CARPATHIAN J. MATH.
23 (2007), No. 1-2, 32-41
Non-steady Navier-Stokes Equations with
Homogeneous Mixed Boundary Conditions and
Arbitrarily Large Initial Condition
M ICHAL B ENE Š , P ETR K U ČERA
A BSTRACT. Let Ω ⊂ R2 be a bounded domain, ∂Ω ∈ C 0,1 and ∂Ω = Γ1 ∪ Γ2 such that Γ1 and Γ2
are closed, sufficiently smooth, 1-dimensional measure of Γ1 ∩ Γ2 is zero and 1-dimensional measure
of Γ1 is positive. Further let (0, T ) be a time interval. We prescribe the non-slip boundary conditions
on Γ1 × (0, T ) and the boundary condition
∂u
=0
∂n
on Γ2 × (0, T ). Here u = (u1 , u2 ) is velocity, P represents pressure and n = (n1 , n2 ) is an outer
normal vector.
Our aim is to prove the existence and uniqueness of this problem on some time interval (0, T ∗ )
for sufficiently small T ∗ , 0 < T ∗ ≤ T .
−Pn +
1. I NTRODUCTION
0,1
|
Let Ω be a bounded domain in R2 with a Lipschitz boundary, ∂Ω ∈ C
and
let Γ1 , Γ2 be open disjoint subsets of ∂Ω such that ∂Ω = Γ1 ∪ Γ2 , Γ1 6= ∅ and
the 1-dimensional measure of ∂Ω − (Γ1 ∪ Γ2 ) is zero. The domain Ω represents a
channel filled up with a fluid, Γ1 is a fixed wall and Γ2 is both the input and the
output of the channel.
The authors in [2] and [14] use the Neumann condition on the output of the
channel. Some qualitative properties of the Navier–Stokes equations with the
mixed boundary conditions are studied in [3], [4], [5], [6], [7], [8], [9], [10].
Let T ∈ (0, ∞], Q = Ω × (0, T ). The classical formulation of our problem is as
follows:
∂u
(1.1)
− ν∆u + (u · ∇)u + ∇P = g in Q,
∂t
(1.2)
div u = 0 in Q,
(1.3)
0
in Γ1 × (0, T ),
=
0
in Γ2 × (0, T ),
(1.5)
u(0) =
γ
in Ω,
(1.6)
γ
0
on Γ1 .
(1.4)
u =
−Pn + ν
∂u
∂n
=
Received: date; In revised form: date
2000 Mathematics Subject Classification. : 76D05, 76D03.
Key words and phrases. Navier-Stokes equations, mixed boundary conditions, regularity.
32
Non-steady Navier-Stokes Equations with Homogeneous Mixed Boundary Conditions
33
Functions u, P, g, γ are smooth enough, u = (u1 , u2 ) is velocity, P represents
pressure, ν denotes the viscosity, g is a body force and n = (n1 , n2 ) is an outer
normal vector. The problem (1.1) − (1.6) will be called the nonsteady Navier−Stokes
problem with the mixed boundary conditions. For simplicity we suppose that
ν = 1 throughout this paper.
We solve also the problem, in which (1.2)−(1.6) hold and (1.1) is replaced with
∂u
− ∆u + ∇P = g in Q.
∂t
The problem (1.2)−(1.6) and (1.7) will be called the nonsteady Stokes problem
with the mixed boundary conditions.
(1.7)
The Dirichlet boundary condition (1.3) expresses a non-slip behaviour of the
fluid on fixed walls of the channel.
Our aim is to prove the existence and uniqueness of this problem on some time
interval (0, T ∗ ) for sufficiently small T ∗ , 0 < T ∗ ≤ T .
2. D EFINITION OF SOME FUNCTION SPACES AND GENERALIZED FORMULATION
OF THE PROBLEM
We shall denote by c a generic constant, i.e. a constant whose value may change
from line to line. On the other hand, numbered constants will have a fixed value
throughout the paper. Constants will always depend only on domain Ω.
