Existence, regularity and uniqueness of weak solutions for a class of incompressible generalized Navier-Stokes system with slip boundary conditions in $\mathbb{R}^3_+$

0.22 MB·English

by  Aibin Zang

Existence, regularity and uniqueness of weak
solutions for a class of incompressible
generalized Navier-Stokes system with slip
3
1 boundary conditions in R3 ∗
0 +
2

n
a
Aibin Zang
J

1
1

]
P The School of Mathematics and Computer Science, Yichun University,
A
Yichun, Jiangxi, P.R.China, 336000
.
h Email: [email protected]
t
a
m Abstract
[ Weobtaintheexistence,regularity,uniquenessofthenon-stationary

1 problems of a class of non-Newtonian fluid is a power law fluid with
v p > 9 in the half-space under slip boundary conditions.
5 5
Keywords: Non-NewtonianFluid;Navier’sslipboundaryconditions;
2
4 weak solution;
2
MathematicsSubjectClassification(2000): 76D05,35D05,54B15,
.
1 34A34.
0
3
1
1 Introduction
:
v
i
X
Let Ω ⊂ R3 be an open set. For any T < ∞, set Q = Ω × (0,T). The
T
r
a motion of a homogeneous, incompressible fluid through Ω is governed by the
following equations
∂ u−divS +(V ·∇)u+∇π = f, in Q
t T

∇·u = 0, in QT (1.1)

