Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 47, pp. 1–35.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.47

GLOBAL GRADIENT ESTIMATES FOR SHEAR

THINNING-TYPE STOKES SYSTEM ON NON-SMOOTH

DOMAINS

NAMKYEONG CHO

Abstract. This article presents global Lq estimates for the weak solution of

the steady p-Stokes equations, which describe the motion of shear-thinning
flow under the nonslip boundary condition. We focus on non-smooth domains

whose boundaries extend beyond the Lipschitz category, with coefficients be-

longing to the BMO (Bounded Mean Oscillation) space having a sufficiently
small BMO semi-norm.

1. Introduction

The Navier-Stokes equations with shear-dependent viscosity are extensively stud-
ied in the literature. The stationary generalized Navier-Stokes equations with shear-
dependent viscosity are expressed as

div T (u, π)− (u · ∇)u = f,

div u = 0,
(1.1)

where

T (u, π) = −πI + νT (u)Du and Du :=
1

2
(∇u+∇uT ). (1.2)

Here, ∇u ∈ Rn×n denotes a matrix-valued function where (∇u)ij = ∂iu
j , and

MT ∈ Rm×n represents the transpose of the matrix M ∈ Rn×m. If νT (u) =

(|Du|+ µ)
p−2

, where µ ≥ 0 and p ̸= 2, then equations (1.1) and (1.2) are referred
to as the p-Navier-Stokes equations. Ladyzhenskaya’s fundamental works initiate
the systematic study of the p-Navier-Stokes equations [25, 26]. For further details on
the p-Navier-Stokes equations, readers are referred to [27, 17, 31] and the references
therein.

To simplify the problem, we shall consider the p-Stokes equations, which are
the equations without the convection term in (1.1). A systematic study of the p-
Stokes equations is meaningful because, in many cases, the regularity of a solution
to the p-Stokes equations is closely related to the regularity of a solution to the
p-Navier-Stokes equations.

2020 Mathematics Subject Classification. 35Q35, 35J25, 35J92.
Key words and phrases. Stokes system; Reifenberg flat; symmetric gradient;

non-Newtonian fluid.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted May 8, 2024. Published August 26, 2024.

1



2 N. CHO EJDE-2024/47

The interior regularity results for p-Stokes equations are well-established in many
cases. However, when we want to extend the regularity results up to the boundary,
the problem becomes more difficult. The difficulty arises from the existence of a
pressure term and the fact that the equation depends only on the symmetric part
of the gradient Du and not on the full gradient, ∇u. This may lead to the loss of
regularity in reconstructing the normal derivatives.

To simplify the boundary value problem, a cubical domain is considered for shear
thickening and shear thinning cases in [2, 3], respectively. These papers study the
reconstruction of the normal direction derivative. These results are extended to
C2,1 domains in [4] for the shear thickening fluid and in [6] for the shear thinning
fluid. In [5, 3], the shear thinning case p < 2 is considered, but the range of
p is restricted to be greater than 3/2 because of the technical issue. Later, this
restriction is removed, and the full range p ∈ (1, 2) is considered in [6]

This article studies gradient Lq estimates of the shear-thinning fluid with the
nonslip boundary condition.

In addition, we assume that the domain Ω belongs to the δ-Reifenberg flat do-
main and that the BMO semi-norm of the coefficients is sufficiently small. The
global gradient Lq estimates on non-smooth domains are initiated in [9] for linear
elliptic equations. Other types of equations on non-smooth domains have been
studied subsequently. For instance, the linear Stokes equations are explored in
[8, 18], p-Laplacian type equations are studied in [7], and shear-thickening p-Stokes
equations are studied in [12].

The problem under consideration is the stationary Stokes equations

divA(x,Du)−∇π = div (φ′′(|F |)F ) i nΩ,

div u = 0 in Ω,

u = 0 on ∂Ω,

(1.3)

where φ ∈ C2(R+) is a function defined as

φ(t) :=

∫ t

0

(µ+ s)p−2s ds+
κ2
2
t2. (1.4)

for some constant κ2 > 0, p ∈ (1, 2) and µ ∈ (0, 1). In particular, if n = 2, we
restrict the range of p to p ∈ (4/3, 2). It is readily checked that the following
properties hold:

p− 1 ≤ tφ′′(t)

φ′(t)
≤ 1,

φ′′(0) = µp−2 + κ2 > 0,

φ′′(t) is a decreasing function.

(1.5)

We shall denote by Rn×n
sym the class of symmetric matrices and 0 as the zero

matrix. For A,B ∈ Rn×n, we denote A : B =
∑n

i,j=1AijBij . The nonlin-

earity A(x, P ) : Ω × Rn×n
sym → Rn×n

sym is a given matrix-valued function such that

A(x, ·) ∈ C0
(
Rn×n

sym

)
∩ C1

(
Rn×n

sym \ {0}
)
for each x ∈ Ω and satisfies the following



EJDE-2024/47 SHEAR THINNING FLUIDS 3

basic structural conditions
n∑

i,j,k,l=1

∂A(x, P )

∂Pkl
QijQkl ≥ νφ′′(|P |)|Q|2,

|∂klAij(x, P )| ≤ Lφ′′(|P |),
A(x,0) = 0,

(1.6)

for all P,Q ∈ Rn×n
sym and for some 0 < ν ≤ L.

Definition 1.1. For F ∈ Lφ(Ω), we say that (u, π) ∈ W 1,φ
0 (Ω) × Lφ∗

(Ω) is the
weak solution pair to (1.3) if∫

Ω

A(x,Du) : Dξ dx−
∫
Ω

π div ξ dx =

∫
Ω

φ′′(|F |)F : Dξ dx,

for all ξ ∈W 1,φ
0 (Ω). Here, φ∗ is the conjugate function of φ, and the details of the

Orlicz functions and related function spaces shall be specified in Section 2.1.

For each open set U ⊂ Ω, let us denote the integral average by

(f)U =

∫
−Uf(x) dx =

1

|U |

∫
U

f(x) dx.

For notational convenience, we write

β(A, Br(y)) := sup
P∈Rn×n

sym \{0}

|A(x, P )−
(
A(·, P )

)
Br(y)

|
φ′(|P |)

,

where Br(y) is the ball of radius r centered at y. By (1.6), we obtain∣∣β(A, Br(y))
∣∣ ≤ c := c(L, p). (1.7)

We assume the domain Ω, is a bounded δ-Reifenberg flat domain for a small δ ∈
(0, 1/16). We further assume that A has a small BMO (Bounded Mean Oscillation)
semi-norm. The precise assumptions are stated below.

Assumption 1.2. (i) There exist R1 > 0 and δ ∈ (0, 1/16) such that for all
x ∈ ∂Ω and for all r ∈ (0, R1], there exists a coordinate system (z1, z2, . . . , zn)
which depends on r and x so that in this coordinate system, x is the origin
and

Br(0) ∩ {zn > 0} ⊂ Br(0) ∩ Ω ⊂ Br(0) ∩ {zn > −δr}.

(ii) There exists R2 ≤ R1 such that for all x ∈ ∂Ω and for all r ∈ (0, R2], there
exists a coordinate system (z1, z2, . . . , zn) which depends on r and x so that
in this coordinate system, x is the origin and

Br(0) ∩ {zn > 0} ⊂ Br(0) ∩ Ω ⊂ Br(0) ∩ {zn > −δr},

and

sup
0<r≤R2

sup
y∈Rn

∫
−Br(y)β

(
A, Br(y)

)
dx ≤ δ, (1.8)

where δ is the same as the one chosen in (i).



4 N. CHO EJDE-2024/47

In this work, δ > 0 only shows the flatness of Ω and the smallness of the BMO
semi-norm of A in this paper. The detailed reason for imposing two separate
assumptions on the domain and coefficients shall be specified later in Remark 4.9.

For simplifying the notation, let us introduce the maximal exponent q̄, depending
on n, κ2 and p, as

q̄ =

{
any number in (1,∞) if n = 2 and 3/4 ≤ p < 2,
n

n−2 if n ≥ 3.
(1.9)

We are ready to state our main results.

Theorem 1.3. We assume that p ∈ (1, 2) if n ≥ 3 and p ∈ [ 43 , 2). Let A satisfy
(1.6), Ω be a bounded open set, κ2 > 0 and q ∈ [1, q̄) with q̄ as in (1.9). Suppose
that φ(|F |) ∈ Lq(Ω). There exists a δ > 0 depending only on |Ω|, R1, q, n, ν, L, p
and κ2 such that if A and Ω satisfy Assumption 1.2, then the weak solution pair to
(1.3), (u, π) ∈W 1,φ

0 (Ω)× Lφ∗
(Ω), satisfy∫

Ω

φ(|Du|)q dx+

∫
Ω

φ∗(|π|)q dx ≤ c

∫
Ω

φ(|F |)q + 1 dx

where the constant c only depends on µ, ν, L, p, κ2, |Ω|, R1 and q.

This article is organized as follows. Section 2 provides the required preliminary
materials. Section 3 presents the regularity results of the p-Stokes equations on the
local flat boundary. Section 4 presents comparison estimates. Finally, Section 5
has the proof of our main theorem.

2. Preliminaries

This section consists of preliminary materials and notation. Throughout this
paper, the universal constant c may vary from line to line, but it only depends on
the data,

data := {n, ν, L, p,R1/ diam(Ω)}.
If there exists a universal constant c := c(data) > 1 such that 1

cf ≤ g ≤ cf , then
we shall denote f ∼ g. When the constant depends on other quantities, we shall
specify them. In this paper, ε > 0 is a small number, and instead of writing cε,
we shall denote it as ε. When it is clear from the context, we shall not distinguish
between scalar, vector-valued, or tensor-valued functions and their function spaces.
If u ∈ Rm and v ∈ Rn, then u⊗ v ∈ Rm×n is defined as (u⊗ v)ij := uivj .

2.1. Function spaces. Let us introduce the definition of the Orlicz spaces and
Sobolev-Orlicz spaces. A convex function G : R+ → R+ is an N-function if G′(t) is
a non-decreasing function, limt→∞G(t)/t = ∞ and limt→0G(t)/t = 0. For a given
N -we function, we define a conjugate of G by

G∗(t) := sup
s>0

{st−G(s)}.

For all t > 0, we uniformly

G∗(G′(t)) ∼ G(t). (2.1)

Let us define ∆2(G) by the smallest constant M > 0 satisfying

G(2t) ≤MG(t) for all t ≥ 0. (2.2)



EJDE-2024/47 SHEAR THINNING FLUIDS 5

If such a M > 0 does not exist, we denote ∆2(G) = ∞. If ∆2(G),∆2(G
∗) <∞, we

define the Orlicz space by

LG(Ω) := {f ∈ L1(Ω) :

∫
Ω

G(|f |) dx <∞},

equipped with the Luxemburg norm

∥f∥LG(Ω) := inf
λ>0

{λ > 0 :

∫
Ω

G
( |f |
λ

)
dx ≤ 1}.

If ∆2(G),∆2(G
∗) <∞, we have

tG′(t) ∼ G(t) and tG′′(t) ∼ G′(t), (2.3)

and for each ε > 0, we have

st ≤ cεG(s) + εG∗(t) and st ≤ cεG
∗(s) + εG(t) (2.4)

for some constant cε > 0 depending only on ∆2(G),∆2(G
∗) and ε > 0. By the

convexity of G and (2.2), there exists a constant c > 0 depending only on ∆2(G)
and ∆2(G

∗) satisfying

G(t1 + t2) ≤ cG(t1) + cG(t2) for all t1, t2 ≥ 0, (2.5)

A shifted N -function is of G is defined by

G′
a(t) := G′(a+ t)

t

a+ t
, for a > 0. (2.6)

For the shifted N function, we have

G|P |(t) ≤ cεG|Q|(t) + εG|P |(|P −Q|), (2.7)

for the details, we refer to [28]. The Sobolev-Orlicz space W 1,G(Ω) is defined as

W 1,G(Ω) := {f ∈W 1,1(Ω) : |f |, |∇f | ∈ LG(Ω)}

equipped with the norm

∥f∥W 1,G(Ω) := ∥f∥LG(Ω) + ∥∇f∥LG(Ω).

