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

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized NavierStokes equation modeling by pðxÞ-power law with Dirichlet boundary condition under the restriction ð3n/ðn + 2Þn + 2Þ < p ðxÞ < ð2ðn + 1ÞÞ/ðn − 1Þ. In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction ð2ðn + 1ÞÞ/ðn + 3Þ < pðxÞ < ð2ðn + 1ÞÞ/ðn − 1Þ, which arises from the nonflatness of domain.


Introduction
In this note, we consider the steady flows of non-Newtonian fluids in ℝ n , n = 2, 3, which is modeled by the following system: where u is the velocity, π the pressure, f the external force, Du ≔ ð∇u+∇u T Þ/2, Ω a bounded domain, and pðxÞ > 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].
Global higher differentiability for weak solutions to the problem (1) with pðxÞ = const have been studied by several authors; for example, see [8][9][10][11][12][13][14][15][16][17][18][19] under the condition f ∈ L p∧′ ðΩÞ with p∧ ′ =p/ðp − 1Þ andp ≔ min fp, 2g. 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 W 2,ð4p−2Þ/ðp+1Þ ∩ W 1,4p−2 -regularity for ð20/11Þ < 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 W 2,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 ∈ L αp∧′ðxÞ ðΩÞ, where p∧′ ðxÞ = ðpðxÞÞ/ðpðxÞ − 1Þ,pðxÞ ≔ min fpðxÞ, 2g, and α > 1. This is slightly stronger rather than the standard condition f ∈ L p∧′ ðΩÞ 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 C 1,γ ð ΩÞ ∩ W 2,2 ðΩÞ-solution corresponding to small data are proved, without further restrictions on the bounds on pðxÞ.
If one assumes the condition f ∈ L p∧′ðxÞ ðΩÞ for the problem (1), then when applying difference quotient, due to the pðxÞ-dependence of leading term, the additional term will appear: 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 < pðxÞ < 6 in [1] (chapter 3) by Ruzicka. In [28], Ettwein and Růžička showed the existence of W 2,ð3pðxÞÞ/ðpðxÞ+1Þ loc -solutions without the artificial upper bound pðxÞ < 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 ∈ L p∧′ðxÞ ðΩÞ. But in that case, we consider the following the boundary condition: for Ω = ð0, 1Þ n , n = 2, 3, and Γ ≔ fx ∈ ∂Ω : |x 1 |,| x n−1 |<1, x n = 0 or x n = 1g, where by x′-periodic, we mean periodic of period 1 both in x 1 and x n−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 of C 1,α ð ΩÞ-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 bound-ary condition (3). For C 1,α ðΩÞ-regularity in 2D, we refer to [18,29,30,[33][34][35][36]. Set For n = 3, we define where μ is arbitrarily close to 0 for p ≠ const, and for n = 2, where q is arbitrary real number such that 1 < q < ∞ and μ 0 > 0 arbitrary close to 0. The main results are as follows.
But due to pðxÞ-dependence of the leading term, we cannot obtain W 2,q -regularity of weak solution to (1) provided f ∈ L p∧′ðxÞ ðΩÞ. 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 2 Advances in Mathematical Physics Dirichlet boundary condition. More precisely, due to them, there appears the term in deriving of W 2,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 ∈ ð2, 4Þ in 2D. Due to pðxÞ-dependence of the extra stress tensor, we consider an approximate problem with ðð1 + jDu λ j 2 Þ/ð1 + λjDu λ j 2 ÞÞ ðpðxÞ−2Þ/2 Du λ for λ ∈ ð0, 1Þ instead of ð1 + jDuj 2 Þ ðpðxÞ−2Þ/2 Du. It is easy to see that the approximate solution belongs to V 2 ðΩÞ ∩ W 2,2 ðΩÞ owing to the trivial inequality 1 ≤ ð1 + a 2 Þ/ð1 + λa 2 Þ ≤ 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 Section 2, we give preliminaries. Section 3 is devoted to prove the main results.
For n × n-matrices F, H denote F : For p ∈ L ∞ ðΩÞ, p ≥ 1, the variable exponent Lebesgue space L pðxÞ ðΩÞ is defined by endowed with the norm kuk pðxÞ,Ω ≔ inf f λ > 0 | ρ pðxÞ ðu/λÞ ≤ 1g. Then, we define the variable exponent Sobolev space by ðΩÞ. We do not distinguish between scalar, vector-valued, and tensor-valued function spaces in the notations. Define Definition 3. We say that function u is a weak solution to the problem (1) if u ∈ V pðxÞ and it satisfies We refer to the term strong solution as a weak solution which additionally satisfies u ∈ W 2,q ðΩÞ for some 1 ≤ q < ∞.

