On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid
In the present work, the properties of root elements of the problem on small motions of a viscous relaxing fluid completely filling a bounded domain are studied. A multiple p-basis property of special system of elements is proven for the case where the system is in weightlessness. The solution of th...
Gespeichert in:
| Datum: | 2017 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2017
|
| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/140583 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid / D. Zakora // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 402-413. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-140583 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1405832018-07-11T01:23:40Z On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid Zakora, D. In the present work, the properties of root elements of the problem on small motions of a viscous relaxing fluid completely filling a bounded domain are studied. A multiple p-basis property of special system of elements is proven for the case where the system is in weightlessness. The solution of the evolution problem is expanded with respect to the corresponding system. 2017 Article On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid / D. Zakora // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 402-413. — Бібліогр.: 6 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.04.402 Mathematics Subject Classification 2000: 45K05, 58C40, 76R99 http://dspace.nbuv.gov.ua/handle/123456789/140583 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In the present work, the properties of root elements of the problem on small motions of a viscous relaxing fluid completely filling a bounded domain are studied. A multiple p-basis property of special system of elements is proven for the case where the system is in weightlessness. The solution of the evolution problem is expanded with respect to the corresponding system. |
| format |
Article |
| author |
Zakora, D. |
| spellingShingle |
Zakora, D. On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid Журнал математической физики, анализа, геометрии |
| author_facet |
Zakora, D. |
| author_sort |
Zakora, D. |
| title |
On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid |
| title_short |
On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid |
| title_full |
On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid |
| title_fullStr |
On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid |
| title_full_unstemmed |
On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid |
| title_sort |
on properties of root elements in the problem on small motions of viscous relaxing fluid |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140583 |
| citation_txt |
On Properties of Root Elements in the Problem on Small Motions of Viscous Relaxing Fluid / D. Zakora // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 4. — С. 402-413. — Бібліогр.: 6 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT zakorad onpropertiesofrootelementsintheproblemonsmallmotionsofviscousrelaxingfluid |
| first_indexed |
2025-07-10T10:47:59Z |
| last_indexed |
2025-07-10T10:47:59Z |
| _version_ |
1837256657751506944 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No. 4, pp. 402–413
doi:10.15407/mag13.04.402
On Properties of Root Elements in the
Problem on Small Motions of Viscous
Relaxing Fluid
D. Zakora
Voronezh State University
1 University Sq., Voronezh 394006, Russia
E-mail: dmitry.zkr@gmail.com
Received October 20, 2015, revised May 11, 2016
In the present work, the properties of root elements of the problem on
small motions of a viscous relaxing fluid completely filling a bounded domain
are studied. A multiple p-basis property of special system of elements is
proven for the case where the system is in weightlessness. The solution of
the evolution problem is expanded with respect to the corresponding system.
Key words: viscous fluid, compressible fluid, basis.
Mathematical Subject Classification 2010: 45K05, 58C40, 76R99.
1. Introduction
This paper is connected with [1] and deals with the problem on small motions
of a viscous relaxing fluid completely filling a bounded domain. We study root
elements of the corresponding spectral problem. A multiple p-basis property of
special system of elements is proven for the case where the system is in weight-
lessness. The solution of an evolution problem is expanded with respect to the
corresponding system.
This work was supported by the grant of the Russian Foundation for Basic Research (project
no. 14-21-00066), Voronezh State University.
c© D. Zakora, 2017
On Properties of Root Elements . . . Small Motions of Viscous Relaxing Fluid
2. Statement of the Problem
2.1. Statement of the boundary-value problem. Consider a container
completely filled by a viscous inhomogeneous fluid. The fluid is said to fill a
bounded region Ω ⊂ R3. We introduce a system of coordinates Ox1x2x3 which
is toughly connected with the container so that the origin of coordinates is in
the region Ω. In what follows, we suppose the hydrodynamical system to be in
weightlessness.