Let us present the Poincaré inequality in the setting of the Orlicz space.

Lemma 2.1. Let G be an N-function with ∆2(G),∆2(G
∗) < ∞. Then for all

v ∈ W 1,G(Br), there exists 0 < θ < 1 depending only on ∆2(G) and ∆2(G
∗), such

that ∫
−BrG

( |v − (v)Br
|

r

)
dx ≤ c

(∫
−Br

G(|∇v|)θ dx
)1/θ

. (2.8)

Furthermore, if Z = {x ∈ Br : v(x) = 0} has a positive measure satisfying

|Z| ≥ c0|Br| (2.9)

for some constant α0 ∈ (0, 1), then we have∫
−Br

G
( |v|
r

)
dx ≤ c(c0)

(∫
−Br

G(|∇v|)θ dx
)1/θ

. (2.10)

Proof. The proof of (2.8) can be found in [14, Theorem 7]. For second term, (2.10),
we proceed as in [22, Theorem 2.5]. Since v = 0 on Z, we have

|Z|G
( |(v)Br |

r

)
=

∫
Z

G
( |v − (v)Br |

r

)
dx ≤

∫
Br

G
( |v − (v)Br |

r

)
dx, (2.11)



6 N. CHO EJDE-2024/47

which implies

G
( |(v)Br

|
r

)
≤ 1

|Z|

∫
Br

G
( |v − (v)Br

|
r

)
dx. (2.12)

Using the convexity of G and (2.12), we have∫
Br

G
( |v|
r

)
dx ≤ c

∫
Br

G
( |v − (v)Br

|
r

)
dx+ c

∫
Br

G
( |vBr

|
r

)
dx

≤ c
(
1 +

|Br|
|Z|

)∫
Br

G
( |v − (v)Br |

r

)
dx.

Inequality (2.10) holds by (2.11), (2.9) and (2.8). □

From assumptions (1.4) and (1.5), the following inequalities are well-known:

min(λ2, λp)φ(t) ≤ φ(λt) ≤ max(λ2, λp)φ(t), (2.13)

min(λ2, λ
p

p−1 )φ∗(t) ≤ φ∗(λt) ≤ max(λ2, λ
p

p−1 )φ∗(t). (2.14)

From inequalities (2.13), (2.14) and (2.3), it is straightforward to check that

tp ≤ φ(t)

φ(1)
+ 1 and t2 ≤ φ∗(t)

φ∗(1)
+ 1. (2.15)

From (2.15)1, it follows that

φ−1(t) ≤
( t

φ(1)

)1/p

+ 1. (2.16)

Now, let us denote

V (P ) =

√
φ′(|P |)
|P |

P,

then the following estimates are known:(
A(x, P )−A(x,Q)

)
:
(
P −Q

)
∼ φ′′(|P |+ |P −Q|)|P −Q|2

∼ |V (P )− V (Q)|2,
(2.17)

and

|A(x, P )−A(x,Q)| ≤ φ′
|P |(|P −Q|) ∼ φ′′(|P |+ |P −Q|)|P −Q|. (2.18)

We refer to [23] for a proof of the relations above. Furthermore, we have the
following result, since φ′′(t) is a decreasing function.

Lemma 2.2. For φ defined in (1.4), we have

φ(|P −Q|) ≤ cε|V (P )− V (Q)|2 + εφ(|Q|). (2.19)

Proof. We use (2.3), the increasing property of φ′(t), Young’s inequality, (2.17) and
decreasing property of φ′′(t) to have

φ(|P −Q|) ≤ cφ′(|P −Q|)|P −Q|
≤ cφ′(|P −Q|+ |Q|)|P −Q|
≤ cφ′′(|P −Q|+ |Q|)(|P −Q|2 + |Q||P −Q|)
≤ cφ′′(|P −Q|+ |Q|)(cε|P −Q|2 + ε|Q|2)
≤ cε|V (P )− V (Q)|2 + εφ′′(|Q|)|Q|2

≤ cε|V (P )− V (Q)|2 + εφ(|Q|). □



EJDE-2024/47 SHEAR THINNING FLUIDS 7

The subscript “div” implies an additional divergence-free condition. For exam-
ple, we have:

C∞
0,div(Ω) = {u ∈ C∞

0 (Ω) : div u = 0 in Ω},

W 1,G
0,div(Ω) = {u ∈W 1,G

0 (Ω) : div u = 0 in Ω}.

The subscript “0” with Orlicz spaces and Lebesgue spaces implies that additional
integral zero condition holds, that is,

Lq
0(Ω) := {f ∈ Lq(Ω) :

∫
Ω

f dx = 0},

LG
0 (Ω) := {f ∈ LG(Ω) :

∫
Ω

f dx = 0}.

2.2. Existence and uniqueness of the solution. The existence and uniqueness
of the solution pairs of (1.3) shall be studied in this subsection.

Lemma 2.3 ([29, Theorem 2.2]). Let V be a separable reflexive Banach space
and F ∈ V ∗, the dual space of V . Assume that Γ : V → V ∗ is monotone and
demicontinuous. We further assume that there exists ρ > 0 such that Γ(v)(v) >
F(v) for all v ∈ V with ||v|| > ρ. Then there exists u ∈ V satisfying Γ(u) = F .

Remark 2.4. By setting V = W 1,φ
0,div(Ω) and using Lemma 2.3, it is a standard

process to show the existence and uniqueness of u ∈W 1,φ
0,div(Ω) satisfying

⟨A(x,Du), Dξ⟩ = ⟨φ′′(|F |)F,∇ξ⟩,

for all ξ ∈ W 1,φ
0,div(Ω) if F ∈ Lφ(Ω). Next, by testing (1.3) with the function

u ∈W 1,φ
0,div(Ω), we find

⟨A(x,Du), Du⟩ = ⟨φ′′(|F |)F,∇u⟩.

Using (2.4), (2.17), (2.1) and Korn’s inequality (2.22), we discover that∫
Ω

φ(|Du|) dx ≤ 1

2c

∫
Ω

φ(|∇u|) dx+ c

∫
Ω

φ∗(φ′′(|F |)|F |) dx

≤ 1

2c

∫
Ω

φ(|∇u|) dx+ c

∫
Ω

φ(|F |) dx

≤ 1

2

∫
Ω

φ(|Du|) dx+ c

∫
Ω

φ(|F |) dx,

which implies ∫
Ω

φ(|Du|) dx ≤ c

∫
Ω

φ(|F |) dx. (2.20)

Let us define a linear functional F :W 1,φ
0 (Ω) → R by

F(ξ) :=

∫
Ω

A(x,Du) : Dξ dx−
∫
Ω

φ′′(|F |)F : ∇ξ dx ∀ξ ∈W 1,φ
0 (Ω).

By [20, Theorem III.5.3] and Lemma 2.7, there exists a unique function π ∈ Lφ∗

0 (Ω)
satisfying

F(ξ) =

∫
Ω

π div ξ dx ∀ξ ∈W 1,φ
0 (Ω).



8 N. CHO EJDE-2024/47

2.3. Known results in a John domain. A John domain is a very generalized
domain. In particular, it is well-known that (1/600, R1)-Reifenberg flat domain
is an α John domain for some α = α(R1/ diam(Ω)); see [1, 24] for details. Any
two points in a John domain can be connected by a curve that is not too close
to the boundary, see Definition 2.5. This subsection introduces the definition and
auxiliary results of an α John domain.

Definition 2.5 ([15]). A domain Ω ⊂ Rn is called an α John domain, for some
α > 0, if for all x, y ∈ Ω, there exists a rectifiable path γ⃗ such that

dist(γ⃗(t), ∂Ω) ≥ 1

α
min{t, |γ⃗(t)| − t} ∀t ∈ [0, |γ⃗|].

Here, we assume that γ⃗ is parameterized by its arclength. We also emphasize that
α is a scaling invariant.

For each open set Ω, there exist an universal constants k1 = k1(n) and k2 = k2(n)
satisfying 1 < k1 < k2, a positive number N = N(n) > 0 and a family of cubes
{Qj}j∈N such that

(C1) Ω = ∪j∈Nk1Qj = gcupj∈N2k1Qj ,

(C2)
1

2
k1 diam(Qj) ≤ dist(Qj , ∂Ω) ≤ k2 diam(Qj),

(C3)
∑
j∈N

χ2k1Qj ≤ NχΩ.

(2.21)

We refer to this decomposition as a Whitney covering of Ω. We present a decom-
position theorem in an α John domain.

Lemma 2.6 (Decomposition Theorem). Let Ω ⊂ Rn be an α John domain and
1 < q < ∞. Let us denote W := {Wi :=

3
2k1Qi} where Qi is a Whitney covering

of Ω. Then there exists a constant k = k(α) > 0 and a family of continuous linear
operators Ti : L

q
0(Ω) → Lq

0(Wi) satisfying the following:

(1) For each i ∈ N, we have |Tif | ≤ ck0χWiMf almost everywhere, where Mf
is a maximal function of f .

(2) For all f ∈ Lq
0(Ω), we have

f =
∑
i∈N

Tif in Lq
0(Ω).

This series converges for every permutation of the sequence.
(3) There exists c = c(k0) such that

1

c
∥f∥Lq(Ω) ≤

(∑
i∈N

∥Tif∥qLq(Wi)

)1/q

≤ c∥f∥Lq(Ω).

(4) If f ≡ 0 in Wi, then Tif ≡ 0.

Proof. Statements (1)-(3) follow from [15, Theorem 4.2]. Statement (4) follows
from the construction of Tif in [15, (4.9)-(4.14)], i.e., we have

|Tif | ≤ c
(∫

Wi

|f |dx
)χWi

|Wi|
= 0.

□



EJDE-2024/47 SHEAR THINNING FLUIDS 9

Lemma 2.7 ([16, Theorem 4.2]). Let Ω ⊂ Rn, n ≥ 2, be a bounded α John domain
and G be an N-function with ∆2(G),∆2(G

∗) < ∞. For each f ∈ LG
0 (Ω), there

exists at least one v ∈W 1,G
0 (Ω) satisfying

div v = f and

∫
Ω

G(|∇v|) dx ≤ c

∫
Ω

G(|f |) dx,

for some c = c(α,∆2(G),∆2(G
∗)) > 0.

Next, we present Korn’s inequality in the Orlicz spaces whose proof can be found
in [15, Theorem 6.10, Theorem 6.13].

Lemma 2.8. Let Ω ⊂ Rn be a bounded α John domain and G be an N-function

with ∆2(G),∆2(G
∗) <∞. Then for all u ∈W 1,G

0 (Ω)n, we have∫
Ω

G(|∇u|) dx ≤ c

∫
Ω

G(|Du|) dx, (2.22)∫
Ω

G(|∇u− (∇u)Ω|) dx ≤ c

∫
Ω

G(|Du− (Du)Ω|) dx (2.23)

for some c = c(α,∆2(G),∆2(G
∗)) > 0.

The proof of the next lemma can be found in [16, Lemma 4.3].