Let
©
ª
E(Ω) = u ∈ C ∞ (Ω)2 ; divu ≡ 0, supp u ∩ Γ2 ≡ ∅ .
Let V k,p be a closure of E(Ω) in the norm of W k,p (Ω)2 , k ≥ 0 (k need not be an
integer) and 1 ≤ p < ∞. Then V k,p is a Banach space with the norm of the space
W k,p (Ω)2 . For simplicity, we denote V 1,2 and V 0,2 , respectively, as V ¡¡and ¢¢H.
Note, that V and H, respectively, are Hilbert spaces with scalar products . , . V
¡¡ ¢¢
and . , . H ,
Z
Z
¡¡ ¢¢
¡¡
¢¢
∂Φi ∂Ψi
. , . V = Φ, Ψ V =
∇Φ · ∇Ψ d(Ω) =
d(Ω)
Ω
Ω ∂xj ∂xj
and
¡¡ ¢¢
¡¡
¢¢
. , . H = Φ, Ψ H =
Z
Z
Φ · Ψ d(Ω) =
Ω
Φi Ψi d(Ω)
Ω
2
and they are closed subspaces of spaces W 1,2 (Ω)2 and L (Ω)2 .
Let
(2.8)
¡¡
¢¢
¡¡
¢¢
D = {w ∈ V ; there exists f ∈ H such that w, v V = f , v H for every v ∈ V }
and
kwkD = kf kH ,
where w, f are corresponding functions via (2.8). Let wi and f i are corresponding functions via (2.8). Note that D is the Hilbert space with the scalar product
¡¡
., .))D such that
¡¡
¡¡
¢¢
w1 , w2 ))D = f 1 , f 2 H .
34
Michal Beneš, Petr Kučera
Similarly it can be shown as in ([16], Chapter I.,2.6) that there exist functions
φ1 , φ2 , . . . , φk , · · · ∈ V ⊂ H and real positive numbers λ1 , λ2 , . . . , λk , · · · → ∞ for
k → ∞, such that
¡¡
¢¢
¡¡
¢¢
φk , v V = λk φk , v H
for every v ∈ V . φ1 , φ2 , . . . is a system that is complete in both H and V , orthonormal in H and orthogonal in V . Note that
∞
∞
n
o
X
X
(2.9)
H = v; v =
ak φk , ak ∈ R and
a2k < ∞ ,
k=1
k=1
∞
∞
n
o
X
X
V = v; v =
ak φk , ak ∈ R and
λk a2k < ∞
(2.10)
k=1
k=1
and
∞
∞
n
o
X
X
D = v; v =
ak φk , ak ∈ R and
λ2k a2k < ∞ .
(2.11)
k=1
k=1
Let X be an arbitrary Banach space and p ∈ [1, ∞). As usual Lp (α, β; X ) and L∞ (α, β; X )
(for −∞ ≤ α < β ≤ ∞) denote the Banach spaces
Z β
o
n
φ; φ(t) ∈ X for almost every t ∈ (α, β),
kφ(t)kpX dt < ∞
α
and
n
o
φ; φ(t) ∈ X for almost every t ∈ (α, β), ess sup kφ(t)kX < ∞
t∈(α,β)
with the norms
³Z
kφk
Lp (α,β; X )
β
=
α
´1/p
kφ(t)kpX dt
and
kφkL∞ (α,β; X ) = ess sup kφ(t)kX .
t∈(α,β)
Note that ([11]) and ([1]) yield existence of γ, 1 < γ < 2 such that
D ,→,→ W γ,2 (Ω)2 ,→ W 1,2+2δ (Ω)2 ,
(2.12)
where δ =
γ−1
2−γ .
Definition 2.1. Let 0 < T ∗ ≤ T , f ∈ L2 (0, T ∗ ; H), u0 ∈ V . Then u is called a
generalized solution of the Stokes problem with the mixed boundary conditions and with
data f and u0 on (0, T ∗ ) (problem (1.2)−(1.7) for T = T ∗ ) if u ∈ L2 (0, T ∗ ; D) ∩
L∞ (0, T ∗ ; V ), u0 ∈ L2 (0, T ∗ ; H) and
¡¡
¢¢
¡¡
¢¢
¡¡ 0
¢¢
(2.13)
u (t), v H + u(t), v V = f (t), v H
holds for every v ∈ V and for almost every t ∈ (0, T ∗ ).