The problem on small motions of a rotating viscous relaxing fluid is described
by the following initial-boundary value problem (see [1, 2]):
∂~u(t, x)
∂t
− ρ−1
0
(
µ∆~u(t, x) + (η + 3−1µ)∇ div ~u(t, x)
)
+ a∞ρ
−1/2
0 ∇ρ(t, x)−
m∑
l=1
∫ t
0
e−bl(t−s)a∞ρ
1/2
0 kl∇ρ(s, x) ds = ~f(t, x),
∂ρ(t, x)
∂t
+ a∞ρ
1/2
0 div ~u(t, x) = 0 (in Ω), ~u(t, x) = ~0 (on S := ∂Ω),
~u(0, x) = ~u0(x), ρ(0, x) = ρ0(x), (1)
where ρ0 = const is the given stationary density of the fluid, a∞ is the given
sound velocity, ~f(t, x) is a weak field of external forces, ~u(t, x) is the field of the
velocities in the fluid, a−1
∞ ρ
1/2
0 ρ(t, x) is the dynamic density of the fluid, µ and η
are the dynamic and the second viscosity of the fluid. The numbers b−1
l are used
as the times of relaxation in the system (0 < bl < bl+1 (l = 1,m− 1)), and kl >
0 are some structural constants.
2.2. Operator formulation of the problem. To pass to the operator
formulation of the problem, we introduce basic spaces and a number of operators
(see [2]).
Let us introduce a vector Hilbert space ~L2(Ω, ρ0) with the scalar product and
the norm
(~u,~v)~L2(Ω,ρ0) :=
∫
Ω
ρ0~u(x) · ~v(x) dΩ, ‖~u‖2~L2(Ω,ρ0)
=
∫
Ω
ρ0|~u(x)|2 dΩ.
We introduce a scalar Hilbert space L2(Ω) of the square summable functions
in the region Ω, and also its subspace L2,Ω :=
{
f ∈ L2(Ω) | (f, 1)L2(Ω) = 0
}
.
Lemma 1. The following statements hold.
1. Let the boundary S of the region Ω belong to the class C2. Consider a bound-
ary-value problem
−ρ−1
0
(
µ∆~u+ (η + 3−1µ)∇ div ~u
)
= ~w (in Ω), ~u = ~0 (on S).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 403
D. Zakora
For every field ~w ∈ ~L2(Ω, ρ0), the generalized solution of the problem given by
the formula ~u = A−1 ~w exists and it is unique. The operator A is selfadjoint
and positive definite in ~L2(Ω, ρ0). The operator A−1 belongs to the class Sp
(p > 3/2). In addition, D(A1/2) =
{
~u ∈ ~W 1
2 (Ω) | ~u = ~0 (on S)
}
.
2. Let us define an operator B~u := a∞ρ
1/2
0 div ~u, D(B) := D(A1/2). The ad-
joint operator B∗ρ = −a∞ρ−1/2
0 ∇ρ, D(B∗) = W 1
2 (Ω) ∩ L2,Ω, B∗BA−1 ∈
L(~L2(Ω, ρ0)).
3. The operator Q := BA−1/2 is bounded: Q ∈ L(~L2(Ω, ρ0), L2,Ω). The operator
Q+ := A−1/2B∗ admits extension to the bounded operator Q∗: Q+ = Q∗.
Using the operators introduced above, we can write problem (1) as a system
of two equations with initial-value conditions in a Hilbert space H0 := ~L2(Ω, ρ0)⊕
L2,Ω:
d~u
dt
+ 2ω0iS~u+A~u−B∗ρ+
m∑
l=1
∫ t
0
e−bl(t−s)B∗ρ0klρ(s) ds = ~f(t),
dρ
dt
+B~u = 0, (~u(0); ρ(0))τ = (~u0; ρ0)τ .
(2)
The upper index τ denotes the transposition.
Definition 1. A strong solution to problem (2) is said to be a strong solution
to initial-value problem (1). A function ζ(t) := (~u(t); ρ(t))τ is a strong solution to
problem (2) if ζ(t) ∈ D(A)⊕D(B∗) for all t ∈ R+, (A~u(t);B∗ρ(t))τ ∈ C(R+, H0),
(~u(t); ρ(t))τ ∈ C1(R+, H0), and ζ(t) satisfies (2) for all t ∈ R+ := [0,+∞).