Lemma 2.9. Let Ω be an α John domain and let G be an N-function satisfying
∆2(G),∆2(G

∗) < ∞. Then for all π ∈ LG∗

0 (Ω) there exists a constant c∗ :=
c∗(α,∆2(G),∆2(G

∗) > 0 such that

∥π∥LG∗ (Ω) ≤ c∗ sup
∥ξ∥

W
1,G
0 (Ω)

≤1

⟨π,div ξ⟩

and ∫
Ω

G∗(|π|) dx ≤ sup
ξ∈W 1,G

0 (Ω)

(∫
Ω

π div ξ dx− 1

c∗

∫
Ω

G(|∇ξ|) dx
)
. (2.24)

Remark 2.10. For the solution pair (u, π) ∈ W 1,φ(Ω) × Lφ∗

0 (Ω), we use (2.17),
(2.4), (2.5) and (2.1) to have∫

Ω

π div ξ dx =

∫
Ω

A(x,Du) : Dξ dx+

∫
Ω

φ′′(|F |)F : Dξ dx

≤ c

∫
Ω

φ′(|Du|)|Dξ| dx+ c

∫
Ω

φ′(|F |)|Dξ| dx

≤ c(c∗)

∫
Ω

φ∗ (φ′(|Du|) + φ′(|F |)) dx+
1

c∗

∫
Ω

φ(|∇ξ|) dx

≤ c(c∗)

∫
Ω

φ(|Du|) + φ(|F |) dx+
1

c∗

∫
Ω

φ(|∇ξ|) dx

(2.25)

for all ξ ∈W 1φ(Ω). Using (2.25) and (2.20), we have∫
Ω

π div ξ dx− 1

c∗

∫
Ω

φ(|∇ξ|) dx ≤ c

∫
Ω

φ(|Du|) dx+ c

∫
Ω

φ(|F |) dx

≤ c

∫
Ω

φ(|F |) dx, ∀ξ ∈W 1,φ
0 (Ω).

(2.26)



10 N. CHO EJDE-2024/47

Finally, by (2.24) and (2.26), we obtain∫
Ω

φ∗(|π|) dx ≤ sup
ξ∈W 1,φ

0 (Ω)

(∫
Ω

π div ξ dx− 1

c∗

∫
Ω

φ(|∇ξ|) dx
)

≤ c

∫
Ω

φ(|F |) dx.

2.4. Technical lemmas. In this subsection, we present auxiliary lemmas. Let us
begin with the following scaling property in the Reifenberg flat domain.

Lemma 2.11 ([13, Lemma 3.1]). Let (u, π) be the solution pair of (1.3) satisfying
(1.4) and (1.6). Suppose that Ω satisfies Assumption 1.2-(i) for some δ and R1

and A satisfies Assumption 1.2-(ii) for some δ and R2. For λ > 0, M > 0, r > 0,
P ∈ Rn×n and x, y ∈ Ω, denote

ũ(x) =
u(y + rx)

λr
, F̃ (x) =

F (y + rx)

λ
,

π̃(x) =
λπ(y + rx)

Mr
, Ω̃ =

{x− y

r
: y ∈ Ω

}
,

Ã(x, P ) =
λA(y + rx, λP )

M
, φ̃(t) =

φ(λt)

M
.

Then the following statements hold:

(1) We have ∆2(φ) = ∆2(φ̃) and ∆2(φ
∗) = ∆2(φ̃

∗).

(2) Ã satisfies (1.6) with φ replaced by φ̃.

(3) Ω̃ satisfies Assumption 1.2-(i) with δ and R1

r .

(4) Ã satisfies Assumption 1.2-(ii) with δ and R2

r ,

(5)
(
ũ, π̃

)
is the solution of

div Ã(x,Dũ)−∇π̃ = div
(
φ′′(|F̃ |)F̃

)
in Ω̃,

div ũ = 0 in Ω̃,

ũ = 0 on ∂Ω̃.

Lemma 2.12 ([9]). Let C and D be a measurable sets with C ⊂ D ⊂ Ω. Assume
that Ω is (δ, 1)-Reifenberg flat for some small δ > 0. Furthermore, assume that
following two conditions are satisfied.
First, there exists ϵ > 0 such that

|C ∩B1(x)| < ϵ|B1|. (2.27)

Second, for all x ∈ Ω and r ∈ (0, 1),

|C ∩Br(x)| > ϵ|Br(x)| implies Br(x) ∩ Ω ⊂ D. (2.28)

Then, there exists C1 := C(n) satisfying

|C| ≤
( 20

1− δ

)
ϵ|D| := C1ϵ|D|.

We would like to mention that (δ,R)-Reifenberg flat domain has a measure
density condition, see [9]:

sup
0<r≤R

sup
y∈Ω

|Br(y)|
|Ω ∩Br(y)|

≤
( 20

1− δ

)n

. (2.29)



EJDE-2024/47 SHEAR THINNING FLUIDS 11

3. Regularity of the Limiting system up to the boundary

In this section, we focus on the following limiting equations:

div Ā(Dw)−∇πw = 0 in B+
8 ,

divw = 0 in B+
8 ,

w = 0 on {xn = 0} ∩B8,

(3.1)

where

Ā(·) =
∫
−B+

8
A(x, ·) dx.

In this section, we revisit the method outlined in [6], providing additional details
for the sake of completeness. Throughout this section, we shall assume that η ∈
C∞(B+

8 ) is a cut-off function satisfying

χB4(x) ≤ η(x) ≤ χB6(x) and |η|+ |∇η|+ |∇2η| ≤ c, (3.2)

for some c = c(n) > 0. We denote ∂τw for the derivative in the tangential direction.
For x = (x′, xn) ∈ Rn−1 × R and h ∈ Rn−1 with 0 < |h| < 1/16, we denote the
tangential translation by

fτ (x) :=f(x
′ + h, xn)− f(x′, xn) and f−τ (x) := f(x′ − h, xn)− f(x′, xn),

and the difference operator by

∆+f := fτ − f and ∆−f := f−τ − f

and the difference quotient by

d+f :=
∆+f

|h|
and d−f :=

∆−f

|h|
.

For all g ∈ W 1,1(B+
8 ), we have d+g → ∂τg as h → 0 almost everywhere in B+

8 .
Moreover, using the difference quotient method presented in [19, Chapter 5.8.2], we
have

∥d+g∥Lq(B+
6 ) ≤ c∥∇g∥Lq(B+

8 ). (3.3)

Conversely, if supp(g) ⊂ B+
6 and ∥d+g∥Lq(B+

8 ) ≤ c for all h ∈ Rn−1 satisfying

|h| ∈ (0, 1/2), then

∥∂τg∥Lq(B+
8 ) ≤ c.

For any real-valued functions f, g satisfying supp(f)∪supp(g) ⊂ B+
6 with 0 < |h| <

1/2, we have ∫
B+

8

d+fg dx =

∫
B+

8

fd−g dx. (3.4)

During the process of the proof, we occasionally use specific numbers such as
φ(1), φ′′(0) or φ∗(1). By the definition of φ in (1.4) and (1.5), these constants
depend only on κ2, µ, p and n. Now, let us begin with the following lemmas.

Lemma 3.1. Let w ∈W 1,φ(B+
8 ) be a weak solution of (3.1). Then,∫

B+
8

φ(η|d+Dw|) dx ≤ c

∫
B+

8

η2|d+V (Dw)|2 dx+ c

∫
B+

8

φ(|∇w|) dx, (3.5)∫
B+

8

|πw|2 dx ≤ c

∫
B+

8

φ(|∇w|)
φ∗(1)

+ 1 dx. (3.6)



12 N. CHO EJDE-2024/47

Proof. It is directly checked that the following inequalities hold:

φ(η|d+Dw|)
(2.7)

≤ cφ|Dw|+|∆+Dw|(η|d+Dw|) + cφ(|Dw|+ |∆+Dw|)
(2.3)

≤ cφ′′
|Dw|+|∆+Dw|(η|d

+Dw|)η2|d+Dw|2 + cφ(|Dw|+ |∆+Dw|)
(2.6)

≤ cφ′′(|Dw|+ |∆+Dw|+ η|d+Dw|)η2|d+Dw|2

+ cφ(|Dw|+ |∆+Dw|)
(1.5)3
≤ cφ′′(|Dw|+ |∆+Dw|)η2|d+Dw|2 + cφ(|Dw|+ |∆+Dw|)

(2.17)

≤ cη2|d+V (Dw)|2 + cφ(|Dw|+ |∆+Dw|).

Integrating on both sides over B+
8 , we have (3.5).

Next, we proceed as in Remark 2.4 and Remark 2.10 to find∫
B+

8

φ∗(|πw|) dx ≤ c

∫
B+

8

φ(|∇w|) dx. (3.7)

Using (2.15)2 and (3.7), we find∫
B+

8

|πw|2 dx ≤ c

∫
B+

8

φ∗(|πw|)
φ∗(1)

+ 1 dx ≤ c

∫
B+

8

φ(|∇w|)
φ∗(1)

+ 1 dx,

which implies (3.6). □

Next, we present the anisotropic embedding theorem whose proof can be found
in [4, Theorem 2.1] and [30].

Lemma 3.2. Suppose that F ∈ W 1,1(B+
8 ) and supp(F ) ⊂ supp(η) ∩ B+

8 . Let
∂τF ∈ Lq1(B+

8 ) and ∂nF ∈ Lq2(B+
8 ) for some q1, q2 > 1. Then F ∈ Lq3(B+

8 ) with

1 +
n

q3
=
n− 1

q1
+

1

q2

and the inequality

∥F∥Lq3 ≤ c (∥∂τF∥Lq1 + ∥F∥Lq1 )
n−1
n (∥∂nF∥Lq2 + ∥F∥Lq2 )

1/n
,

holds.

3.1. Regularity in tangential direction. This subsection presents the regularity
results in the tangential direction near the boundary.

Proposition 3.3. Suppose that w ∈ W 1,φ(B+
8 ) is a solution of (3.1) and the

inequality (3.29) holds. Then∫
B+

8

φ(η|∂τDw|) dx+

∫
B+

8

η2|∂τV (Dw)|2 dx ≤ c

∫
B+

8

φ(|∇w|) dx (3.8)∫
B+

8

η2|∂τπw|2 dx ≤ c

(
φ′′(0) +

1

φ∗(1)
+ 1

)∫
B+

8

φ(|∇w|) + 1 dx. (3.9)

Proof. By testing d−(ηψ) to the (3.1) and using (3.4), we find∫
B+

8

d+Ā(Dw) : D(ηψ) dx =

∫
B+

8

πw div
(
d−(ηψ)

)
dx. (3.10)



EJDE-2024/47 SHEAR THINNING FLUIDS 13

Now, taking ψ = ηd+w and using (3.4), we have∫
B+

8

η2|d+V (Dw)|2 dx

≤
∫
B+

8

η2d+Ā(Dw) : d+Dwdx

=

∫
B+

8

Ā(Dw) : d−(2η∇η
sym
⊗ d+w) dx+

∫
B+

8

πwd
−(2η∇η · d+w) dx

=: I1 + I2.

(3.11)

We estimate the first term as

I1 ≤ c

∫
B+

8

|Ā(Dw)||d−(η∇η
sym
⊗ d+w)| dx

(1.6)

≤ c

∫
B+

8

φ′(|Dw|)|d−(η∇η
sym
⊗ d+w)| dx

(2.4)

≤ cε

∫
B+

8

φ∗(φ′(|∇w|)) + ε

∫
B+

8

φ(|d−(η∇η
sym
⊗ d+w)|) dx

(3.3)

≤ cε

∫
B+

8

φ∗(φ′(|∇w|)) + ε

∫
B+

8

φ(|∇(η∇η
sym
⊗ d+w)|) dx

(2.22)

≤ cε

∫
B+

8

φ∗(φ′(|∇w|)) + ε

∫
B+

8

φ(|D(η∇η
sym
⊗ d+w)|) dx

(2.1)
(3.3)

≤ cε

∫
B+

8

φ(|∇w|) + ε

∫
B+

8

φ(η|Dd+w|) dx.