Non-steady Navier-Stokes Equations with Homogeneous Mixed Boundary Conditions
35
2
Theorem 2.1. Let f ∈ L (0, T ; H), u0 ∈ V . There exists a unique generalized solution
u of the Stokes problem with the mixed boundary conditions and with data f and u0 on
(0, T ). Moreover u ∈ C([0, T ]; V ) and
(2.14) kukL2 (0,T ; D) + kukL∞ (0,T ; V ) + ku0 kL2 (0,T ; H) ≤ c1 (kf kL2 (0,T ; H) + ku0 kV ).
Proof: Since f ∈ L2 (0, T ; H) and u0 ∈ V , we have
(2.15)
f=
∞
X
µk (t)φk ,
u0 =
k=1
where
T
0
k=1
ak φk ,
k=1
∞ Z
X
(2.16)
∞
X
µ2k (t) dt +
∞
X
a2k < ∞.
k=1
Let ϑk be a solution of the ordinary differential equation
ϑ0k (t) + λk ϑk (t) = µk (t)
(2.17)
with the initial condition
(2.18)
ϑk (0) = ak
for k = 1, 2, . . . Then
Z
t
ϑk (t) =
eλk (s−t) µk (s) ds + ak e−λk t
0
for almost every t ∈ (0, T ). Hence ϑk ∈ W 1,2 ((0, t)).
Multiplying (2.17) by 2ϑ0k and integrating over (0, t) we get
Z t
Z t
2
2
ϑ0k (s) ds + λk ϑ2k (t) = λk ϑk 2 (0) + 2
µk (s)ϑk 0 (s) ds ≤
0
0
Z t
Z t
02
2
≤ ϑk (0) +
ϑk (s) ds +
µ2k (s) ds
0
0
for k = 1, 2, . . . and for almost every t ∈ (0, T ) and therefore
Z t
Z t
2
(2.19)
ϑ0k (s) ds + λk ϑ2k (t) ≤ ϑ2k (0) +
µ2k (s) ds.
0
0
Thus (2.19) yields
(2.20)
∞ Z
X
k=1
∞
X
2
t
0
2
ϑ0k (s) ds +
ϑ2k (0) + 2
k=1
∞
X
k=1
T
∞ Z
X
k=1
0
λk ϑ2k (t) ≤
∞ Z
X
k=1
0
T
2
ϑ0k (s) ds +
∞
X
λk ϑ2k (t) ≤
k=1
µ2k (s) ds
for almost every t ∈ (0, T ) (remind that k doesn’t depend on t) and therefore we
get
(2.21)
u=
∞
X
k=1
ϑk (t)φk ∈ L∞ (0, T ; V ),
u0 ∈ L2 (0, T ; H)
36
Michal Beneš, Petr Kučera
and
kukL∞ (0,T ; V ) + ku0 kL2 (0,T ; H) ≤ 2kf kL2 (0,T ; H) + 2ku0 kV .
(2.22)
(2.17) yields also inequalities
2
λ2k ϑ2k (t) ≤ 2µ2k (t) + 2ϑ0 k (t)
for every k = 1, 2, . . . and for almost every t ∈ (0, T ). Therefore we get
Z t
Z T
∞
∞
∞ Z T
∞ Z T
X
X
X
X
2
2
2
2
2
2
λk
ϑk (s) ds ≤
λk
ϑk (s) ds ≤ 2
µk (s) ds+2
ϑ0 k (s) ds.