2.3. Reduction to the first-order differential equation. Let us sup-
pose that problem (2) has a strong solution ~u(t), ρ(t) and ρ0 ∈ D(B∗). From
Lemma 1, it follows that ~u(t) and ρ(t) satisfy the system
d~u
dt
+A1/2
{
A1/2~u−Q∗
[
1−
m∑
l=1
ρ0kl
bl
]
ρ
−
m∑
l=1
Q∗
∫ t
0
e−bl(t−s)
ρ0kl
bl
dρ(s)
ds
ds
}
= ~f(t) +
m∑
l=1
e−blt
ρ0kl
bl
B∗ρ0,
dρ
dt
+QA1/2~u = 0, (~u(0); ρ(0))τ = (~u0; ρ0)τ .
(3)
In what follows, we suppose the physical parameters satisfy the condition
ϕ0 := 1−
m∑
l=1
ρ0kl
bl
> 0. (4)
404 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Properties of Root Elements . . . Small Motions of Viscous Relaxing Fluid
This condition supposes that the times of relaxation b−1
l and the structural con-
stants kl (l = 1,m) are sufficiently small.
Let us define the following objects connected with the function ρ(t):
ϕ
1/2
0 ρ(t) =: r(t),
∫ t
0
e−bl(t−s)
[
ρ0kl
bl
]1/2 dρ(s)
ds
ds =: rl(t) (l = 1,m). (5)
The functions r(t) and rl(t) (l = 1,m) are continuously differentiable. From
(3), it follows that these functions satisfy the system
d~u
dt
+A1/2
{
A1/2~u− ϕ1/2
0 Q∗r −
m∑
l=1
[
ρ0kl
bl
]1/2
Q∗rl
}
= ~f0(t),
dr
dt
+ ϕ
1/2
0 QA1/2~u = 0,
drl
dt
+
[
ρ0kl
bl
]1/2
QA1/2~u+ blrl = 0 (l = 1,m),
~f0(t) := ~f(t) +
m∑
l=1
e−blt
ρ0kl
bl
B∗ρ0.
(6)
Let us write system (6) as a first-order differential equation in a Hilbert space
H := H ⊕H0 (H := ~L2(Ω, ρ0), H0 := ⊕m+1
l=1 L2,Ω):
dξ
dt
+ Aξ = F(t), ξ(0) = ξ0, (7)
where
ξ := (~u;w)τ , w := (r; r1; . . . ; rm)τ , ξ0 := (~u0;w0)τ , (8)
w0 := (ϕ
1/2
0 ρ0; 0; . . . ; 0)τ , F(t) := (~f0(t); 0; . . . ; 0)τ . (9)
The operator A satisfies the following formulae:
A= diag
(
A1/2,I0
)( I Q∗
−Q G
)
diag
(
A1/2,I0
)
, (10)
A=
(
I 0
−QA−1/2 I0
)
diag (A, G+ QQ∗)
(
I A−1/2Q∗
0 I0
)
, (11)
D(A) =
{
ξ = (~u;w)τ ∈ H | ~u+A−1/2Q∗w ∈ D(A)
}
, (12)
where I and I0 are the identity operators in H and H0, respectively,
Q :=
(
−ϕ1/2
0 Q,−
[
ρ0k1
b1
]1/2
Q, . . . ,−
[
ρ0k1
bm
]1/2
Q
)τ
,
G := diag (0, b1I, . . . , bmI) .
From [1], it follows that A is a maximal sectorial (and accretive) operator.
From this and (7) one can obtain the theorem on strong solvability of problem (1).
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 405
D. Zakora
Theorem 1 (see [1]). Let the field ~f(t, x) satisfy the Hölder condition ∀τ ∈
R+ ∃K = K(τ) > 0, k(τ) ∈ (0, 1] :
∥∥∥~f(t)− ~f(s)
∥∥∥
~L2(Ω,ρ0)
≤ K|t− s|k for all 0 ≤
s, t ≤ τ . Then for any ~u0 ∈ D(A) and ρ0 ∈ D(B∗) there exists a unique strong
solution to initial-boundary value problem (1).