(3.12)

For the second term I2, we use (2.4), (2.22) and (3.3) to find that for any ε > 0,

I2 ≤
∫
B+

8

|πw||d−(2η∇η · d+w)| dx

≤ cε

∫
B+

8

φ∗(|πw|) dx+ ε

∫
B+

8

φ(|∇(2η∇η · d+w)|) dx

≤ cε

∫
B+

8

φ∗(|πw|) dx+ ε

∫
B+

8

φ(|D(2η∇η · d+w)|) dx

≤ cε

∫
B+

8

φ∗(|πw|) dx+ ε

∫
B+

8

φ(|d+w|) dx+ ε

∫
B+

8

φ(η|d+Dw|) dx

≤ cε

∫
B+

8

φ∗(|πw|) dx+ ε

∫
B+

8

φ(|∇w|) dx+ ε

∫
B+

8

φ(η|d+Dw|) dx.

(3.13)

Using (3.5) (3.11), (3.12) and (3.13), we have∫
B+

8

φ(η|d+∇w|) + η2|d+V (Dw)|2 dx

≤ c

∫
B+

8

η2|d+V (Dw)|2 dx+ c

∫
B+

8

φ(|∇w|) dx

≤ c

∫
B+

8

φ(|∇w|) dx+ ε

∫
B+

8

φ(η|d+∇w|) dx+ cε

∫
B+

8

φ∗(|πw|) dx.



14 N. CHO EJDE-2024/47

Now, we take ε = 1/2 and then use (3.7) to find∫
B+

8

φ(η|d+Dw|) + η2|d+V (Dw)|2 dx

≤ c

∫
B+

8

φ(|∇w|) dx+ c

∫
B+

8

φ∗(|πw|) dx

≤ c

∫
B+

8

φ(|∇w|) dx.

(3.14)

Finally, (3.8) holds by (3.14) and different quotient argument.
Let us present the estimate of ∂τπw. Rewriting (3.10), we have∫

B+
8

ηd+πw divψ dx

=

∫
B+

8

ηd+Ā(Dw) : Dψ dx+

∫
B+

8

d+Ā(Dw) : ∇η
sym
⊗ ψ dx

−
∫
B+

8

πwd
−(∇η · ψ) dx,

(3.15)

for all ψ ∈ W 1,φ
0 (B+

8 ). By Lemma 2.7, there exists ψ ∈ W 1,φ
0 (B+

8 ) satisfying
divψ = ηd+πw − (ηd+πw)B+

8
and

∥∇ψ∥L2(B+
8 ) ≤ c∥ηd+πw −

(
ηd+πw

)
B+

8
∥L2(B+

8 ). (3.16)

Testing ψ to (3.15) yields∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx

=

∫
B+

8

ηd+Ā(Dw) : Dψ dx+

∫
B+

8

d+Ā(Dw) : ∇η
sym
⊗ ψ dx

−
∫
B+

8

πwd
−(∇η · ψ) dx

:= J1 + J2 + J3.

(3.17)

By (2.18), Young’s inequality, (1.5) and (3.16), we have

|J1| ≤ c

∫
B+

8

ηφ′′(|Dw|+ |∆+Dw|)|d+Dw||Dψ| dx

≤ c

ε

∫
B+

8

η2φ′′(|Dw|+ |∆+Dw|)2|d+Dw|2 dx+ ε

∫
B+

8

|∇ψ|2 dx

≤ cφ′′(0)

ε

∫
B+

8

η2|d+V (Dw)|2 dx+ ε

∫
B+

8

|∇ψ|2 dx

≤ cφ′′(0)

ε

∫
B+

8

η2|d+V (Dw)|2 dx

+ ε

∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx.

(3.18)



EJDE-2024/47 SHEAR THINNING FLUIDS 15

For the second term, J2, we use (3.4), (1.6)2, Young’s inequality, Poincaré inequal-
ity, (3.3), (3.2), (2.3), (1.5) and (3.16) to find

|J2| ≤ c

∫
B+

8

|Ā(Dw)||d−(∇η ⊗ ψ)| dx

≤ c

∫
B+

8

φ′(|Dw|)|d−(∇η ⊗ ψ)| dx

≤ c

ε

∫
B+

8

φ′(|Dw|)2 dx+ ε

∫
B+

8

|d−(∇η ⊗ ψ)|2 dx

≤ c

ε

∫
B+

8

φ′(|Dw|)2 dx+ ε

∫
B+

8

|ψ|2 + |∇ψ|2 dx

=
c

ε

∫
B+

8

φ′(|Dw|)|Dw|φ
′(|Dw|)
|Dw|

dx+ ε

∫
B+

8

|∇ψ|2 dx

≤ c

ε

∫
B+

8

φ(|Dw|)φ′′(|Dw|) dx+ ε

∫
B+

8

|∇ψ|2 dx

≤ cφ′′(0)

ε

∫
B+

8

φ(|∇w|) dx+ ε

∫
B+

8

|∇ψ|2 dx

≤ cφ′′(0)

ε

∫
B+

8

φ(|∇w|) dx+ ε

∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx.

(3.19)

Similarly, we use Young’s inequality, (3.3) and Poincaré inequality to find

|J3| ≤ ε

∫
B+

8

|d−(∇η · ψ)|2 dx+
c

ε

∫
B+

8

|πw|2 dx

≤ ε

∫
B+

8

|∇ψ|2 dx+
c

ε

∫
B+

8

|πw|2 dx

≤ ε

∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx+

c

ε

∫
B+

8

|πw|2 dx.

(3.20)

Combining (3.17) ∼ (3.20), we have∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx

≤ cφ′′(0)

ε

∫
B+

8

η2|d+V (Dw)|2 dx+
cφ′′(0)

ε

∫
B+

8

φ(|∇w|) dx

+
c

ε

∫
B+

8

|πw|2 dx+ ε

∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx.

(3.21)

Taking ε = 1/2, the last term in the right hand side of (3.21) can be absorbed to
the left hand side. Consequently,∫

B+
8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx

≤ cφ′′(0)

∫
B+

8

η2|d+V (Dw)|2 dx

+ cφ′′(0)

∫
B+

8

φ(|∇w|) dx+ c

∫
B+

8

|πw|2 dx.



16 N. CHO EJDE-2024/47

By (3.8), we have ∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx

≤ cφ′′(0)

∫
B+

8

φ(|∇w|) dx+ c

∫
B+

8

|πw|2 dx.
(3.22)

By a direct calculation, it is clear that∫
B+

8

η2|d+πw|2 dx ≤ 2

∫
B+

8

|ηd+πw −
(
ηd+πw

)
B+

8
|2 dx

+ c
∣∣ ∫

B+
8

ηd+πw dx
∣∣2. (3.23)

Using (3.4), Hölder’s inequality (3.3) and (3.2), we have∣∣ ∫
B+

8

ηd+πw dx
∣∣2 =

∣∣ ∫
B+

8

d−ηπw dx
∣∣2 ≤ c

∫
B+

8

|πw|2 dx. (3.24)

Combining (3.22), (3.23), (3.24) and (3.6), one finds that∫
B+

8

η2|d+πw|2 dx ≤ c
(
φ′′(0) +

1

φ∗(1)
+ 1

)(∫
B+

8

φ(|∇w|) dx+ 1
)

which implies (3.9). □

Remark 3.4. Proceeding as in the Proposition 3.3, the interior regularity holds∫
B+

8

φ(η|∇Dw|) dx+

∫
B+

8

η2|∇V (Dw)|2 dx ≤ c(η)

∫
B+

8

φ(|∇w|) dx,

where η ∈ C∞
0 (B+

8 ) with supp(η) ⊂⊂ B+
8 . Therefore,∇V (Dw) belongs to L2

loc(B
+
8 ).

Remark 3.5. As a result of Proposition 3.3, we have the following inequalities:∫
B+

8

φ(η|∂τ∇w|) dx ≤
∫
B+

8

φ(|∇(η∂τw)|+ |∇η||∂τw|) dx

(3.2)
(2.5)

≤ c

∫
B+

8

φ(|∇(η∂τw)|) dx+ c

∫
B+

8

φ(|∇w|) dx

(3.2)

≤ c

∫
B+

8

φ(|∇(η∂τw)|) dx+ c

∫
B+

8

φ(|∇w|) dx

(2.22)

≤ c

∫
B+

8

φ(|D(η∂τw)|) dx+ c

∫
B+

8

φ(|∇w|) dx

(3.2)
(2.5)

≤ c

∫
B+

8

φ(η|D∂τw|) dx+ c

∫
B+

8

φ(|∇w|) dx

(3.8)

≤ c

∫
B+

8

φ(|∇w|) dx.

(3.25)



EJDE-2024/47 SHEAR THINNING FLUIDS 17

3.2. Regularity in the normal direction. For simplicity of notation, we use D
instead of Dw. Rewriting (3.1) for each α = 1, . . . , n− 1, we have

−
n∑

β=1

∂βĀαβ(D) = ∂απw a.e. in B+
8 .

From a direct calculation, we have

−
n∑

i,j,β=1

∂ijĀαβ(D)∂βDij = ∂απw

For brevity of notation, let us use

Ajα := ∂njĀαn, bj := ∂nDnj

fα := ∂απw + ∂nnAαn(D)∂nDnn +
n−1∑
i,j=1

∂ijĀαn(D)∂nDij

+ 2

n−1∑
i,β=1

∂inĀαβ(D)∂βDin +

n−1∑
i,j,β=1

∂ijĀαβ(D)∂βDij .

Since both Ā and D are symmetric, we find that

2

n−1∑
j=1

Ajαbj = fα a.e. in B+
8

for α = 1, . . . , n− 1. Multiplying by bα on both side and then taking a summation
from α = 1 to n− 1, we obtain

λφ′′(|D|)|b|2 ≤ 2

n−1∑
α,j=1

Ajαbjbα ≤ |f ||b| a.e. in B+
8 . (3.26)

Using the divergence-free condition on w, we have

∂nDnn = ∂n∂nwn =

n−1∑
i=1

∂n∂iwi.

From the structural condition (1.6), we find that

|f | ≤ c|∂τπw|+ cφ′′(|D|)|∂τ∇w|. (3.27)

Using the divergence-free condition on w again, we have

bα =
1

2
∂n∂nwα − 1

2

n−1∑
β=1

∂α∂βwβ .

Denoting b̃α = ∂n∂nwα, it is readily checked that

2|b| ≥ |b̃| − |∇∂τw| and |∇2w| ≤ c(|b̃|+ |∇∂τw|). (3.28)

Let us now assume that ∫
B+

8

φ(|∇w|) dx ≤ K (3.29)

for some constant K > 1.



18 N. CHO EJDE-2024/47

From a direct calculation, we have

νφ′′(|D|)|∇2w|2
(3.28)2
≤ νφ′′(|D|)(|b|2 + |∇∂τw|2)

(3.26)

≤ |f ||b|+ cφ′′(|D|)|∇∂τw|2

(3.27)

≤ c|∂τπw||b|+ cφ′′(|D|)|∂τ∇w||b|+ cφ′′(|D|)|∇∂τw|2

Y oung′s

≤ c

ε1
|∂τπw|2 + ε1|b|2

+
c

ε2
φ′′(|D|)|∂τ∇w|2 + ε2φ

′′(|D|)|b|2

+ cφ′′(|D|)|∇∂τw|2

≤ c

ε1
|∂τπw|2 + ε1|∇2w|2

+
c

ε2
φ′′(|D|)|∂τ∇w|2 + ε2φ

′′(|D|)|∇2w|2

+ cφ′′(|D|)|∇∂τw|2.