u| = u (x), in Ω,
t=0 0



∗ 
This research is partially supported by NSFC(11201411) and Natural Science Funds
of Jiangxi Science and Technology (20122BAB211004)
1
where u is the velocity, π is pressure and f is the force, V is chosen a solenoid
vector function and tangential to the boundary of Ω, u is initial velocity and
0
S = (s )n is stress tensor. The above system (1.1) has to be completed
ij i,j=1
by boundary conditions except that Ω is the whole space and by constitutive
assumptions for the extra tensor. Concerning the former we can impose the
following Navier slip boundary conditions
u·n = 0,(S ·n) −αu = 0,on ∂Ω×(0,T), (1.2)
τ τ
where α is the frictional constant.
Many extra tensors are characterized by Stoke’s law S = νD(u), where
D(u) is the symmetric velocity gradient, i.e.
1 ∂u ∂u
i j
D (u) = + .
ij
2 (cid:18)∂x ∂x (cid:19)
j i
Assume that ν is a constant and V = u, (1.1) is called incompressible Navier-
Stokes equations.
However, there are phenomena that can be described by ν = ν(|D(u)|)
with power-law ansatz to model certain non-Newtonian behavior of the fluid
flows, and they are frequently used engineering literature. We can refer the
book by Bird, Armstrong and Hassager [15] and the survey paper due to
M´alek and Rajagopal [30]. Typical examples for this constitutive relations
are
S(D(u)) = µ(δ +|D(u)|)q−2D(u)
(1.3)
S(D(u)) = µ(δ +|D(u)|2)q−22D(u),
with 1 < q < ∞,δ ≥ 0, and µ > 0.
The mathematical analysis of these models started with the work of
Ladyˇzhenska [33], [34], [35]. She investigated the well-posedness of the initial
boundary value problem with non-slip boundary conditions, associated with
the stress tensor (1.3). In 1969, J.L. Lions [36] proved some existence results
for p−Laplacian equation with p ≥ 1+ 2n and the uniqueness for p ≥ n+2
n+2 n
under no-slip boundary conditions. In those papers, the authors applied the
properties of monotone operator and Minty trick theory for the stress tensor
satisfies the strict monotonicity and coercivity.
Overthese years, Ladyˇzhenska’s andLions’workwere improvedinseveral
directions by different authors. In particular, for the steady problem, there
are several results proving existence of weak solution in bounded domain [22,
25,26], interior regularity [1,37] and very recently regularity up to boundary
for the Dirichlet problem [5–11,19,20,40]. Concerning the time-evolution
Dirichlet problem in a 3D domain, J. M´alek, J. Ne˘cas, and M. R˚u˘zi˘cka [29]
2
study the weak solution for p ≥ 2. Later, L. Diening et.al have recent
advances on the existence of weak solutions in [22] for p > 8 and in [23] for
5
p > 6. There are also many papers dealing with regularity of for evolution
5
Dirichlet boundary problems and we refer instance to [3,4,8–11,16,17]. In
the three-dimensional cube with space periodic boundary conditions, there
are a lot of literatures for the well-posedness of this model, we refer to the
monograph [28] and papers [14,21].
Itshouldbeemphasized thattheoretical contributionsmostlyconcern the
homogenous boundary condition and space periodic boundary conditions.
However, many other boundary conditions are important for engineer exper-
iment and computation science. Commonly used boundary conditions are
Navier-type boundary conditions, which were introduced by Navier in [38].
Newtonian fluid under Navier slip boundary conditions was studied by many
mathematician, [12,13] and [42]. However, there are not too many results
for non-Newtonian fluid. In [5,24], the authors investigated the regularity of
steady flows with shear-dependent viscosity on the slip boundary conditions.
M. Bul´ıˇcek, J.Ma´lek and K.R. Rajagopal [18] obtained the weak solution
for the evolutionary generalized Navier-stokes-like system of pressure and
shear-dependent viscosity on the Navier-type slip boundary conditions in
the bounded domain.
In this paper, we consider the problem (1.1) with stress tensor S induced
by p−potential as in Definition 2.1, when Ω = R3, under the following slip
+
boundary conditions
u·n| = 0, ((S(D(u))·n)−(n·S(D(u))·n)n)| = 0. (1.4)
x3=0 x3=0
In fact, this problem corresponds to the free boundary problem for the non-
Newtonian fluids with free surface supposed invariable.