0
k=1
0
k=1
k=1
The last inequality and (2.20) yield
Z t
Z
∞
∞
X
X
(2.23)
λ2k
ϑ2k (s) ds ≤
λ2k
k=1
0
k=1
0
T
0
k=1
ϑ2k (s) ds ≤ 6
∞ Z
X
k=1
0
T
0
µ2k (s) ds + 4
∞
X
ϑ2k (0)
k=1
for almost every t ∈ (0, T ). Therefore we get
u ∈ L2 (0, T ; D)
(2.24)
and
(2.25)
kukL2 (0,T ; D) ≤ c(kf kL2 (0,T ; H) + ku0 kV ).
The last inequality and (2.22) imply (2.14). It is easy to see that
¡¡ 0
¢¢
¡¡
¢¢
¢¢
u (t), v H + u(t), v V = hf (t), v H
for every v ∈ V and for almost every t ∈ (0, T ) and that
Since
d
2
dt (ku(s)kV
)=
u(0) = u0 .
P∞
k=1
2λk ϑk (t)ϑ0 k (t) for almost every t ∈ (0, T ) and
Z
∞
X
2
k=1
≤
λ2k
T
|λk ϑk (s)ϑ0 k (s)| ds ≤
0
Z
T
Z
2
ϑk (s) ds +
0
0
T
2
ϑ0 k (s) ds < ∞
we get
(2.26)
ku(.)kV ∈ C([0, T ]).
(2.21), (2.26) and the fact that V is a Hilbert space imply u ∈ C([0, T ]; V ).
Suppose that u1 , u2 are two solutions of our problem. Denote u = u2 − u1 .
Then
¡¡
¢¢
¡¡ 0
¢¢
u (t), v H + u(t), v V = 0
for every v ∈ V and for almost every t ∈ (0, T ) and
u(0) = 0V .
Then
¡¡
¢¢
¡¡ 0
¢¢
1 d
ku(t)k2H + ku(t)k2V = 0
u (t), u(t) H + u(t), u(t) V =
2 dt
Non-steady Navier-Stokes Equations with Homogeneous Mixed Boundary Conditions
for almost every t ∈ (0, T ). Therefore
kuk2L2 (0,T ; V )
37
= 0. The theorem is proved.
If θ, ψ, v ∈ V , then b(θ, ψ, v) denotes the trilinear form
Z
∂ψi
(2.27)
b(θ, ψ, v) =
θj
vi d(Ω).
∂x
j
Ω
Remark 2.1. Let θ, ψ ∈ D. Then (2.12) yields that b(θ, ψ, .) ∈ H. If u, w ∈
L2 (0, T ∗ ; D) ∩ L∞ (0, T ∗ ; V ) then b(u, w, .) = b(u(t), w(t), .) ∈ L2 (0, T ∗ ; H).
Now we define a generalized formulation of the Navier-Stokes problem.
Definition 2.2. Let 0 < T ∗ ≤ T , f ∈ L2 (0, T ; H), u0 ∈ V . Then u is called a
generalized solution of the problem (1.1) − (1.6) on (0, T ∗ ) (a generalized solution of
the Navier-Stokes problem with the mixed boundary conditions) with data f and u0 if
u ∈ L2 (0, T ∗ ; D) ∩ L∞ (0, T ∗ ; V ), u0 ∈ L2 (0, T ∗ ; H),
¡¡ 0
¢¢
¡¡
¢¢
¡¡
¢¢
(2.28)
u (t), v H + u(t), v V + b(u, u, v) = f (t), v H
holds for every v ∈ V and for almost every t ∈ (0, T ∗ ), and
(2.29)
u(0) = u0 .
3. T HE MAIN RESULT
Our aim is to prove the following result:
Theorem 3.2. Let u0 ∈ D, f ∈ L2 (0, T ; H). Then there exists T ∗ and
u ∈ L2 (0, T ∗ ; D) ∩ L∞ (0, T ∗ ; V ), u0 ∈ L2 (0, T ∗ ; H) such that u is a generalized
solution of the Navier-Stokes problem on (0, T ∗ ).
Let 0 < T ∗ ≤ T . We make use the following reflexive Banach spaces.