Now, we take ε2 = ν
2 to find that

ν

2
φ′′(|D|)|∇2w|2 ≤ c

ε1
|∂τπw|2 + ε1|∇2w|2 + cφ′′(|D|)|∂τ∇w|2.

Then we use (1.4) and take ε1 = νκ2

4 to obtain

νκ2
4

|∇2w|2 ≤ c

κ2
|∂τπw|2 + cφ′′(|D|)|∂τ∇w|2 a.e. in B+

8 . (3.30)

We multiply both sides by η2, defined as in (3.2) and then take an integral on the
both side of (3.30) to find

∫
B+

8

η2|∇2w|2 dx ≤ c

κ22

∫
B+

8

η2|∂τπ|2 dx

+
c

κ2

∫
B+

8

η2φ′′(|Dw|)|∂τ∇w|2 dx.
(3.31)

By (3.9) and (3.29), we have

c

κ22

∫
B+

8

η2|∂τπ|2 dx ≤ c(µ, κ2)K. (3.32)



EJDE-2024/47 SHEAR THINNING FLUIDS 19

For the second term, we have

c

κ2

∫
B+

8

η2φ′′(|Dw|)|∂τ∇w|2 dx

(1.5)3
≤ cφ′′(0)

κ2

∫
B+

8

η2|∂τ∇w|2 dx

(1.5)2
≤ cφ′′(0)

κ22

∫
B+

8

η2φ′′(η|∂τ∇w|)|∂τ∇w|2 dx

(2.3)

≤ cφ′′(0)

κ22

∫
B+

8

φ(η|∂τ∇w|) dx

(3.25)

≤ cφ′′(0)

κ22

∫
B+

8

φ(|∇w|) dx

(3.29)

≤ c(µ, κ2)K.

(3.33)

From the estimates (3.31), (3.32) and (3.33), one can see to it that∫
B+

8

η2|∇2w|2 dx ≤ c(µ, κ2)K. (3.34)

When n ≥ 3, we use the Poincaré-Sobolev inequality, (3.34) and (3.29), to find(∫
B+

8

(η|∇w|)
2n

n−2 dx
)n−2

n ≤ c

∫
B+

8

η2|∇w|2 dx+ c

∫
B+

8

η2|∇2w|2 dx

≤ c(µ, κ2)K.

(3.35)

Using Hölder’s inequality and the fact that K > 1, one has(∫
B+

8

(η|∇w|)
pn

n−2 dx
)n−2

n ≤
(∫

B+
8

(η|∇w|)
2n

n−2 dx
) p(n−2)

2n

≤ c(µ, κ2)K
p
2

≤ c(µ, κ2)K.

(3.36)

Using (2.13), (3.35) and (3.36), we have(∫
B+

4

φ(|∇w|)
n

n−2 dx
)n−2

n

≤ cφ(1)
(∫

B+
8

(η|∇w|)
2n

n−2 dx+

∫
B+

8

(η|∇w|)
pn

n−2 dx
)n−2

n

≤ c(µ, κ2)K.

(3.37)

If n = 2, the above inequality holds with q for all 1 < q <∞ instead of n/(n− 2).

3.3. Conclusion. In this subsection, we conclude the results in the previous sub-
sections.

Theorem 3.6. Suppose that (w, πw) ∈ is a solution pair of (3.1) satisfying (3.29).
Let q̄ > 1 be a constant as in (1.9). Then there exists a constant depending only on
µ > 0, q̄ and data satisfying∫

B+
4

φ(|∇w|)q̄ dx ≤ c(µ, κ2, q̄)K
q̄. (3.38)



20 N. CHO EJDE-2024/47

4. Comparison estimates near the boundary

To prove the main theorem, we need comparison estimates between the solution
of the localized original problem and the solutions of the associated homogeneous
problems. The arguments in this section are similar to those in [12]. However, we
have provided detailed explanations to make this document self-contained. Let us
introduce some notation:

Ωρ := Bρ(x0) ∩ Ω, B+
ρ (x0) := {x ∈ Bρ(x0) : xn > 0},

∂wΩρ(x0) := Bρ(x0) ∩ ∂Ω, Tρ := {x ∈ Bρ(x0) : xn = 0},

where, as usual, Bρ(x0) is the ball with center x0 ∈ Rn and radius ρ > 0. We shall
omit a point x0 when it is clear from the context. For simplicity, assume that we
are under the following geometric setting:

B+
16 ⊂ Ω16 ⊂ B16 ∩ {xn > −32δ}.

Without further clarification, we assume that the constants in the rest of this article
also depend on R1. We specify issues regarding dependency later in Remark 4.9.

Let us introduce the localized original equations and the corresponding homoge-
neous equations as follows:

divA(x,Du)−∇π = div (φ′′(|F |)F ) i nΩ16,

div = 0 in Ω16,

u = 0 on ∂wΩ16,

(4.1)

and

divA(x,Dv)−∇πv = 0 in Ω16,

div v = 0 in Ω16,

v = u on ∂Ω16.

(4.2)

To introduce the limiting equations, we introduce auxiliary functions. Let ψδ =
ψδ(xn) ∈ C∞(R+) be a smooth cut-off function satisfying

(i) 0 ≤ ψδ ≤ 1,
(ii) ψδ(xn) = 1 on [2δ, 8],
(iii) |∇ψδ| ≤ 2

δ ,
(iv) ψδ(xn) = 0 on [0, δ].

Next, we choose ξ0 ∈W 1,φ
0 (B+

8 ) that satisfies

div ξ0 = −∇ψδ · v +
∫
−B+

8
∇ψδ · v dx.

The existence of ξ0 follows from Lemma 2.7 with the estimate∫
B+

8

φ(|∇ξ0|) dx ≤ c

∫
B+

8

φ (|∇ψδ · v|) dx ≤ c

∫
B+

8 ∩{xn≤2δ}
φ
(∣∣∣v
δ

∣∣∣) dx. (4.3)

We next define h(x) as h(x) = (0, 0, . . . ,−
(
∇ψδ · v

)
B+

8
xnχ{xn≥0}). From a direct

calculation, we have

|∇h(x)| ≤ c|∇ψδ · v| ≤ c
∣∣∣v
δ

∣∣∣. (4.4)



EJDE-2024/47 SHEAR THINNING FLUIDS 21

Now, let us introduce the limiting system by

div Ā(Dw)−∇πw = 0 in B+
8 ,

divw = 0 in B+
8 ,

w = ψδv + ξ0 + h on ∂B+
8 .

(4.5)

We shall start with the standard estimates for (4.1) and (4.2).

Lemma 4.1. Let (u, π) be a solution of (4.1) and (v, πv) be the solution of (4.2).
Then we have the estimate∫

−Ω16
φ(|Dv|) dx ≤ c

∫
−Ω16

φ(|Du|) dx.

Proof. We test (4.2) with u− v to have∫
Ω16

A(x,Dv) : Dv dx =

∫
Ω16

A(x,Dv) : Dudx.

By (2.17) and (2.4), we find that∫
Ω16

φ(|Dv|) dx ≤ cε

∫
Ω16

φ(|Du|) dx+ ε

∫
Ω16

φ(|Dv|) dx.

Taking ε = 1
2 , we complete our proof. □

Next we present a higher integrability result of φ(|∇v|) near the boundary. We
are motivated by the arguments presented in [10, Theorem 5.5].

Lemma 4.2. Let (v, πv) be the solution of (4.2). Then there exists θ > 1 such that(∫
−Ω8φ(|∇v|)θdx

)1/θ

≤ c

∫
−Ω16φ(|∇v|)dx.

Proof. Let us extend v to be zero on B16 \ Ω16 and still denote it as v. We intend
to show ∫

−Bρ(x0)φ(|∇v|) dx ≤ c
(∫
−B2ρ(x0)φ(|∇v|)

σ dx
)1/σ

dx (4.6)

holds for all B2ρ(x0) ⊂ B8 and for some σ ∈ (0, 1). Once inequality (4.6) holds, we
use Gerhings Lemma, [21, Proposition V.1.1] to complete our proof.

First, we consider interior case B2ρ(x0) ⊂⊂ Ω8. Now choose a cut-off function
η ∈ C∞

0 (B2ρ) where η ≡ 1 on Bρ and |∇η| ≤ c/ρ. We define

f(x) := pηp−1∇η · (v − (v)B2ρ(x0))−
∫
−B2ρ(x0)pη

p−1∇η · (v − (v)B2ρ(x0)) dx

= pηp−1∇η · (v − (v)B2ρ(x0)).

Using that η = 0 on ∂B2ρ(x0), div v = 0 and Gauss-Green theorem, we have∫
B2ρ(x0)

pηp−1∇η · (v − (v)B2ρ(x0)) dx =

∫
B2ρ(x0)

div(ηp(v − (v)B2ρ(x0))) dx = 0.

(4.7)

By Lemma 2.7, there exists ψ ∈W 1,φ
0 (B2ρ(x0)) satisfying divψ = f in B2ρ(x0) and∫

B2ρ(x0)

φ(|∇ψ|) dx ≤ c

∫
B2ρ(x0)

φ(|f |) dx

≤ c

∫
B2ρ(x0)

φ
( |v − (v)B2ρ(x0)|

ρ

)
dx.

(4.8)



22 N. CHO EJDE-2024/47

Testing (4.2) with ξ = ηp(v − (v)B2ρ(x0))− ψ gives∫
Ω16

ηpA(x,Dv) : Dv dx

= −
∫
Ω16

A(x,Dv) : (pηp−1∇η
sys
⊗ (v − (v)B2ρ(x0)))−Dψ) dx.

Now, we use (2.17), (2.18), (2.4) and (4.8) to find∫
Bρ(x0)

φ(|Dv|) dx

≤ c

∫
B2ρ(x0)

ηpA(x,Dv) : Dv dx

= −c
∫
B2ρ(x0)

A(x,Dv) : (pηp−1∇η
sys
⊗ (v − (v)B2ρ(x0))−Dψ) dx

≤ c

∫
B2ρ(x0)

φ′(|Dv|)
( |v − (v)B2ρ(x0)|

ρ
+ |∇ψ|

)
dx

≤ ε

∫
B2ρ(x0)

φ(|Dv|) dx+ cε

∫
B2ρ(x0)

φ
( |v − (v)B2ρ(x0)|

ρ

)
+ φ(|∇ψ|) dx

≤ ε

∫
B2ρ(x0)

φ(|Dv|) dx+ cε

∫
B2ρ(x0)

φ
( |v − (v)B2ρ(x0)|

ρ

)
dx

Taking ε = 1
2 and using (2.8), we have∫

−Bρ(x0)φ(|Dv|) dx ≤ c

∫
−B2ρ(x0)φ

( |v − (v)B2ρ(x0)|
ρ

)
dx

≤ c
(∫
−B2ρ(x0)φ(|∇v|)

σ dx
)1/σ

.

(4.9)

Next consider the case when B2ρ(x0) ̸⊂ Ω8. If B 3
2ρ
(x0) ⊂⊂ Ω8, then we proceed

as in the interior case. If B 3
2ρ
(x0) ̸⊂ Ω8, we may further assume thatBρ(x0) ̸⊂

B8 \ Ω8. Otherwise, (4.6) holds, and there is nothing to show. Because of (2.29),
we can guarantee that

|B2ρ((x0)) \ Ω8|
|B2ρ(x0)|

≥ c(n).