Since we choose the stress tensor induced by a p−potential, and then we
will obtain the equivalent conditions:
∂u
i
u | = = 0 (i = 1,2). (1.5).
3 x3=0 ∂x (cid:12)
3(cid:12)x3=0
(cid:12)
(cid:12)
From these conditions, we extend to the external force term f and initial
velocity u to whole space by mirror reflection method and change (1.1)
0
into a Cauchy problem. Hence, we can focus on the regularity estimates,
uniqueness and existence of this Cauchy problem. Then one can obtain the
existence of the solution by Galerkin Method in the half space.
The paper is organized as follows. In section 2, after recalling the no-
tation and presenting some preliminary results, we give the definitions of
the p−potential and weak solutions. We also present the existence of the
3
divergence-free base with boundary conditions (1.5) in W2,2. In section 3,
we show some theorems for the existence, regularity and uniqueness of weak
solutions for the system (1.1) with boundary conditions (1.5).
2 Preliminaries
Inthissection, wewill givesomeassumptions, functionspaces anddefinitions
for weak solution. We will show the Korn-type inequalities for unbounded
domain and construction of the basis with boundary conditions (1.6). Let
Mn×n be the vector space of all symmetric n × n matrices ξ = (ξ ). We
ij
equip Mn×n with scalar product ξ : η = ni,j=1ξijηij and norm |ξ| = (ξ : η)21.
P
Definition 2.1. Let p > 1 and let F : R {0} → R {0} be a convex
+ +
function, which is C2 on the R {0}, sucSh that F(0) S= 0, F′(0) = 0.
+
Assume that the induced functionSΦ : Mn×n → R+ {0}, defined through
Φ(B) = F(|B|), satisfies S
(∂jk∂lmΦ)(B)CjkClm ≥ γ1(1+|B|2)p−22|C|2, (2.1)
jXklm
|(∇2n×nΦ)(B)| ≤ γ2(1+|B|2)p−22 (2.2)
for all B,C ∈ Mn×n with constants γ ,γ > 0. Such a function F, resp. Φ,
1 2
is called a p−potential.
We define the extra stress S induced by F, resp. Φ, by
B
S(B) = ∇2 Φ(B) = F′(|B|)
n×n |B|
for all B ∈ Mn×n\{0}. From (2.1), (2.2) and F′(0) = 0, it easy to know that
S can be continuously extended by S(0) = 0.
As in the [21] and [28], one can obtain from (2.1) and (2.2) the following
properties of S.
Theorem 2.2. There exist constants c ,c > 0 independent of γ ,γ such
1 2 1 2
4
that for all B,C ∈ Mn×n there holds
S(0) = 0, (2.3)
(Sij(B)−Sij(C))(Bij −Cij) ≥ c1γ1(1+|B|2 +|C|2)p−22|B −C|2,
Xi,j
Sij(B)Bij ≥ c1γ1(1+|B|2)p−22|B|2, (2.4)
Xi,j
|S(B)−S(C)| ≤ c2γ2(1+|B|2 +|C|2)p−22|B −C|,
|S(B)| ≤ c2γ2(1+|B|2)p−22|B|. (2.5)
In the following part of this section, we will give some function spaces
and the definition of weak solutions for the system (1.1).
D(R3) = {u ∈ C∞(R3) : u = 0 on x = 0},V (R3) = {u ∈ D(R3) : ∇·u = 0}k∇·kLp,
+ 0 + 3 3 p + +
H = D(R3)k·kL2.
+
Denote Ω = {x ∈ R3 : |x| ≤ R} for R > 0 , then we have the corresponding
R +
spaces for domain Ω as follows,
R
D(Ω ) = {u ∈ C∞(Ω ) : u = 0 on x = 0}, V (Ω ) = {u ∈ D(Ω ) : ∇·u = 0}k∇·kLp,
R 0 R 3 3 p R R
H(Ω ) = D(Ω )k·kL2.
R R
Let Γ1 = B ∩ {x = 0}, Γ2 = ∂B ∩ R3 and Q = R3 × [0,T], QR =
R R 3 R R + + T
Ω ×[0,T].
R
Definition 2.3. Let 6 ≤ p < ∞, under the assumption of Definition 2.1. Let
5
f ∈ H or f ∈ V∗, which is the dual space of V , and u ∈ H with ∇·u = 0
p p 0 0
in the sense of distribute. A vector function u ∈ L∞(0,T;H)∩Lp(0,T;V )
p
is called a weak solution to (1.1) if the following identity
− (u·∂ φ)dxdt+ (S(x,t,D(u))−V ⊗u) : D(φ)dxdt
t
Z Z
QT QT (2.6)
= f ·φdxdt+ u ·φ(0)dx
0
ZQT ZR3+
holds for all φ ∈ C∞(R3 ×[0,T]) with divφ = 0, φ | = 0, and suppφ ⊂
+ 3 x3=0
R3 ×[0,T).
+
We need to recall the following Korn-type inequality (see Theorem 3-2
in [32].)
5
Theorem 2.4. Let K be cone in Rn and p > 1. If |D(u)|pdx < +∞,
K
then there is a skew-symmetric matrix A with constantRcoefficients such that
|∇(u(x)−Ax)|pdx ≤ C |D(u)|pdx
Z Z
K K
where the constant C does not depend on u.
The previous result leads to the following.
Corollary 2.5. There exists a constant C depending only on p such that
k∇uk ≤ CkD(u)k
p p
for all u ∈ C∞(R3).
0 +
Proof. Since the domain R3 is a special cone in R3, therefore, along the proof
+
Corollary 1 in [27], it is easy to get the result by Theorem 2.5.
To construct the basis in W2,2(Ω ) with the boundary conditions (1.5),
R
we consider the following problem
−∆u+∇p = f, in Ω
R