©
¡
ª
X1,T ∗ = ϕ; ϕ ∈ L2 (0, T ∗ , W 1+2δ,2 (Ω)2 ) ∩ L9 0, T ∗ ; V )
and
©
ª
X2,T ∗ = ϕ; ϕ ∈ L2 (0, T ∗ ; D), ϕ0 ∈ L2 (0, T ∗ ; H) ,
respectively, with norms
kϕkX1,T ∗ = kϕkL2 (0,T ∗ ,W 1+2δ,2 (Ω)2 ) + kϕkL9 (0,T ∗ ;V )
and
kϕkX2,T ∗ = kϕkL2 (0,T ∗ ; D) + kϕ0 kL2 (0,T ∗ ; H) .
Since
X2,T ∗ ,→,→ L2 (0, T ∗ , W 1+2δ,2 (Ω)2 )
and
¡
X2,T ∗ ,→,→ L9 0, T ∗ ; V )
the embedding
(3.30)
X2,T ∗ ,→,→ X1,T ∗
38
Michal Beneš, Petr Kučera
holds.
Let u be a generalized solution of problem (1.1)−(1.6) with a right hand side
f ∈ L2 (0, T ; H) and initial condition
(3.31)
u0 ∈ D.
Let
(3.32)
u = u0 + w.
Then w ∈ L2 (0, T ; D) ∩ L∞ (0, T ; V ), w0 ∈ L2 (0, T ; H), the form
¡¡ 0 ¢¢
¡¡
¢¢
w , v H + w, v V =
(3.33)
¡¡
¢¢
¡¡
¢¢
= f , v H − u0 , v V − b(u0 , u0 , v) − b(w, u0 , v) − b(u0 , w, v) − b(w, w, v)
holds for every v ∈ V and for almost every t ∈ (0, T ) and
(3.34)
w(0) = 0.
Let F : X1,T ∗ → L2 (0, T ; H) be an operator such that
¡¡
¢¢
¡¡
¢¢
¡¡
¢¢
¡¡
¢¢
F (φ), v H = F (φ)(t), v H =
f (t), v H − u0 , v V − b(u0 , u0 , v) −
−
b(φ(t), u0 , v) − b(u0 , φ(t), v) − b(φ(t), φ(t), v).
Remark 3.2. Note that u = w + u0 is a generalized solution of the problem (1.1)−(1.6)
if and only if the equality
¡¡ 0 ¢¢
¡¡
¢¢
¡¡
¢¢
(3.35)
w , v H + w, v V = F (w), v H
holds for every v ∈ V and for almost every t ∈ (0, T ) and (3.34) holds.
We prove the following lemma:
Lemma 3.1. F is a continuous operator from X1,T ∗ into L2 (0, T ∗ ; H). Moreover there
exists K > 0 such that the inequality
(3.36)
kF (ϕ)kL2 (0,T ∗ ; H) ≤ c2 (T ∗ )1/12 kϕk2X1,T ∗ + c3 (T ∗ )1/6 kϕkX1,T ∗ + K
holds for every ϕ ∈ X1,T ∗ .
Proof of Lemma 3.1: It is easy to see that there exists K > 0 such that
¡¡
¢¢
¡¡
¢¢
(3.37)
k f (t), . H − u0 , v V − b(u0 , u0 , v)kL2 (0,T ∗ ; H) ≤ K.
Further the inequality
³Z
(3.38)
kb(ϕ, u0 , .)kL2 (0,T ∗ ; H) ≤ ku0 kW 1,2+2δ (Ω)2
³Z
≤ ku0 kW 1,2+2δ (Ω)2
0
T∗
kϕk3Lp (Ω)2
´1/3
≤ cku0 kW 1,2+2δ (Ω)2 kϕkX1,T ∗ (T ∗ )1/6
0
T∗
kϕk2Lp (Ω)2
(T ∗ )1/6 ≤
´1/2
≤
Non-steady Navier-Stokes Equations with Homogeneous Mixed Boundary Conditions
39
holds for sufficiently large p. Similarly we obtain the inequality
´1/2
³Z T ∗
(3.39)
kb(u0 , ϕ, .)kL2 (0,T ∗ ; H) ≤ ku0 kL∞ (Ω)2
kϕk2V
≤
³Z
≤ ku0 kW 1,2+2δ (Ω)2
T∗
kϕk9V
0
´1/9
0
(T ∗ )7/9 ≤
≤ cku0 kW 1,2+2δ (Ω)2 kϕkX1,T ∗ (T ∗ )7/9 .