Now, we take ρ0 ≤ 1
1600(k2+1) , where k2 = k2(n) is a universal constant in

(2.21). We assume that 0 < ρ ≤ ρ0. If B2ρ(x0) ∩ Wi ̸= ∅ for some Wi ∈ W,
then dist(x,Wi) ≤ 4ρ0. By (C2)-(2.21), diam(Wi) ≤ 4ρ0k2. Therefore for all
x1 ∈ B2ρ(x0) and x2 ∈ Wi, we have dist(x1, x2) ≤ 16(1 + k2)ρ0 ≤ 1

100 . Thus,
Wi ⊂ B1(x0) and Wi ⊂ B12. Since Wi ⊂ Ω, it is clear that Wi ⊂ Ω12. Note that
the choice of ρ0 only depends on n. Now, we denote

f(x) := pηp−1∇η · v −
∫
−B2ρ(x0)pη

p−1∇η · v dx = pηp−1∇η · v in B2ρ(x0) ∩ Ω8.

We extend f(x) = 0 in B2ρ(x0) \ Ω8. Proceeding as in (4.7), we have f ∈ Lφ
0 (Ω).



EJDE-2024/47 SHEAR THINNING FLUIDS 23

Using Lemma 2.6-(2) and Lemma 2.6-(3), we can decompose f =
∑
Tif , where

Tif ∈ L2
0(Wi) and

1

c
∥f∥2L2(Ω) ≤

∑
i∈N

∥Tif∥2L2(Wi)
≤ c∥f∥2L2(Ω). (4.10)

Now, for each Tif , we choose ψi ∈W 1,p̄
0 (Wi) satisfying

divψi = Tif in Wi and ∥ψi∥W 1,p̄(Wi) ≤ c∥Tif∥Lp̄(Wi). (4.11)

By Lemma 2.6-(4), Wi ∩ B2ρ = ∅ implies Tif ≡ 0 and again by (4.11), we have

ψi ≡ 0. After prolongation by zero outsideWi, we may assume that ψi ∈W 1,p̄
0 (Ω12)

from the choice of ρ0, Wi ∩ B2ρ ̸= ∅ implies Wi ⊂ Ω12, if not ψi ≡ 0. Let
ψ =

∑
i∈N ψi. By (4.11) and (4.10), the summation converges and therefore

ψ ∈W 1,p̄
0 (Ω12). Moreover, we have

divψ = div
∑
i∈N

ψi =
∑
i∈N

divψi =
∑
i∈N

Tif = f,

∥∇ψ∥p̄Lp̄(Ω) ≤
∑
i∈N

∥∇ψi∥p̄Lp̄(Wi)
≤ c

∑
i∈N

∥Tif∥p̄Lp̄(Wi)
≤ c∥f∥p̄Lp̄(Ω12)

.

The rigorous verification of these inequalities can be found in [15, Theorem 5.2].
So we can choose ξ = ηpv − ψ as a test function, to find∫

Bρ(x0)

φ(|Dv|) dx ≤ c

∫
B2ρ(x0)

ηpA(x,Dv) : Dv dx

= −c
∫
B2ρ(x0)

A(x,Dv) : (pηp−1∇η
sys
⊗ v −Dψ) dx

≤ c

∫
B2ρ(x0)

φ′(|Dv|)
(
|v|
ρ

+ |∇ψ|
)
dx

≤ ε

∫
B2ρ(x0)

φ(|Dv|) dx+ cε

∫
B2ρ(x0)

φ
( |v|
ρ

)
+ φ(|∇ψ|) dx

Taking ε = 1/2 and using (4.8) and (2.10), we find that∫
−Bρ(x0)φ(|Dv|) dx ≤ c

∫
−B2ρ(x0)φ

( |v|
ρ

)
dx ≤ c

(∫
−Br

φ(|∇v|)σ dx
)1/σ

. (4.12)

To conclude, we use (4.9) for interior case and use (4.12), to find∫
−Bρ(x0)φ(|∇v|) dx ≤ c

∫
−B2ρ(x0)φ(|∇v − (∇v)Bρ(x0)|) dx+ cφ

(
(∇v)B2ρ(x0)

)
≤ c

∫
−B2ρ(x0)φ(|Dv − (Dv)Bρ(x0)|) dx+ cφ

(
(∇v)B2ρ(x0)

)
≤ c

∫
−B2ρ(x0)φ(|Dv|) dx+ cφ

(
(∇v)B2ρ(x0)

)
≤ c

(∫
−B2ρ(x0)φ(|∇v|)

σ dx
)1/σ

+ cφ
(
(∇v)B2ρ(x0)

)
.

Let Φ(t) := φ(t)σ where σ ∈ (0, 1) is a small number defined in Lemma 2.1 which
is chosen so that that Φ(t) is also an N-function. Then by the Jensen’s inequality,

φ
(∫
−B2ρ(x0)|∇v| dx

)
≤ φ ◦ Φ−1

(∫
−B2ρ(x0)Φ(|∇v|) dx

)



24 N. CHO EJDE-2024/47

=
(∫
−B2ρ(x0)φ(|∇v|)

σ dx
)1/σ

.

Combining the last two inequalities, we have (4.6) and complete our proof. □

Lemma 4.3. Let (v, πv) be the solution of (4.2) and (w, πw) be the solution of
(4.5). We shall extend w by zero on Ω8 \ B+

8 and still denote it as w. Then, we
have ∫

Ω8

φ(|Dw|) dx ≤ c

∫
Ω16

φ(|Dv|) dx+ c

∫
Ω16∩{xn≤2δ}

φ(|∇v|) dx. (4.13)

Proof. We may assume that w is also defined on Ω8 by the zero extension. Test
(4.5) with w − ψδv − ξ0 − h to have the equality∫

B+
8

Ā(Dw) : Dwdx =

∫
B+

8

Ā(Dw) : D(ψδv +Dξ0 +Dh) dx.

We use (2.17), properties of ψ, Young’s inequality, (1.6), (4.3) and (4.4) to discover
that ∫

B+
8

φ(|Dw|) dx

≤
∫
B+

8

Ā(Dw) : (v ⊗∇ψδ) dx,

+

∫
B+

8

Ā(Dw) : Dξ0 dx+

∫
B+

8

Ā(Dw) : Dhdx,

≤ ϵ

∫
Ω8

φ(|Dw|) dx+ cϵ

∫
Ω8

φ(|Dv|) dx+ cϵ

∫
Ω8

φ(|v ⊗∇ψδ|) dx

+ cϵ

∫
Ω8

φ(|∇ξ0
∣∣) dx+ cϵ

∫
Ω8

φ(|Dh|) dx,

≤ ϵ

∫
Ω8

φ(|Dw|) dx+ cϵ

∫
Ω8

φ(|Dv|) dx+ cϵ

∫
Ω8∩{xn≤2δ}

φ
( |v|
δ

)
dx.

(4.14)

Consequently, after taking ϵ = 1/2, we have∫
Ω8

φ(|Dw|) dx ≤ c

∫
Ω8

φ(|Dv|) dx+ c

∫
Ω8∩{xn≤2δ}

φ
( |v|
δ

)
dx. (4.15)

We need to estimate the last term. Using Jensen’s inequality, we have the following
inequalities ∫

Ω8∩{xn≤2δ}
φ
( |v|
δ

)
dx

≤
∫
Ω8∩{xn≤2δ}

φ
(∑

i

∫ xn

−2δ

∣∣∣∣∂nvi(x′, y)δ

∣∣∣∣ dy) dxndx′,
≤

∫
Ω8∩{xn≤2δ}

φ
(
c

∫
− 2δ

−2δ|∇v(x′, y)| dy
)
dxn dx

′,

≤ c

∫
Ω8∩{xn≤2δ}

∫
−2δ

−2δφ(|∇v(x′, y)|) dydxndx′,

≤ c

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx.

(4.16)



EJDE-2024/47 SHEAR THINNING FLUIDS 25

By (4.15) and (4.16), the required estimate (4.13) follows. □

Lemma 4.4. Let (u, π) be a solution of (4.1) and (w, πw) be the of solution of
(4.5), we then discover ∫

−Ω8
φ(|∇w|) dx ≤ c

∫
−Ω16

φ(|∇u|) dx.

Proof. By (2.22) and similar calculation as in (4.14), we see that∫
−Ω8

φ(|∇w|) dx

≤ c

∫
−Ω8φ(|∇w −∇(ψδv + ξ0 + h)|) dx+ c

∫
−Ω8φ(|∇(ψδv + ξ0 + h)|) dx,

≤ c

∫
−Ω8

φ(|Dw −D(ψδv + ξ0 + h)|) dx+ c

∫
−Ω8

φ(|∇(ψδv + ξ0 + h)|) dx,

≤ c

∫
−Ω16

φ(|∇v|) dx+ c

∫
−Ω16∩{xn≤2δ}φ

( |v|
δ

)
dx.

(4.17)

Following the same procedure by (4.16), we have∫
−Ω16∩{xn≤2δ}φ

( |v|
δ

)
dx ≤ c

∫
−Ω16φ(|∇v|) dx. (4.18)

By (2.22), Lemma 4.1, (4.3) and (4.4) we achieve∫
−Ω8

φ(|∇v|) dx ≤ c

∫
−Ω8

φ(|∇v −∇u|) dx+ c

∫
−Ω8

φ(|∇u|) dx,

≤ c

∫
−Ω8

φ(|Dv −Du|) dx+ c

∫
−Ω8

φ(|∇u|) dx,

≤ c

∫
−Ω16

φ(|∇u|) dx.

(4.19)

Combining (4.17), (4.18) and(4.19), the lemma is proved. □

The following lemma states the regularity of the limiting system near the bound-
ary, which is the result of Theorem 3.6.

Lemma 4.5. Let (u, π) be the solution of (4.1) with∫
−Ω16φ(|∇u|) dx ≤ cK.

Let (w, πw) be the solution of (4.5). For q̄, as in (1.9), we have∫
−B+

4
φ(|∇w|)q̄ dx ≤ c(µ, κ2)K

q̄.

For the rest of the paper, we extend (w, πw), the solution pair of (4.5), to be zero
on Ω8 \ B+

8 , and we still denote it as (w, πw) for simplicity of the notation. Since
a Reifenberg flat domain is an extension domain, we have (w, πw) ∈ W 1,φ(B8) ×
Lφ∗

(B8). Also, we shall assume that K > 1.

Lemma 4.6. Suppose that Ω is a (δ,R1)-Reifenberg flat domain, (v, πv) is the weak
solution of (4.2) and the following inequalities hold:∫

−Ω16
φ(|∇u|) dx ≤ K and

∫
−Ω16

β(A, B16) dx ≤ δ. (4.20)



26 N. CHO EJDE-2024/47

Then for any 0 < ϵ < 1, there exists a sufficiently small δ = δ(ϵ,data, R1) > 0 such
that if (w, πw) is the weak solution of (4.5), then we have∫

−Ω8 |V (Dw)− V (Dv)|2 dx ≤ ϵK. (4.21)

Proof. We first test w − (ψδv + ξ0 + h) with (4.5) and (4.2). Then, we obtain

0 =

∫
−Ω8

(
Ā(Dw)−A(x,Dv)

)
: D

(
w − (ψδv + ξ0 + h)

)
dx.

From a direct calculation, we find that∫
−Ω8

(
Ā(Dw)− Ā(Dv)

)
: (Dw −Dv) dx

=

∫
−Ω8

(
Ā(Dw)− Ā(Dv)

)
: D((ψδ − 1)v) dx

+

∫
−Ω8

(
Ā(Dw)− Ā(Dv)

)
: D(ξ0 + h) dx

−
∫
−Ω8

(
Ā(Dv)−A(x,Dv)

)
: D(w − ψδv − ξ0 − h) dx

:= I + II + III.

With the help of Lemma 2.17, one has

1

c

∫
−Ω8

|V (Dw)− V (Dv)|2 dx ≤ I + II + III.