∇·u = 0, in Ω ,
 R



u = 0, ∂u1 = ∂u2 = 0 on Γ1, (2.7)
 3 ∂x ∂x R
3 3




u = 0 on Γ2R,



The following definition 2.6, Lemma 2.7 and its proof will be found in [31].
Definition 2.6. By a weak solution of the problem (2.7) we mean a function
u(x) ∈ V (Ω ) such that
2 R
(D(u),D(v)) = (f,v), ∀v ∈ V (Ω )
2 R
Lemma 2.7. Assumef ∈ H(Ω ),thenthere exists aunique solution (u(x),p(x))
R
to problem (2.7) such that u ∈ V (Ω )∩W2,2(Ω ). Moreover, the following
2 R R
estimates hold:
kD(u)k ≤ Ckfk
L2(ΩR) H(ΩR)
k∇2uk +k∇pk ≤ C kfk +kuk
L2(ΩR) L2(ΩR) H(ΩR) V2(ΩR) (2.8)
1 (cid:0)1 (cid:1)
k∇uk ≤ C kfk2 k∇uk2 +kuk ,
L3(ΩR) (cid:16) H(ΩR) L2(ΩR) V2(ΩR)(cid:17)
where C is independent of u, f.
6
With the aid of previous lemma, one can prove the following proposition
Proposition 2.8. The eigenvalue problem
−∆u+∇p = λu, in Ω
R

∇·u = 0, in Ω ,
 R



 ∂u ∂u
u = 0, 1 = 2 = 0 on Γ1,
 3 ∂x ∂x R
3 3




u = 0 on Γ2R,



λ ∈ R, u ∈ V (Ω ) admits a denumberable positive eigenvalue {λ } clustering
2 R i
at infinity. Moreover, the corresponding eigenfunctions {a } are in W2,2(Ω ),
i R
and associate pressure fields p ∈ W1,2(Ω ). Finally, {a } are orthogonal and
i R i
complete in H(Ω ) and V (Ω ).
R 2 R
Proof. The mapping A : f −→ u defined by Lemma 2.7 is linear and
continuous from H(Ω ) onto V (Ω ), into W1,2(Ω ). Since Ω is bounded,
R 2 R R R
byRellichTheorem, weknowthatW1,2(Ω ) ֒→ L2(Ω )iscompact. Itiseasy
R R
to know that operator A is a positive symmetric and self-adjoint operator on
L(Ω ). Therefore, A possess an sequence of eigenfunctions a :
R i
Aa = λ a k ≥ 0,λ > 0, λ → ∞ as k → ∞
i i i i i
(a ,a ) = δ , (D(a ),D(a )) = λ δ .
i j L2 i,j i j L2 k i,j
By Lemma 2.7, we can get for each i, there exists p with the estimates
i
(2.8).
3 Main results and their proofs
To study the well-posedness of problem (1.1), we define a reflection as follows
(u (x ,x ,x ),u (x ,x ,x ),u (x ,x ,x )) if x ≥ 0;
1 1 2 3 2 1 2 3 3 1 2 3 3
u∗(x) = 
(u (x ,x ,−x ),u (x ,x ,−x ),−u (x ,x ,−x ), if x < 0.
1 1 2 3 2 1 2 3 3 1 2 3 3
(3.1)