2.4. The basic spectral problems and the theorem on recalcula-
tion of root elements. We consider the spectral problem to the evolution
problem (7). Assuming F(t) ≡ 0, a dependence on time for the unknown func-
tion be of the form ξ(t) = exp(−λt)ξ, where λ is a spectral parameter and ξ is
an amplitude element, we have
Aξ = λξ, ξ ∈ D(A) ⊂ H. (13)
Let ξ = (~u;w)τ ∈ D(A). Changing the sought element in problem (13),
diag(A1/2,I0)ξ = ζ =: (~v;w)τ , using factorization (10), we have the following
spectral problem:
A(λ)ζ :=
(
I − λA−1 Q∗
−Q G− λ
)(
~v
w
)
= 0, ζ ∈ H= H ⊕H0. (14)
Let λ /∈ {0, b1, . . . , bm} = σ(G). From (14) it follows that
L(λ)~v :=
[
I − λA−1 + Q∗(G− λ)−1Q
]
~v = 0, ~v ∈ H. (15)
From [1], we obtain the theorem on the spectrum of the operator A.
Theorem 2 (see [1, Theorems 2, 4]). The following statements hold.
1. σess(A) = ΛE ∪ ΛL, where
ΛE :=
{
λ ∈ C | 1−
m∑
l=1
ρ0kl
bl − λ
= λ
4µ+ 3η
3a2
∞ρ0
}
,
ΛL :=
{
λ ∈ C | 1−
m∑
l=1
ρ0kl
bl − λ
= λ
7µ+ 3η
3a2
∞ρ0
}
.
The set C\σess(A) consists of regular points and isolated eigenvalues of finite
multiplicity of the operator A.
2. λ = 0 is not an eigenvalue of the operator A. If ‖A−1/2‖ < b
−1/2
q , then λ =
bq is not an eigenvalue of the operator A. Otherwise the point λ = bq can be
an eigenvalue of finite multiplicity of the operator A.
406 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Properties of Root Elements . . . Small Motions of Viscous Relaxing Fluid
3. If λ0 is a non-real eigenvalue of the operator A, then
γ1 :=
[
2‖A−1/2‖2
]−1
< Reλ0 < bm + ‖Q‖2 + ‖Q‖
(
bm + ‖Q‖2
)1/2
=: γ2,
|λ0|2 <
(
bm + 2‖Q‖2 + 2‖Q‖
(
bm + ‖Q‖2
)1/2) (
2bm + ‖Q‖2
)
.
The spectrum of the operator A is real if the following condition holds:
2
∥∥∥A−1/2
∥∥∥2
6
(
bm + ‖Q‖2 + ‖Q‖
(
bm + ‖Q‖2
)1/2)−1
. (16)
Let us introduce the following definition.
Definition 2 (see [3, p. 61]). Let λ0 and ~v0 be an eigenvalue and an eigenvec-
tor of the operator pencil L(λ), i.e., L(λ0)~v0 = 0. The elements ~v1, ~v2, . . . , ~vn−1
are said to be adjoint to ~v0 if
∑j
k=0(k!)−1L(k)(λ0)~vj−k = 0 (j = 1, 2, . . . , n − 1).
The number n is called the length of the chain ~v0, ~v1, . . . , ~vn−1 of root elements.
From [4], we obtain the following theorem on connection between root ele-
ments of the operator A and the operator pencil L(λ).
Theorem 3. Let {ξk = (~uk;wk)
τ}n−1
k=0 denote a chain of root elements of
the operator A and this chain corresponds to λ0 (λ0 6= 0, b1, . . . , bm). Then
{~vk}n−1
k=0 :=
{
A1/2~uk
}n−1
k=0
is a chain of root elements of the operator pencil L(λ)
and this chain corresponds to λ0.
