’s inequalities, it follows that for α=n+2p^1−3n/2p^1,
(85)I51≤13Cζ∂τuλ2p^1′2≤Cζ∂τuλp^12αζ∂τuλp∧1∗21−α≤Cζ∂τuλp^12+Cζ∂τuλp^12αζ∂τDuλp^x21−α≤C+Cζ∂τuλp^12αMλ−1/22px/2−px,Ω121−α+1Aλ1−α≤14C+cp−24Aλ,where, in the last estimate, we use Young’s inequality with a pair 1/1−α,1/α. From (33), (25), and (26), it follows that
(86)I53≤uλ2p^′12∇ψp^1≤C+CDζ∂τuλp^x≤C+CMλ−1/22px/2−px,Ω1+1Aλ1/2≤C+cp−24Aλ.Thus, identity (82) together with (83)-(86) yields
(87)I5≤C+cp−8Aλ.Thus, the estimate (29) continues to hold for the full problem (23).Next, let us prove that there also holds (30). Indeed, for the full problem (23), there will be additionally added the term
(88)−∫uλ·∇uλ·∂τζϕdx,on the right hand side of (49). It is clear that
(89)npxn+1px−2n≤2∗,x∈Ω1,4≤p∗x,x∈Ω2.Recalling thatW1,p˘xΩ°LqΩ for any q<∞ and using Hölder’s inequality, we obtain
(90)∫Ωuλ·∇uλ·∂τζϕdx=∫Ω∂τuλ·∇uλ·ζϕdx+∫Ωuλ·∇∂τuλ·ζϕdx≤c∂τuλζp∗x,Ω1∇uλpx,Ω1ϕnpx/n+1px−2n,Ω1+∂τuλζ4,Ω2∇uλ2,Ω2ϕ4,Ω2+cuλp∗x,Ω1∇∂τuλζpx,Ω1ϕnpx/n+1px−2n,Ω1+uλ4,Ω2∇∂τuλζ2,Ω2ϕ4,Ω2≤13,14CAλ1/2ϕ1,p˘x.Thus, there also holds (30) for the problem (23).Step 2.
Estimates of the approximate solutions independent of parameter in all directions.Here, we prove the validity of (79).For the problem (23), there will be additionally added the term
(91)cMλ−1/2uλ·∇uλζr1xon the right hand side of (61) and hence,
(92)VpxλDuλm1x≤c1+fp∧′x+Tλ1/2+Sλp∧′x+Mλ−1/2uλ·∇uλζr1x,where m1x is from (67).By (25) and (26) and Korn’s, Hölder’s, and Young’s inequalities, we have
(93)Mλ−1/2uλ·∇uλζr1x,Ω1≤Mλ−1/22px/2−px,Ω1uλnpx/n−px,Ω1∇uλζ2npx/n+2px−2n,Ω1≤CDuλζ2npx/n+2px−2n,Ω1+C≤CMλ−1/22px/2−px,Ω1VpxλDuλζnpx/n+1px−2n,Ω1+C≤ε0VpxλDuλm1x+C,where, to perform the last estimate, we use the fact
(94)npxn+1px−2n<2npxpxn−3+2=m1x,forn=2,3,x∈Ω1,which holds by the condition px>3n/n+2.On the other hand, by the same line above, we obtain that(95)Mλ−1/2uλ·∇uλζr1x,Ω2≤uλ2n/n−2,Ω2∇uλζ2npx/n+2px−2n,Ω2≤CDuλζ2npx/n+2px−2n,Ω2+C≤Mλ1/2≥1CVpxλDuλζ2npx/n+2px−2n,Ω2+C≤ε0VpxλDuλm1x+C,where, to perform the last estimate, we use the fact
(96)2npxn+2px−2n<2npxpxn−1−2=m1x,forn=2,3,x∈Ω2,which also holds by the condition px>3n/n+2.Inserting (93) and (95) into (92) and noting (69), (72), and (77), we conclude that
(97)VpxλDuλm1x≤C1+fp∧′x.This is just the estimate (79).Step 3.
The rest is the same as the previous subsection. Thus, Theorem1 is completely proved.



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*Source: 6755411-2021-12-07.xml*