For the first term I, we have

I =

∫
−Ω8

(
Ā(Dw)− Ā(Dv)

)
: D((ψδ − 1)v) dx,

=

∫
−Ω8

(
Ā(Dw)− Ā(Dv)

)
: ((ψδ − 1)∇v + v ⊗∇ψ) dx.

By Young’s inequality, properties of ψ and (1.6) we find that

I ≤ ϵ1
|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|Dw|) dx+
cϵ1
|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) + φ
( |v|
δ

)
dx,

for some ϵ1 > 0. Applying (4.16), Lemma 4.1, Lemma 4.3 and (4.20) in order
provides

I ≤ ϵ1K +
cϵ1
|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx.

For the second term, we apply (2.4), (4.3), (4.4) and Jensen’s inequality to obtain

II ≤ ϵ1

∫
−Ω8

φ(|Dv|) + φ(|Dw|) dx+ cϵ1

∫
−Ω8

φ(|v · ∇ψδ|) dx,

and then we follow the same process as in estimate of I, to derive

II ≤ ϵ1K +
Cϵ1

|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx.

From Definition 1.2 and properties of ψ, we have

III ≤
∫
−Ω8

β(A, B8)φ
′(|Dv|)

∣∣Dw −Dv −D((ψδ − 1)v + ξ0 + h)
∣∣ dx,



EJDE-2024/47 SHEAR THINNING FLUIDS 27

≤
∫
−Ω8

β(A, B8)φ
′(|∇v|)(|∇w|+ |∇v|) dx

+
c

|Ω8|

∫
Ω8∩{xn≤2δ}

β(A, B8)φ
′(|∇v|)

(
|∇v|+

∣∣v
δ

∣∣+ |∇ξ0|+ |∇h|
)
dx

:= III1 + III2.

Using Young’s inequality and (2.17), we obtain

III1 ≤ cϵ1

∫
−Ω8

β(A, B8)φ(|∇v|) dx+ ϵ1

∫
−Ω8

β(A, B8)φ(|∇w|) dx.

Using Hölder’s inequality, (1.7), Lemma 4.1, (4.20), Lemma 4.2, Lemma 4.3 and
Assumption 1.2, we have

III1 ≤ cϵ1

(∫
−Ω8

β(A, B8)
θ

θ−1 dx
) θ−1

θ
(∫
−Ω8

φ(|∇v|)θ dx
)1/θ

+ ϵ1K,

≤ (cϵ1δ + ϵ1)K.

For the second term, we apply Young’s inequality, (4.4), (4.3) and (1.7) to obtain

III2 ≤ c

|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx. (4.22)

Combining the previous inequalities, we derive∫
−Ω8 |V (Dw)− V (Dv)|2 dx ≤ (ϵ1 + δ)K +

c

|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx,

≤ 2δK +
c

|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx
(4.23)

by taking ϵ1 = δ. We are left to show that∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx

can be sufficiently small if we choose δ > 0 small enough.
Using Lemma 4.2, Lemma 4.4 and Hölder’s inequality, we discover that∫

Ω8∩{xn≤2δ}
φ(|∇v|) dx

≤
(∫

Ω8∩{xn≤2δ}
φ(|∇v|)θ dx

)1/θ∣∣Ω8 ∩ {xn ≤ 2δ}
∣∣ θ−1

θ ,

≤ Cδ
θ−1
θ |B8|c

∫
Ω8

φ(|∇u|) dx,

≤ Cδ
θ−1
θ |B8|cK,

(4.24)

for some constant c = c(θ, n). We first combine (4.23) and (4.24), then choose δ > 0
small enough depending only on ϵ > 0 and data to reach (4.21). □

Lemma 4.7. Suppose that (u, π) is a weak solution of (4.1) with the following
normalization condition and the small BMO condition:∫

−Ω16
φ(|∇u|) dx ≤ K and

∫
−Ω16

β(A, B16) dx ≤ δ (4.25)



28 N. CHO EJDE-2024/47

where Ω is a (δ,R1)-Reifenberg flat domain. Then, for any 0 < ϵ < 1, there exists
a sufficiently small δ = δ(ϵ,data, R1) > 0 such that if (w, πw) is the weak solution
of (4.5), we have∫

−Ω8
|V (Du)− V (Dw)|2 dx ≤ ϵK + cϵ

∫
−Ω16

φ(|F |) dx. (4.26)

Proof. We subtract (4.2) from (4.1) to obtain the following:

div(A(x,Du)−A(x,Dv))−∇(π − πv) = div(φ′′(|F |)F ) in Ω16,

div
(
u− v

)
= 0 in Ω16,

u− v = 0 on ∂Ω16.

We test u − v with previous system of equation, then use (2.17), (2.4), (4.25) and
Lemma 4.1, to have∫

−Ω16
|V (Du)− V (Dv)|2 dx ≤ cϵ1

∫
−Ω16

φ(|F |) dx+ ϵ1

∫
−Ω16

φ(|Du|) dx,

≤ cϵ1

∫
−Ω16

φ(|F |) dx+ ϵ1K

(4.27)

for any ϵ1 > 0. By Lemma 4.6, there exists δ(ϵ2,data) > 0 and the solution of
(4.5), (w, πw) satisfying ∫

−Ω8 |V (Dw)− V (Dv)|2 dx ≤ ϵ2K. (4.28)

provided that Ω is a (δ,R)-Reifenberg flat for such δ. Combining (4.27) and (4.28),
we have ∫

−Ω8
|V (Du)− V (Dw)|2 dx

≤ C

∫
−Ω8 |V (Du)− V (Dv)|2 dx+ C

∫
−Ω8 |V (Dv)− V (Dw)|2 dx,

≤ C

∫
−Ω16

|V (Du)− V (Dv)|2 dx+ C

∫
−Ω8

|V (Dv)− V (Dw)|2 dx,

≤ Cϵ1

∫
−Ω16φ(|F |) dx+ (ϵ1 + ϵ2)K.

By taking ϵ1 = ϵ2 = ϵ
2 > 0, we obtain (4.26) with some small δ = δ(data, ϵ) > 0. □

To establish a global Calderón-Zygmund estimate, we need a comparison es-
timate of full gradients of the localized original solution, (4.1), and the limiting
system, (4.5), rather than just the symmetric gradient.

Lemma 4.8. Under the assumptions in Lemma 4.7, we have∫
−Ω8

φ(|∇u−∇w|) dx ≤ ϵK + cϵ

∫
−Ω16

φ(|F |) dx. (4.29)

Proof. By (2.22), we find that∫
−Ω8

φ(|∇u−∇w|) dx

≤ c

∫
−Ω8

φ(|∇u−∇v|) dx+ c

∫
−Ω8

φ(|∇v −∇(w − ψδv − ξ0 − h)|) dx



EJDE-2024/47 SHEAR THINNING FLUIDS 29

+ c

∫
−Ω8

φ(|∇((ψδ − 1)v + ξ0 + h)|) dx

≤ c

∫
−Ω16φ(|Du−Dv|) dx+ c

∫
−Ω8φ(|Dv −D(w + ψδv + ξ0 + h)|) dx

+ c

∫
−Ω8

φ(|∇((ψδ − 1)v + ξ0 + h)|) dx

≤ c

∫
−Ω16φ(|Du−Dv|) dx+ c

∫
−Ω8φ(|Dv −Dw)|) dx

+ c

∫
−Ω8

φ(|∇((ψδ − 1)v + ξ0 + h)|) dx

=: I + II + III.

Now, using (2.19), (4.25) and (4.27), we have following estimates

I ≤ cε

∫
−Ω16

|V (Du)− V (Dv)|2 dx+ ε

∫
−Ω16

φ(|Du|) dx

≤ (cεε1 + ε)K + cε1

∫
−Ω16φ(|F |) dx.

We choose ε1 > 0 small enough so that cεε1 < ε we have

I ≤ εK + cε

∫
−Ω16φ(|F |) dx.

Similarly, we use (2.19), (4.21), Lemma 4.1 and (4.25) to discover

II ≤ cε

∫
−Ω8

|V (Dv)− V (Dw)|2 dx+ ε

∫
−Ω8

φ(|Dv|) dx

≤ (cεε1 + ε)K ≤ 2εK

after choosing ε1 > 0 small enough so that cεε1 < ε. For the last term III, we
proceed as in (4.22) to discover

III ≤ c

|Ω8|

∫
Ω8∩{xn≤2δ}

φ(|∇v|) dx
(4.24)

≤ εK.

Combining estimates for I, II and III, we have (4.29). □

Remark 4.9. (Dependency of the constants and the regularity of the coefficient)
We want to end this section with an important issue regarding the dependency of
the constant. The constants in Lemma 2.7 and Lemma 2.8 depend on the domain
Ω and δ, and again, the choice of δ depends the on constants c in Lemma 2.7
and Lemma 2.8. This implies that there is a circular reasoning in choosing δ.
As a consequence, the class of the coefficient of A(·, P ) is restricted, and even a
continuous coefficient may not be allowed. Thus we need an additional assumption
on the coefficient.

Note that if Ω is a (δ,R)-Reifenberg flat domain, the constants in Lemma 2.7
and Lemma 2.8 are decreasing functions of δ and increasing functions on R. Now,
to overcome such a dependency issue in our Assumption 1.2-(i) and Assumption
1.2-(ii) have different radii, R1 and R2. Then δ may depend on R1 but does not
dependent on R2. After δ is chosen, we choose R2 small enough to satisfy (1.8). In
this way, we can include the wider class of the coefficients of A(·, P ). This argument
follows from [18, Remark 2.3].



30 N. CHO EJDE-2024/47

Remark 4.10. For simplicity, we only presented estimates near the boundary.
With the same procedure, we have similar results.

Suppose that B16 ⊂⊂ Ω and u ∈ W 1,φ
0,div(Ω) is a solution of (1.3). We then

consider the following equations:

divA(x,Dv)−∇πv = 0 in B16,

div v = 0 in B16,

v = u on ∂B16,

and

div Ā(Dw)−∇πw = 0 in B8,

divw = 0 in B+
8 ,

w = v on ∂B8.

Then there exists δ = δ(ε,data) such that if∫
−B16

φ(|∇u|) dx ≤ K and

∫
−B16

β(A, B16) dx ≤ δ

holds, then∫
B16

φ(|∇u−∇w|) dx ≤ ϵK + Cϵ

∫
−Ω16φ(|F |) dx

∫
−B4φ(|∇w|)q̄ dx ≤ c(µ, κ2)K

q̄.

for q̄ defined in (1.9).

5. Proof of the main theorem

The following lemmas were originally introduced in [11, Lemma 2.7] and [9].
Since we are considering a very general system with a pressure term, we do not have
a Lipschitz regularity of the limiting equations, (4.5). Moreover, the equations only
depend on the symmetric gradient, which makes the problem harder. Thus, we
need some modification. The comparison estimate in Section 4 shall carry out the
proof of the main theorem, the scaling argument, Lemma 2.11, and a Vitali-type
covering lemma, Lemma 2.12. To do this, let us begin with the definition of the
Hardy-Littlewood maximal function, which is defined below:

M(f)(x) := sup
x∈B

∫
−BχΩ|f | dx,

where B represents a ball. We drop out the index Ω when it is clearly from the
context.

Now let us define the sets

C = {y ∈ Ω :M(φ(|∇u|)) > Ki},
D = {y ∈ Ω :M(φ(|∇u|)) > Ki−1} ∪ {x ∈ Ω :M(φ(|F |)) > σKi−1}.

We need to verify that assumptions of Lemma 2.12 are satisfied for C and D.