Let {~vk}n−1
k=0 denote a chain of root elements of the operator pencil L(λ) and
this chain corresponds to λ0. Then
{
ξk = (A−1/2~vk;wk)
τ
}n−1
k=0
, where wk :=∑k
l=0(G− λ0)−(k−l+1)Q~vl, is a chain of root elements of the operator A.
3. A Multiple p-Basicity of Special System of Elements
3.1. The theorem on completeness and p-basicity of root elements
of the operator A. The following considerations are based on the theory of
spaces with indefinite metrics (see [5, 6]). Therefore we assume that H = H+ ⊕
H−, where H+ := H = ~L2(Ω, ρ0), H− := H0. We recall here some concepts and
facts of this theory.
Define J := diag(I,−I0) and introduce in H an indefinite scalar product
by the formula [ξ1, ξ2] := (Jζ1, ζ2)H = (~v1, ~v2)H+ − (w1, w2)H− . Denote by P+
and P− orthogonal projectors of the space H onto the subspace H+ and H−,
respectively: P+H= H+, P−H= H−.
A subspace L+ is called nonnegative if [ξ, ξ] > 0 for all ξ ∈ L+, and maximal
nonnegative (L+ ∈ M+) if it is not a principle part of any other nonnegative
subspace. In the same way, we can define nonpositive subspace L−.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 407
D. Zakora
From [5, p. 70]), it follows that L+ ∈ M+ if there exists a restriction K+ :
H+ → H− (‖K+‖ 6 1) such that L+ = {ξ = ξ+ +K+ξ+ : ξ+ ∈ H+}. This re-
striction is called an angular operator of the subspace L+.
A positive subspace L+ is said to be uniformly positive if it is a Hilbert space
with respect to the scalar product generated by the indefinite metric.
We say that the subspace L+ belongs to the class h+ if it can be repre-
sented as the J-orthogonal sum of the uniformly positive subspace and the finite-
dimensional neutral subspace. In particular, L+ ∈ h+ if K+ ∈ S∞ (see [5, p. 84]).
Let L± ∈M±. If L+ and L− are J-orthogonal, then we say that L+ and L−
form a dual pair {L+, L−}. We can write {L+, L−} ∈ h, if L± ∈ h±.
We say that a continuous J-selfadjoint operator B belongs to Helton’s class
(B ∈ (H)) if there exists at least one dual pair {L+, L−} ∈ h of invariant with
respect to B-subspaces and every B-invariant dual pair belongs to the class h.
Theorem 4. A−1 ∈ (H).
Proof. The Schur–Frobenius factorization (11) of the operator A and Theo-
rem 2 (0 ∈ ρ(A)) imply
A−1 =
(
I −A−1/2Q∗
0 I0
)
diag
(
A−1, (G+ QQ∗)−1
)( I 0
QA−1/2 I0
)
=
(
A−1 −A−1/2Q∗(G+ QQ∗)−1QA−1/2 −A−1/2Q∗(G+ QQ∗)−1
(G+ QQ∗)−1QA−1/2 (G+ QQ∗)−1
)
. (17)
The operator A−1 is J-selfadjoint and bounded. The compactness of the op-
erator A−1/2 implies the compactness of the operator P+A−1P−. From this
and [5, p. 287], it follows that the operator A−1 has a dual invariant pair
{L+(A−1), L−(A−1)}. Let K+ denote the angular operator of invariant non-
negative subspace L+(A−1). Then K+ : H+ → H−, ‖K+‖ 6 1, and
L+(A−1) = {(~u;w)τ ∈ H+ ⊕H− | (~u;w)τ = (~u;K+~u)τ , ~u ∈ H+} .
Let (~u1;w1)τ =(~u1;K+~u1)τ ∈L+(A−1). Then A−1(~u1;K+~u1)τ =(~u2;K+~u2)τ .
From this and (17), we deduce the equation for angular operator K+:
(G+ QQ∗)−1K+ =− (G+ QQ∗)−1QA−1/2
+K+
(
A−1 −A−1/2Q∗(G+ QQ∗)−1QA−1/2
)
−K+A
−1/2Q∗(G+ QQ∗)−1K+. (18)
From A−1/2 ∈ S∞, it follows that K+ ∈ S∞(H+).