Next, we will show the existence, uniqueness of strong solutions to the
problem (1.1). We give the definition of the strong solution for the problem
(1.1) as follows
7
Definition 3.1. We say a couple (u,π) is a strong solution to problem (1.1)
if
u ∈ L∞(0,T;W1,2(Ω))∩Lp(0,T;W2,p(Ω))∩Lp(0,T;V )∩L∞(0,T;H)
loc loc p
∂u
∈ L2(0,T;L2 (Ω));π ∈ Lp′(0,T;Lp′ (Ω).
∂t loc loc
(3.2)
where p′ = p and satisfies the weak formulation
p−1
∂u
ϕdx+ S(D(u)) : D(ϕ)dx+ (V ·∇)u·ϕdx
Z ∂t Z Z
Ω Ω Ω (3.3)
= πdivϕdx+ fϕdx.
Z Z
Ω Ω
holds for all ϕ ∈ C∞(Ω) and almost all t ∈ (0,T), at same time, the boundary
0
conditions hold in the sense of trace.
At first, we provide the definition of difference and recall a well-known
result. Fixed any domain Ω ⊂ R3,Ω′ ⊂⊂ Ω, and we put δ(Ω′,Ω) =
+
dist(Ω′,∂Ω\{x = 0}).
3
Definition 3.2. For g : Ω −→ R3 we set
(∆ g)(x) = g(x+λe )−g(x), x ∈ Ω′,0 < λ < δ(Ω′,Ω),k = 1···3.
λ,k k
where e ,e ,e is the canonical base of R3. We shall omit the dependence on
1 2 3
k where the meaning is clear.
Lemma 3.3. For any u ∈ W1,p(Ω) and 0 < |λ| < δ(Ω′,Ω) it holds
k∆λ,kukp,Ω′ ≤ |λ|ku,kkp,Ω.
For above Ω,Ω′ andδ(Ω′,Ω), then the following theorem and lemma show
that theregularity anduniqueness of theweak solutions to theproblem (1.1).
Theorem 3.4. Let 9 < p < 2,f ∈ Lp′(Q ), V ∈ L∞(0,T;W2,2(Ω)), u ∈
5 T 0
V ∩H satisfy the boundary conditions (1.5), and S be given by a p-potential
2
from Definition 2.1. If u ∈ Lp(0,T;V )∩L∞(0,T;H) is the weak solution for
p
problem (1.1), then this solution is also a unique strong solution to problem
(1.1) such that
kukL∞(0,T;W1,2(Ω′))∩Lp(0,T;W2,p(Ω′)) ≤ C(|Ω′|,u0,f,T,kVkL∞((0,T)×Ω)), (3.4)
T
(1+|D(u)|)p−2|∇D(u)|2dxdt ≤ C(|Ω′|,u0,f,T,kVkL∞((0,T)×Ω)).
Z0 ZΩ′
(3.5)
∂u
k∂tk2L2((0,T)×Ω′) +kΦ(D(u))kL∞(0,T;L1(Ω′)) ≤ C(δ(Ω′,Ω),|Ω′|,u0,f,T,kVkL∞((0,T)×Ω)).
(3.6)
8
Proof. Firstly we extent the u and force term f to the whole space by
0
the reflection defined in (3.1), however, since V ∈ W2,2, we need apply the
extension of Theorem 5.19 in [2], Denote these functions by u∗, f∗, V∗
0
respectively.
We begin to consider the Cauchy problem as follows
∂ v −divS(D(v))+(V∗ ·∇)v +∇π = f∗, in R3 ×(0,T)
t

∇·v = 0, in R3 ×(0,T), (3.7)


