INTERIOR REGULARITY TO THE STEADY INCOMPRESSIBLE SHEAR THINNING FLUIDS WITH NON-STANDARD GROWTH

. We consider weak solutions to the equations of stationary motion of a class of non-Newtonian ﬂuids which includes the power law model. The power depends on the spatial variable, which is motivated by electrorheological ﬂuids. We prove the existence of second order derivatives of weak solutions in the shear thinning cases.


1.
Introduction. Traditionally, the Navier-Stokes equations have received quite a bit of attention. Recently, attention to the behavior of fluids with various viscosities has been increasing dramatically. It is because we can find such fluids everywhere. For example, water, yogurt, lubricants, sand in water, ink, gum solutions, nail polish, ketchup, molasses, ice, paint, custard, paper pulp, even blood in our body. The behavior of many of them can be described in the power law model. In that sense, we are interested in the power law model, which is a generalized Navier-Stokes system.
As mentioned in [16], electrorheological fluids are viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. The motion is governed by a system of partial differential equations consisting of electric field E, polarization, density ρ, velocity u, pressure π, and deviatoric stress tensor S. Refer to [11] for the detail description. In [11] the viscosity is the form of power law model, and the power p depends on the electric field p(|E| 2 ).
In this article we consider the following stationary system related with non-Newtonian fluids: −div {S(x, D(u))} + u · ∇u + ∇π = g, div u = 0 in Ω, In addition, S(x, D(u)) denotes the deviatoric stress. Then T := S − πI is called the full shear stress, such that −div T represents the sum of the internal forces due to friction, which depends mostly on the material of the fluid. The continuous deviatoric stress tensor S : Ω × R n×n → R n×n is assumed to be C 1 -regular in the gradient variable z for every z ∈ R n \ {0}, with S z (·) being Carathéodory regular and satisfying the following nonstandard growth, monotonicity and continuity assumptions: for every z ∈ R n \ {0}, z 1 , z 2 , ξ ∈ R n and x, x 1 , x 2 ∈ Ω, where 0 < ν ≤ L and µ ≥ 0 are fixed numbers. The variable exponent function p(·) : Ω → [0, +∞) is continuous with modulus of continuity ω : where γ 1 > 3/2 if n = 2 and γ 1 > 9/5 if n = 3. We assume that the variable exponent p(·) is Lipschitz continuous, i.e. ω(r) ≤ c p r, for some constant c p > 0. For simplicity, we set p(x) and p 2 := sup Here, we denote the variable exponent Lebesgue space L p(·) (Ω), by the set of all measurable functions f : Ω → R n satisfying For more details, we refer to [6,9]. We say that u ∈ In [13], the existence of weak solutions was provided for constant p > 2n n+2 . In [14,15], the existence of strong solution of (1) was proved when p is constant. In [14], the solution belongs to C 1,α for p > 3 2 when n = 2 (up to boundary), and for p > 6 5 when n = 3 (interior regularity). In [15], for 3n n+2 < p < 2, it is shown that (1 + |D(u)|) p−2 2 ∇D(u) ∈ L 2 loc (Ω) n×n , and u ∈ W 2,t loc (Ω) n for all 1 ≤ t < 2 when n = 2, and u ∈ W 2, 3p 1+p loc (Ω) n when n = 3. In [4], for S(z) = (1 + |z| p−2 )z under slip or no-slip boundary conditions a regularity is provided for shear thickening fluid p > 2.
In case of anisotropic dissipative potential f , where S = ∇f and with exponents 1 < p 1 ≤ p 2 < ∞ and 2 ≤ p 2 < p 1 n+2 n , the existence of weak solutions is given in [3]. For the anisotropic fluid, it is shown in [5] that u ∈ W 1,p1 0 (Ω) n ∩ W 2,s loc (Ω) n for some s > 1 for p 1 > 6/5 when n = 2, and p 1 > 9/5 when n = 3, and p 2 < p 1 n+1 n . Also in that article, there is a good introduction about isotropic and anisotropic cases, and the power depending on x, p(x).
For a variable p(x) depending on E, in [16] it is proved that a weak solution of (1) exists and it has the second weak derivative for γ 1 > 3n n + 2 and µ > 0. Nonetheless, it is not shown there that the second weak derivative of a weak solution belongs to Lebesgue space with variable exponent. It is just shown that a weak solution belongs to W 2,γ1 (Ω) n . The lower bound of p(·), γ 1 , that a weak solution of (1) exists is decreased to 2n n+2 in [10]. But if γ 1 > 2n n+2 , we cannot say that u ⊗ u : D(ϕ) ∈ L 1 for all u, ϕ ∈ W 1,p(·) 0 (Ω) n . In this case, u is not able to be taken as a test function, hence we just consider γ 1 > 3n n+2 . On the other hand, in [7] it was proved that a solution of p(x)-Laplace equation has weak second derivative in L 2 . In that paper, µ = 0 is allowed, but the model is not related with fluid directly. The existence of strong solution of (1) for S(x, D(u)) = (µ + |D(u)|) p(x)−2 D(u) with µ > 0 was proved in [8] for γ 1 > 2. And the same result was proved in [11] for γ 1 > 9 5 . In this paper, we handle non-Newtonian problem (1) where µ is allowed to be 0 and p(·) is Lipschitz continuous. The result of this paper is the following theorem.
2. Proof of main theorem. To simplify the notation, the letter c will always denote any positive constant, which may vary throughout the paper. Moreover, we denote p = p p−1 as the Hölder conjugate exponent of p and p * = np n − p as the Sobolev exponent of p for every p ∈ (1, n).
We recall a useful lemma about higher integrability from [1].
(Ω) n be a weak solution of (1) and assume that S fulfills conditions (2) with Lipschitz continuous variable exponent, p(·). Then there exist constants c, σ > 0, both depending on n, γ 1 , γ 2 , c p such that if B 2R ⊂⊂ Ω, then Here and assume 0 < R ≤ R 0 , throughout this paper. This assumption will be frequently used in the proof, for instance (11) and (12).