408 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Properties of Root Elements . . . Small Motions of Viscous Relaxing Fluid
Remark 1. Theorem 4 is valid for the case where the system is in the gravi-
tational field. In particular, Theorem 4 implies that the nonreal spectrum of the
operator A consists of at most finite number of eigenvalues with regard for their
algebraic multiplicity (see [5, p. 245, Corollary 5.21]).
A system {ξk}∞k=1 is said to be a Riesz basis in a space H if ξk = Tζk, where
T,T−1 ∈ L(H), and {ζk}∞k=1 is an orthonormal basis in the space H. If T =
I+ K, where K ∈ Sp(H), then the system {ξk}∞k=1 is called a p-basis in the
space H.
A basis in the J-space H is said to be almost J-orthonormal if it can be
represented as a unity of a finite number of elements and a set of J-orthonormal
elements, and the sets are J-orthogonal to each other.
Let Lλ(A) denote a root subspace of the operator Awhich corresponds to the
eigenvalue λ (λ ∈ σp(A)). Let us also introduce the following notations:
F(A) := sp{Lλ(A) | λ ∈ σp(A)}, F0(A) := sp{Ker(A− λ) | λ ∈ σp(A)}.
We write λ ∈ s(A) ⊂ R if Ker(A−λ) is degenerate, i.e., there exists ξ0 ∈ Ker(A−
λ) such that [ξ0, ξ] = 0 for all ξ ∈ Ker(A− λ).
Theorem of T.Ya. Azizov (see [5, p. 271, Theorem 2.12]) implies the following
statement.
Theorem 5. The following statements hold.
1. codimF(A) 6 codimF0(A) <∞.
2. F(A) = H⇐⇒ sp{Lλ(A) | λ ∈ σess(A) ∩ (γ1, γ2)} is a non-degenerate sub-
space. The numbers γ1 and γ2 are defined in Theorem 2.
3. F0(A) = H⇐⇒ Lλ(A) = Ker(A− λ) as λ 6= λ and s(A) = ∅. If γ2 6 γ1
then F0(A) = H.
4. If F0(A) = H (F(A) = H), then the eigenelements (root elements) of the
operator A form an almost J-orthonormal p-basis (p > 3) in H. If γ2 6 γ1,
then the eigenelements of the operator A form a J-orthonormal basis in H.
Proof. Theorem 4 implies A−1 ∈ (H). From (18) and A−1 ∈ Sp (p > 3/2) it
follows that K+ ∈ Sp (p > 3). From Theorem 2 we conclude that the spectrum
of the operator A−1 has at most finite number of accumulation points. Therefore
the operator A−1 satisfies all assumptions of T.Ya. Azizov’s theorem.
1. From F(A) = F(A−1), F0(A) = F0(A−1) we obtain the first statement.
2. F(A−1) = H ⇐⇒ sp{Lλ−1(A−1) | λ−1 ∈ s(A−1)} is a non-degenerate
subspace. To prove F(A−1) = H, we should check that Lλ−1(A−1) is a non-
degenerate subspace only for those λ−1 ∈ s(A−1) which are accumulation points
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 409
D. Zakora
of the operator A−1 (see [5, p. 271, Remark 3.8]). From Lλ−1(A−1) = Lλ(A)
we have the following conclusion. To prove F(A−1) = H, we should check that
whether Lλ(A) is a non-degenerate subspace for λ ∈ σess(A) ∩ s(A).
Now we should determine where the set s(A) is positioned. Let λ0 = λ0 ∈
σp(A), λ0 /∈ σ(G), and Ker(A−λ0) be degenerated. By Theorem 3, it is equivalent
to the following. There exists ~v0 ∈ KerL(λ0) such that ξ0 = (A−1/2~v0; (G−
λ0)−1Q~v0)τ is J-orthogonal to all elements ξ = (A−1/2~v; (G− λ0)−1Q~v)τ , where
~v ∈ KerL(λ0), i.e., [ξ0, ξ] = 0. From the above it follows that (L′(λ0)~v0, ~v)H =
0. In particular, we have two equations: (L(λ0)~v0, ~v0)H = 0, (L′(λ0)~v0, ~v0)H = 0.
From this it follows that λ0 is a multiple root of the equation
1− λp− 1
λ
(
q −
m∑
l=1
ql
bl − λ
)
= 0, (19)
p :=
‖A−1/2~v0‖2
‖~v0‖2
, q :=
‖Q~v0‖2
‖~v0‖2
, ql := ρ0klq (l = 1,m).