v| = u∗(x), in R3.
 t=0 0



From[39],Thereexistsaweaksolutionu ∈ Lp(0,T;V (R3))∩L∞(0,T;L2(R3))
p
to problem (3.7), then by interpolation inequality, one follows that u ∈
L53p((0,T) × R3). If p > 95 then p′ < 53p, and V∗ ⊗ u ∈ Lp′((0,t) × Ba),
since V∗ ∈ L∞. Along the proof of Theorem 2.6 in [41], we know that
kπkLp′((0,t)×Ba) ≤ C(kukLp′((0,t)×Ba) +kS +f∗kLp′((0,t)×Ba) +1)
where B is any ball of R3 with radius a, C only depends on p,T,f,u ,a.
a 0
For any ρ > 0 such that 0 < ρ < δ(Ω′,Ω), Set Ω = {x ∈ Ω;dist(x,Ω′) <
ρ
ρ}. Now fix r < 1δ(Ω′,Ω), there exists a ball B , such that Ω ⊂ B Let
4 a a
us choose a cut-off function η such that η ≡ 1 on Ω , η ≡ 0 in R3 \ Ω ,
r 2r
0 ≤ η ≤ 1 and |∇η| < C, |∇2η| < C in Ω , where the constant C depends
r r2 2r
only on the geometry of ∂Ω.
If|λ| < r,itresultsthat∆ (η2∆ u) ∈ L2((0,T)×Ω )∩Lp(0,T;W1,p(Ω )),
−λ λ 3r 0 3r
but it is not divergence free. Take ∆ (η2∆ u) as a test function in the first
−λ λ
equation of (3.7), we obtain
hu ,∆ (η2∆ u)i+(S(D(u)),D(∆ (η2∆ u)))+(V∗ ·∇u,∆ (η2∆ u))
t −λ λ −λ λ −λ λ
= (π,div(∆ (η2∆ u)))+(f,∆ (η2∆ u)).
−λ λ −λ λ
Set
J = hu ,∆ (η2∆ u)i,
1 t −λ λ
J = (S(D(u)),D(∆ (η2∆ u))),
2 −λ λ
J = (V∗ ·∇u,∆ (η2∆ u)),
3 −λ λ
J = (π,div(∆ (η2∆ u))),
4 −λ λ
J = (f,∆ (η2∆ u)).
5 −λ λ
Clearly, J = dkη∆ uk2 . Let I (u) = (1 + |D(u)(x + λe )| +
1 dt λ L2(Ω2r) λ Ω2r k
|D(u)|)p−2|η∆ D(u)|2dx. R
λ
9
J = 2 S(D(u))∆ sym(∆ u⊗η∇η)dx− η2(∆ (S(D(u)))∆ D(u))dx
2 −λ λ λ λ
Z Z
Ω3r Ω2r
:= J −J .
21 22
Since (2.3),
J ≥ C γ (1+|D(u)(x+λe )|+|D(u)|)p−2|η∆ D(u)|2dx = C γ I (u),
22 1 1 k λ 1 1 λ
Z
Ω2r
|J21| ≤ ckS(D(u))kLp′(Ω3r)k∆−λsym(∆λu⊗η∇η)kLp(Ω3r),
Thus
k∆ sym(∆ u⊗η∇η)k ≤ |λ|k∇sym(∆ u⊗η∇η)k ,
−λ λ Lp(Ω3r) λ Lp(Ω3r)
≤ |λ| k∇η|sym(∆ u⊗∇η)|k +kηsym(∆ ∇u ⊗∇η)k
λ Lp(Ω3r) λ xk Lp(Ω3r)
+ksym(cid:2) (∆ u⊗∇η )|k
λ xk Lp(Ω3r)
(cid:3) 1
λ2 2C|λ| p
≤ 4C k∇uk + |η∇(∆ u)|pdx .
r2 Lp(Ω3r) r (cid:18)Z λ (cid:19)
Ω2r
Since η∇(∆ u) = ∇(η∆ u)−(∇η)·∆ u, thus
λ λ λ
1 1
p C|λ| p
|η∇(∆ u)|pdx ≤ k∇uk +C |ηD(∆ u)|pdx
(cid:18)Z λ (cid:19) r Lp(Ω3r) p(cid:18)Z λ (cid:19)
Ω2r Ω2r
However, by H¨older’s inequality we have
1 2−p
|ηD(∆λu)|pdx p ≤ Iλ(u)21 (1+|D(u)(x+λek)|+|D(u)|)pdx 2 .
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
Ω2r Ω2r
Hence
Cλ2 1 2−p
J2 ≤ kS(D(u))kLp′(Ω3r)(cid:18) r2 k∇ukLp(Ω3r) +CIλ(u)2(1+k∇ukLp2(Ω3r))(cid:19)−Cγ1Iλ(u)
≤ (1+k∇ukpL−p(1Ω3r))(cid:18)Crλ22k∇ukLp(Ω3r) +CIλ(u)21(1+k∇ukL2−p2(pΩ3r))(cid:19)−Cγ1Iλ(u)
≤ λ2C(|Ω |,r,ǫ) 1+k∇ukp −(C γ −ǫ)I (u).
3r (cid:16) Lp(Ω3r)(cid:17) 1 1 λ
|J5| = |(f,∆−λ(η2∆λu))| ≤ kfkLp′(Ω3r)|λ|k∇(η2∆λukLp(Ω2r)
≤ C(r,p)λ2kfkLp′(Ω3r)k∇ukLp(Ω3r) +|λ|kη2∆λ∇ukLp(Ω2r)
≤ |λ|Iλ(u)21(1+k∇ukL2−p2(pΩ3r))kfkLp′(Ω3r) +C(r12 + 1r)λ2kfkLp′(Ω3r)k∇ukLp(Ω3r)
≤ C(p,|Ω |)λ2 kfkp′ +k∇ukp +1 +ǫI (u).
3r (cid:16) Lp′(Ω3r) Lp(Ω3r) (cid:17) λ
10