Since
1/2
1/2
kϕkW 1,2+δ (Ω)2 ≤ kϕkV kϕkW 1,2+2δ (Ω)2
we get the inequality
³Z
(3.40)
kb(ϕ, ϕ, .)kL2 (0,T ∗ ; H) ≤
³Z
T∗
0
kϕk2W 1,2+2δ (Ω)2
T∗
0
´1/4 ³Z
kϕk3V kϕkW 1,2+2δ (Ω)2
T∗
0
kϕk9V
´1/6
´1/2
≤
(T ∗ )1/6 ≤ kϕk2X1,T ∗ (T ∗ )1/12 .
The inequalities (3.37)−(3.40) yield F (ϕ) ∈ L2 (0, T ; H) and the inequality (3.36).
Let ϕ1 , ϕ2 ∈ X1,T ∗ and ϕ = ϕ2 − ϕ1 . Then
F (ϕ2 ) − F (ϕ1 ) = b(ϕ, u0 , .) + b(u0 , ϕ, .) + b(ϕ2 , ϕ, .) + b(ϕ, ϕ1 , .)
and
(3.41)
³Z
kb(ϕ2 , ϕ, .)kL2 (0,T ∗ ; H) ≤
³Z
T∗
≤
0
³Z
≤
0
T∗
T∗
0
kϕ2 k2V kϕk2W 1,2+δ (Ω)2
kϕ2 k2V kϕkW 1,2+2δ (Ω)2 kϕkV
kϕ2 k9V
´1/9 ³Z
0
T∗
kϕk9V
´1/2
´1/2
≤
≤
´1/18 ³Z
T∗
0
kϕk2W 1,2+δ (Ω)2
´1/4
(T ∗ )1/6 ≤
≤ kϕkX1,T ∗ kϕ2 kX1,T ∗ (T ∗ )1/6 .
Similarly
(3.42)
kb(ϕ, ϕ1 , .)kL2 (0,T ∗ ; H) ≤ kϕkX1,T ∗ kϕ1 kX1,T ∗ .
Inequalities (3.38), (3.39), (3.41) and (3.42) yield that F is a continuous operator
from X1,T ∗ into L2 (0, T ∗ ; H). The proof is complete.
Definition 3.3. Let T : X1,T ∗ → X2,T ∗ be an operator such that T (ϕ) = w if and
only if
¡¡ 0 ¢¢
¡¡
¢¢
¡¡
¢¢
(3.43)
w , v H + w, v V = F (ϕ), v H ,
holds for every v ∈ V and w(0) = 0.
40
Michal Beneš, Petr Kučera
Lemma 3.2. The operator T is a continuous operator from X1,T ∗ into X2,T ∗ . Moreover,
(3.44)
c1 kT (ϕ)kX1,T ∗ ≤ kT (ϕ)kX2,T ∗ ≤ c2 kF (ϕ)kL2 (0,T ∗ ; H) .
Proof of Lemma 3.2: Inequality (2.14) and Lemma 3.1 imply that T is a continuous operator from X1,T ∗ into X2,T ∗ .
Proof of Theorem 3.2: Let
BR = {ϕ ∈ X1,T ∗ ; kϕkX1,T ∗ ≤ R}.
Lemma 3.1 and Lemma 3.2 imply that for a sufficiently small T ∗ and for a sufficiently large R, T maps BR into itself. By Lemma 3.2 and (3.30), T is totally
continuous operator from X1,T ∗ into X1,T ∗ . Moreover, the Banach space X1,T ∗ is
reflexive. Therefore there exists w ∈ BR such that T (w) = w. Set u = w+u0 . By
Remark 3.2, u is a generalized solution of the problem (1.1)−(1.6). The theorem
is proved.