Let us write (19) in the form
0 =
(
λ− λ2p− q
) m∏
l=1
(bl − λ) +
m∑
l=1
ql
m∏
k 6=l
(bk − λ) = −p(−1)mλm+2
+ (−1)mλm+1
[
1 + p
m∑
l=1
bl
]
− (−1)mλm
q +
m∑
l=1
bl + p
∑
i<j
bibj
+ · · · . (20)
Equation (19) has m real roots λl (λl ∈ (bl−1, bl), l = 1,m, b0 := 0) and a real
double root λ0. Then
0 = −p
m∏
l=1
(λl − λ)(λ− λ0)2 = −p(−1)mλm+2 + (−1)mλm+1p
[
2λ0 +
m∑
l=1
λl
]
− (−1)mλmp
λ2
0 + 2λ0
m∑
l=1
λl +
∑
i<j
λiλj
+ · · · . (21)
Let us compare the coefficients of λm+1 in (20), (21), and the coefficients of
λm in (20), (21). We have
2λ0 +
m∑
l=1
λl =
1
p
+
m∑
l=1
bl, (22)
λ2
0 + 2λ0
m∑
l=1
λl +
∑
i<j
λiλj =
q
p
+
1
p
m∑
l=1
bl +
∑
i<j
bibj . (23)
410 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Properties of Root Elements . . . Small Motions of Viscous Relaxing Fluid
From (22), it follows that 2λ0 = p−1 +
∑m
l=1(bl − λl) > p−1 > ‖A−1/2‖−2.
Hence λ0 >
[
2‖A−1/2‖2
]−1
= γ1.
In what follows, we use the ideas from [6, p. 378]. Define δ := 2−1
∑m
l=1(bl −
λl), ω := (2p)−1. Then λ0 = ω+ δ (see (22)). Extract
∑m
l=1 λl from (22) and use
this expression in (23). It follows that
2δ
m∑
l=1
λl −
∑
i<j
(bibj − λiλj) = −ω2 + 2ω(δ + q)− δ2. (24)
From λl ∈ (bl−1, bl), l = 1,m (b0 := 0), one can obtain the following inequality
(see [6, p. 380, Formula (5.24)]):
∑
i<j
(bibj − λiλj) <
m∑
j=1
(bj − λj)
(
m∑
i=1
λi
)
= 2δ
m∑
l=1
λl. (25)
From (25), it follows that the right-hand part in (24) is positive. Consequently,
ω < δ + q + (2δq + q2)1/2. Therefore, λ0 < 2δ + q + (2δq + q2)1/2 6 bm + ‖Q‖2 +
‖Q‖
[
bm + ‖Q‖2
]1/2
= γ2. Hence, λ0 ∈ (γ1, γ2).
Theorem 2 implies 0 /∈ σp(A). Hence, 0 /∈ s(A).
Let bq /∈ (γ1, γ2). Then from inequalities bq 6 bm < γ2 we obtain bq 6 γ1. Let
us suppose that bq 6 γ1 and Ker(A− bq) is degenerated. This means that there
exists ξ0 ∈ Ker(A− bq) such that [ξ0, ξ] = 0 for all ξ ∈ Ker(A− bq). In particular,
[ξ0, ξ0] = 0.
Let ξ0 ∈ Ker(A− bq). Then (10) implies
A1/2
[
A1/2~u0 − ϕ1/2
0 Q∗r0 −
m∑
l=1
[
ρ0kl
bl
]1/2
Q∗rl0
]
− bq~u0 = 0,
ϕ
1/2
0 QA1/2~u0 − bqr0 = 0,[
ρ0kl
bl
]1/2
QA1/2~u0 + (bl − bq)rl0 = 0, l = 1,m, l 6= q,[
ρ0kq
bq
]1/2
QA1/2~u0 = 0.