Lemma 5.1. Let (u, π) ∈W 1,φ
0,div(Ω)×Lφ∗

(Ω) be a solution of (1.3) and 0 < ϵ < 1.

Then there exists δ = δ(data, ϵ) > 0 so that if Ω and A satisfies Assumption 1.2
with some R1 > R2 = 128, there are some large number K = K(data, ϵ, R1) > 1
and some small number σ = σ(data, ϵ) > 0 such that the following holds:

|C ∩Br(z)| > ϵ|Br(z)| implies Br(z) ∩ Ω ⊂ D
whenever z ∈ Ω and 0 < r < 1.



EJDE-2024/47 SHEAR THINNING FLUIDS 31

Proof. We prove this lemma by contradiction. If the statement is not true, we have

|C ∩Br(z)| > ϵ|Br(z)| (5.1)

and
(
Br(z)∩Ω

)
\D ̸= ∅. Then one can find a point z1 ∈

(
Br(z)∩Ω

)
\D such that

M
(
φ(|∇u|)

)
(z1) ≤ Ki−1 and M

(
φ(|F |)

)
(z1) ≤ σKi−1. (5.2)

Let M∗(|f |) :=M(χB3r(z)∩Ω|f |). Then by (5.2), we have

M
(
φ(|∇u|)

)
(y) ≤ max

{
M∗(φ(|∇u|))(y), 128n} for all y ∈ Br(z). (5.3)

We need to consider two cases: the interior case and the boundary case.
Assume first that B16r(z) ⊂ Ω. As z1 ∈ B16r(z), we have∫

−B16r(z)φ(|∇u|) dx ≤ Ki−1 and

∫
−B16r(z)φ(|F |) dx ≤ σKi−1. (5.4)

We then define the functions

ũ(x) =
u(z + rx)

r
, π̃(z + rx) =

π(z + rx)

r
,

Ã(x, P ) = A(z + rx, P ), F̃ (x) = F (y + rx), φ̃(t) = φ(t).

Then by Lemma 2.11, (ũ, π̃) is a solution of (1.3) with ũ, π̃, F̃ , Ã and φ̃ replacing
u, π, F,A and φ. Then, by (5.4), we have∫

−B16 φ̃(|∇ũ|) dx ≤ Ki−1.

Then one can see that we are under the hypotheses of Remark 4.10 along with the
scaling invariant property. We discover that there exists a δ = δ(ϵ̃, R1,data) such

that if A is (δ, 128)-vanishing, then we can find w ∈W 1,φ
0,div(B16r) satisfying∫

−B8r(z)φ(|∇u−∇w|) dx ≤
(
ϵ̃Ki−1 + cϵ̃

∫
−B16r(z)φ(|F |) dx

)
≤

(
ϵ̃+ cϵ̃σ

)
Ki−1,

(5.5)∫
−B4r(z)φ(|∇w|)

q̄ dx ≤ cKi−1 (5.6)

for small ϵ̃, σ > 0, which shall be determined precisely in Remark 5.2. ForK > 128n,
we have

|{y ∈ Br(z) :M
(
φ(|∇u|)

)
> Ki}|

(5.3)

≤ |{y ∈ Br(z) :M
∗(φ(|∇u|)) > Ki}|,

≤
∣∣∣{y ∈ Br(z) :M

∗(φ(|∇u−∇w|)
)
+M∗(φ(|∇w|)) > Ki

2

}∣∣∣,
≤

∣∣∣{y ∈ Br(z) :M
∗(φ(|∇u−∇w|)

)
>
Ki

4

}∣∣∣
+
∣∣∣{y ∈ Br(z) :M

∗(φ(|∇w|)) > Ki

4

}∣∣∣ =: I + II.

By weak (1,1) estimate and (5.5), we obtain

I ≤ c

Ki

∫
B8r(z)

φ(|∇u−∇w|) dx ≤ C2

K
(ϵ̃+ cϵ̃σ)|Br|, (5.7)



32 N. CHO EJDE-2024/47

where C2 = C2(data). We use Chebysebev inequality, strong (q̄, q̄) estimates and
(5.4) to discover that

II ≤
∣∣∣{y ∈ Br(z) :M

∗
(
φ(|∇w|)

)q̄

>

(
Ki

23p

)q̄ }∣∣∣,
≤ c

K q̄i

∫
B4r(z)

M∗
(
φ(|∇w|)

)q̄

dx,

≤ c

K q̄i

∫
B4r(z)

φ(|∇w|)q̄ dx,

(5.6)

≤ C3(µ, q̄)

K q̄
|Br|.

(5.8)

Note that the constant C3 depends only on data, µ and q̄.
Combining (5.7) and (5.8), one obtains

|C ∩Br(z)| ≤ I + II ≤
(C2

K
(ϵ̃+ Cϵ̃σ) +

C3

K q̄

)
|Br| ≤ ε|Br|, (5.9)

by taking K > 128n large enough and σ, ε̃ > 0 small enough; see below Remark
5.2. Thus we obtain a contradiction to (5.1).

Now, let us consider the boundary case when B16r ̸⊂ Ω. First, we choose y0 ∈
∂Ω∩B16r(z) and by the definition of (δ, 128)-Reifenberg flatness of Ω, there exists
a new coordinate, (x̃1, x̃2, . . . , x̃n), depending on y0 and r, so that origin is y1 :=
y0 + 128δrn⃗0 for some inward unit vector n⃗0. Also, we have

B+
128r ⊂ Ω128r ⊂ {x̃n > −256rδ}

and ∫
−Ω128r

β(A, B128r) dx̃ ≤ δ.

Since 0 < δ < 1/32, from a direct calculation, we have that z1 ∈ Ω64r. Thus by
(5.2), we have∫

−Ω64r
φ(|∇u|) dx ≤ |B128r|

|B+
64r|

∫
−Ω128r

φ(|∇u|) dx ≤ 2nKi−1.

Let us consider the functions

ũ(x̃) =
u(y1 + 4rx̃)

2n(4r)
, π̃(x̃) =

π(y1 + 4rx̃)

4r
,

Ã(x̃, P ) = A(y1 + 4rx̃, 2nP ), F̃ (x̃) =
F (y1 + 4rx̃)

2n
.

As in the interior case, one can see that we are under the hypotheses of Lemma
4.8. Again, we discover that there exists a δ = δ(ϵ̃, R1,data) such that if A and Ω

satisfies Assumption 1.2 with δ and R2 = 128, then we can find w ∈ W 1,φ
0,div(Ω8r)

satisfying for a given ϵ̃ > 0,∫
−Ω8r

φ(|∇u−∇w|) dx̃ ≤ ϵ̃Ki−1 + cϵ̃

∫
−Ω16r

φ(|F |) dx̃ (5.10)∫
−Ω8rφ(|∇w|)q̄ dx̃ ≤ cKi−1 and

∫
−Ω8rφ(|∇w|) dx ≤ cKi−1. (5.11)

Combining (5.10), (5.11) and (5.4), one can see that∫
−Ω24r

φ(|∇u−∇w|) dx ≤
(
ϵ̃+ cϵ̃

∫
−Ω64r

φ(|F |) dx
)
≤

(
ϵ̃+ cϵ̃σ

)
Ki−1. (5.12)



EJDE-2024/47 SHEAR THINNING FLUIDS 33

Once we obtain (5.11) and (5.12), we follow similar steps done in the interior case
to reach a contradiction to (5.1). □

Remark 5.2. To prove our main result, Theorem 1.3, we need to specify the
constants K, ϵ̃ and σ. We want to mention that C2 and C3 differ in the interior
case and the boundary case. We shall choose a bigger one and then choose

K = 2[
2C3

ϵ
]1/q̄. (5.13)

After that, we choose ϵ small enough so that

K > 128n and C1ϵK
q = 2qC1ϵ

1−q/q̄(2C3)
q/q̄ < 1. (5.14)

It is possible since 1 ≤ q < q̄. Then choose ϵ̃ > 0 and σ > 0 small enough to obtain

C2

K
(ϵ̃+ Cϵ̃σ) <

ϵ

2
. (5.15)

Combining (5.13) and (5.15), we have

C2

K
(ϵ̃+ Cϵ̃σ) +

C3

K q̄
≤ ϵ

which is (5.9). Here, we remark that C1, C2 and C3 depend only on data, q, R1, µ, κ2
and ϵ, so are K, σ, ϵ̃.

We are now in a position to prove our main theorem.

Proof of Theorem 1.3. We intend to apply Lemma 2.12. For a non-negative func-
tion g, we denote a upper level function by

Ug(t) := |{x ∈ Ω :M(|g|) > t}|

In view of (5.7), (5.8) and Remark 5.2, we obtain |C ∩ B1(z)| < ϵ|B1|, which
implies (2.27). On the otherhand, one can obtain (2.28) by Lemma 5.1. Therefore,
according to Lemma 2.12 and the fact that γ = 1, we have

Uφ(|∇u|)
(
Ki

)
≤ C1ϵ

(
Uφ(|∇u|)

(
Ki−1

)
+ Uφ(|F |)

(
σKi−1

))
.

As a consequence, we have

Uφ(|∇u|)(K
i) ≤ C1ϵ

(
Uφ(|∇u|)(K

i−1) + Uφ(|F |)(σK
i−1)

)
,

≤ (C1ϵ)
iUφ(|∇u|)

(
1
)
+

i∑
j=1

(C1ϵ)
j
(
Uφ(|F |)

(
σKi−j

))
,

and
∞∑
i=1

Kqi Uφ(|∇u|)
(
Ki

)
≤

∞∑
i=1

(KqC1ϵ)
iUφ(|∇u|)

(
1
)
+

∞∑
i=1

i∑
j=1

Kq(i−j)
(
KqC1ϵ

)jUφ(|F |)
(
σKi−j

)
,

≤
∞∑
i=1

(KqC1ϵ)
iUφ(|∇u|)

(
1
)
+

∞∑
j=1

(
KqC1ϵ

)j ∞∑
i=j

KqiUφ(|F |)
(
σKi

)
.



34 N. CHO EJDE-2024/47

By (5.14), we have
∑∞

i=1(C1ϵK
q)i < ∞ and by the classical measure theory, we

can compute as follows:

S :=

∞∑
i=1

KqiUφ(|∇u|)(K
i),

≤ c|Ω|+ c

∞∑
i=j

KqiUφ(|F |)
(
σKi

)
≤ c|Ω|+ c

∫
Ω

|F |q dx.

Next, we have ∫
Ω

φ(|∇u|)q dx ≤ c|Ω|+ S ≤ c

∫
Ω

|F |q + 1 dx.

By (2.22), we have ∫
Ω

φ(|Du|)q dx ≤ c

∫
Ω

|F |q + 1 dx

for a constant c = c(κ2, µ, q, data). We use Lemma 2.9 with G∗(t) = φ∗(t)q and
following the procedure of Remark 2.10,∫

Ω

φ∗(|π|)q dx ≤ c

∫
Ω

φ(|∇u|)q + φ(|F |)q dx ≤ c

∫
Ω

φ(|F |)q + 1 dx

where c = c(κ2, µ, q, data), which completes the proof. □

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Namkyeong Cho

Center for Mathematical Machine Learning and its Applications (CM2LA), Dept. of
Mathematics, POSTECH, South Korea

Email address: namkyeong.cho@gmail.com


	1. Introduction
	2. Preliminaries
	2.1. Function spaces
	2.2. Existence and uniqueness of the solution
	2.3. Known results in a John domain
	2.4. Technical lemmas

	3. Regularity of the Limiting system up to the boundary
	3.1. Regularity in tangential direction
	3.2. Regularity in the normal direction
	3.3. Conclusion

	4. Comparison estimates near the boundary
	5. Proof of the main theorem
	References