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 ∈ C 2,1 , 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 C 2,1 -function θ P : is the k-dimensional cubic with center 0 and length 2d (which is small enough and will be fixed later), with the following properties: As ∂Ω is a compact, there exist a finite set of points Γ ⊂ ∂Ω and an open set Ω 0 ⊂ ⊂Ω such that Ω ⊂ Ω 0 ∪ S P∈Γ Ω P . We construct a partition of unity fζ 0 , ζ P , P ∈ Γg, corresponding to this covering, such that distðsupp ζ P , ∂Ω P \ ∂ ΩÞ ≥ h 0 for all P ∈ Γ and some suitable small h 0 > 0. Let us fix some P ∈ Γ.
, and a function φ with supp φ ⊂ supp ζ, we define tangential translation through and tangential derivative through Now, we give the two propositions below related to the tangential derivatives.

The Proof of Main Results
We use universal constants c, C > 0, which may vary in different occurrences. In particular, C depends on p − , p + , Ω, n, k∇pðxÞk ∞ , k f kp′ ðxÞ , while c on p − , p + , Ω, n, k∇pðxÞk ∞ .
To begin with, let us define As before, in order to prove the main result, i.e., Theorem 1, we will consider the following approximate problem: where λ ∈ ð0, 1. Let us denote It is known that there exists a weak solution ðu λ , π λ Þ ∈ V 2 ðΩÞ × L 2 0 ðΩÞ to the problem (23) by compactness method if f ∈ L 2 ðΩÞ. Furthermore, the following facts are valid. Proposition 6. Let u λ be a weak solution to the problem (23). Assume that f ∈ L p∧′ðxÞ ðΩÞ and 3/2 ≤ pðxÞ < ∞. Then, the following hold: ð wherepðxÞ is from (8). For the problem (23) without convective term, these are valid for all 1 < pðxÞ < ∞.
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 we can prove by the same line in [20] that if f ∈ L 2 ðΩÞ, pðxÞ ∈ C 0,1 ð ΩÞ, then a weak solution u λ , π λ of the problem (23) satisfies However, the norms of u λ in V 2 ðΩÞ ∩ W 2,2 ðΩÞ and π λ in L 2 0 ðΩÞ ∩ W 1,2 ðΩÞ 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 parameter λ and introduce a shorthand notation S λ ≔ S λ ðx, Du λ Þ if it will be clear from the context.
Let the assumptions of Theorem 1 hold.
The proof is divided into two cases: with and without the convective term.

The Proof of Theorem 1 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 wherepðxÞ is from (8),pðxÞ ≔ max f2, pðxÞg, and Fix P ∈ Γ, and let ζ = ζ P , Ω = Ω P , and θ = θ P be as in Subsection 2.2. For simplicity we will omit the symbol "Ω P " in satisfying where the constant c depends on p − , p + , Ω, n.
Multiplying the first equations in (23) by ϕ = ∂ τ ðζ 2 ∂ τ u λ Þ + ψ, integrating by parts and using Proposition 4, we get We apply Proposition 4 to the left hand side of (34) to get It is clear that From ( [31], (3.6)), we have where cðp − Þ =p − − 1. It is shown in ( [31], (3.7)) that Combining (37) with (38) yields that The terms J 2 , J 3 can be rewritten as Hence these terms can be estimated as follows: by Korn's inequality Combining (39), (41) with (35), we arrive at Since I 1 = −J 3 , the term I 1 can be estimated as J 3 by Note that by Proposition 4 Hence the term I 2 also can be estimated as J 2 + J 3 : We use Hölder's inequality to get
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 Now, we want to derive some estimates on the first derivatives of V λ pðxÞ ðDu λ Þ from (22), independent of parameter λ. This allows us to prove boundedness of T λ and kS λ kp ′ ðxÞ and to improve regularity of solutions to problem (1).
Step 3. The rest is the same as the previous subsection. Thus, Theorem 1 is completely proved.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there is no conflict of interest.