Acknowledgement. The research was supported by the grant CTU0600111 of
the Czech technical university (author one), by the Grant Agency of the Czech
Academy of Sciences (grant No. IAA100190612) (author two) and by the research
plan of the Ministry of Education of the Czech Republic No. MSM 6840770010
(author two).
R EFERENCES
[1] B ENE Š M.: The qualitative properties of the Stokes and Navier-Stokes system for the mixed
boundary problem in a nonsmooth domain. Accepted in journal Mathematics and Computers
in Simulation.
[2] G LOWINSKI R.: Numerical methods for nonlinear variational problems. Springer Verlag,
Berlin-Heidelberg-Tokio-New York, 1984.
[3] K RA ČMAR S., N EUSTUPA J.: Global existence of weak solutions of a nonsteady variational
inequalities of the Navier-Stokes type with mixed boundary conditions. Proc. of the conference
ISNA’92, August-September 1992, Part III, Publ. Centre of the Charles Univ., Prague, 156-157.
[4] K RA ČMAR S., N EUSTUPA J.: Modelling of flows of a viscous incompressible fluid through a
channel by means of variational inequalities. ZAMM 74, 1994, 6, 637-639.
[5] K RA ČMAR S., N EUSTUPA J.: Some Initial Boundary Value Problems of the Navier-Stokes Type
with Mixed Boundary Conditions. Proc. of the seminar Numerical Mathematics in Theory and
Practise, Pilsen, 1993.
[6] K RA ČMAR S., N EUSTUPA J.: Simulation of Steady Flows through Channels by Variational Inequalities. Proc. of the conference Numerical Modelling in Continuum Mechanics. Prague 1994,
publ.1995.
[7] K RA ČMAR S., N EUSTUPA J.: A weak solvability of a steady variational inequality of the NavierStokes type with mixed boundary conditions. Proceedings of the Third World Congress of Nonlinear Analysts, Part 6 (Catania, 2000). Nonlinear Anal. 47 (2001), no. 6, 4169–4180.
[8] K U ČERA P., S KAL ÁK Z.: Solutions to the Navier-Stokes Equations with Mixed Boundary Conditions. Acta Applicandae Mathematicae, 275-288, Kluver Academic Publishers, 1998, vol. 54,
no. 3.
[9] K U ČERA P.: Solution of the Stationary Navier-Stokes Equations with Mixed Boundary Conditions in a Bounded Domain. Conference on Analysis, Numerics and Applications of Differential
and Integral Equations, University of Stuttgart, october 1996, (1998), Pitman Research Notes in
Mathematics Series n.379, Longman, 127-131
Non-steady Navier-Stokes Equations with Homogeneous Mixed Boundary Conditions
41
[10] K U ČERA P.: A structure of the set of critical points to the Navier-Stokes equations with mixed
boundary conditions. Proceedings of the conference Navier-Stokes equations: theory and numerical methods (Varenna, 1997), Pitman Res. Notes Math. Ser., 388, Longman, Harlow, (1998),
201–205. Editor: R.Salvi.
[11] K UFNER A., J OHN O., F U Č ÍK S.: Function spaces. Academia, Prague, 1977.
[12] L IONS J. L., M AGENES E.: Non-Homogeneous Boundary Value Problems and Applications.
Vols. I and II., Springer-Verlag, Berlin 1972.
[13] N E ČAS J., H LAV Á ČEK I.: Mathematical theory of elastic and elastico-plastic bodies: An introduction. Elsevier 1981, Amsterdam, Oxford, New York.
[14] R ANNACHER R.: Numerical analysis of the Navier-Stokes equations. Proceedings of conference
ISNA’92, part I., 1992, 361-380.
[15] TAYLOR A. E.: Introduction to functional analysis. Wiley, 1958.
[16] T EMAM R.: Navier-Stokes Equations, theory and numerical analysis, North-Holland Publishinged edition, Company, Amsterodam, New York, Oxford. Revis (1979).
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