Remark 1.
Although the authors only considered the case µ > 0 in [1], the statement is still valid for µ = 0, since the proof is also available for µ = 0. In this paper, we can remove the dependence of c on γ 2 since γ 2 ≤ 2.
Since lim t→0 + t log t = 0 and lim t→+∞ log t t = 0 for any > 0, we have the following useful estimate.
Our proof is mainly based on that of [15]. We divide our proof of main theorem in three steps. At first, we derive estimation related to finite difference of D(u). After then, applying fractional Sobolev embedding theorem, we can show that u is locally bounded. Finally, we prove that u has second derivatives using difference quotient method.
Combining (7)- (9), we obtain For simplicity, we set ν = 1 since it does not make any trouble for the proof.
Estimation of I 1 . To estimate I 1 , we introduce a measurable set O + 2 : We use the continuity assumption in (2) 3 to see Therefore, Lemma 2.2 and (6) reveals Estimation of I 2 . Noting that (6) implies and we discover Note that Applying (14) and Korn's inequality, we have And the first integral term in the last line is estimated as follows: Combining (13), (15) and (16), we obtain Estimation of I 3 will be postponed to Step 2. Estimation of I 4 . Recalling Lemma 2.1 and the De Rham theory in [10], we see π ∈ L q(·) (Ω) for q(x) = (1 + σ)p (x). Observe that . Via a similar estimating procedure of I 2 , we compute

HYEONG-OHK BAE, HYOUNGSUK SO AND YEONGHUN YOUN
In light of Korn's inequality, it holds that Combining (16), (17) and (18), we obtain Estimation of I 5 . Korn's inequality yields Consequently, it follows from combining the estimates of I 1 , I 2 , I 4 and I 5 , and (10) that Step 2. Boundedness of u.
In this step, we shall prove that u is locally bounded. Note that Lemma 2.1 implies that u is locally Hölder continuous by Morrey embedding in case of n = 2 and γ 1 = 2. So it is not necessary to consider this case. We recall I 3 and apply integration by parts formula for finite difference to see We now assume that u ∈ W 1,s loc (Ω) n for some s ∈ [γ 1 , n). Next we estimate I 3,1 Defining θ := s(n+2)−3n 2n , we see 0 < θ < 1 and where s is the Hölder conjugate exponent of s and s * is the Sobolev exponent of s. It then follows from interpolation that On the other hand, integration by parts formula for finite difference reveals Merging up (20) and (21), we obtain Finally, since p(·) is Lipschitz continuous, (19) and (22) give the boundedness of I 0 : where the constant c depends on u W 1,s (B 3R ) n and π L p 1 (B 2R ) . Set s := 2s s − γ 1 + 2 .
Step 3. Completing the proof of Theorem 1.1.
And we estimate Hence we have Since the integral on the right-hand side above inequality converges to 0 as λ → 0, (31) is valid, and (4) is proved.
Note that following inequality holds by Young's inequality: dx.
Then, we immediately deduce (5) 1 from (32) and Korn's inequality. One can prove (5) 2 in a similar way. This completes our proof.