Multiplying the first equation by ~u0 and using the other equations, we re-
arrange it to the form ‖~v0‖2H = bq‖A−1/2~v0‖2H , where ~v0 := A1/2~u0. Hence bq >
‖A−1/2‖−2 > γ1, which contradicts bq 6 γ1.
From the above, it follows that s(A) ⊂ (γ1, γ2), and the second statement of
the theorem is proven.
3. The first parts of the statements 3 and 4 follow directly from [5, p. 271,
Theorem 2.12]. If γ2 6 γ1, then s(A) = ∅ and the operator A has no non-real
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 411
D. Zakora
eigenvalues. Consequently, F0(A) = H. Hence the eigenelements of the operator
A form a J-orthonormal basis in H.
3.2. Representation of solution of evolution equation. Let us as-
sume that γ2 6 γ1. Then from Theorem 5 it follows that the eigenelements of the
operator A form a J-orthonormal basis in H. This basis is also a p-basis (p > 3)
in H. By Theorem 3, this basis (after being divided into positive and negative
elements) can be represented in the form{
ξ±k :=
(
A−1/2~v±k ; (G− λ±k )−1Q~v±k
)τ}∞
k=1
,
ξ±k ∈ L±(A−1), [ξ+
k , ξ
+
j ] = δkj , [ξ−k , ξ
−
j ] = −δkj , [ξ+
k , ξ
−
j ] = 0. (26)
Let us represent the solution ξ(t) of problem (7) in the following form:
ξ(t) =
∞∑
k=1
c+
k (t)ξ+
k +
∞∑
j=1
c−j (t)ξ−j , c+
k (0) = [ξ0, ξ+
k ], c−j (0) = −[ξ0, ξ−j ]. (27)
From (7), (26), (27), the formulae for ξ0 and F(t), we find that
ξ(t) =
∞∑
k=1
(
e−λ
+
k t
[
ξ0, ξ+
k
]
+
∫ t
0
e−λ
+
k (t−s) [F(s), ξ+
k
]
ds
)
ξ+
k
−
∞∑
j=1
(
e−λ
−
j t
[
ξ0, ξ−j
]
+
∫ t
0
e−λ
−
j (t−s)
[
F(s), ξ−j
]
ds
)
ξ−j , (28)
[
ξ0, ξ±k
]
=
(
~u0, A−1/2~v±k
)
H
− ϕ0
λ±k
(
ρ0, Q~v±k
)
L2,Ω
, k ∈ N,
[F(t), ξ±k ] =
(
~f(t), A−1/2~v±k
)
H
+
m∑
l=1
ρ0kle
−blt
bl
(
ρ0, Q~v±k
)
L2,Ω
, k ∈ N.
From (28), we obtain the representation of the solution ~u(t), ρ(t) of prob-
lem (2) with respect to a system of eigenvectors {~v±k }
+∞
k=1 of the operator pencil
L(λ) (this system is connected with J-orthonormal basis (26) and is normalized
in a special way):(
~u(t)
ρ(t)
)
=
∞∑
k=1
[
T+
k (t)
(
A−1/2~v+
k
(λ+
k )−1Q~v+
k
)
− T−k (t)
(
A−1/2~v−k
(λ−k )−1Q~v−k
)]
,
T±k (t) :=
∫ t
0
e−λ
±
k (t−s)
(
~f(s), A−1/2~v±k
)
H
ds+ e−λ
±
k t
(
~u0, A−1/2~v±k
)
H
+
[
m∑
l=1
ρ0kl(λ
±
k e
−blt − ble−λ
±
k t)
λ±k bl(λ
±
k − bl)
− e−λ
±
k t
λ±k
] (
ρ0, Q~v±k
)
L2,Ω
.
The author thanks Professor N.D. Kopachevsky for attention to this work
and the anonymous referee for useful suggestions.
412 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4
On Properties of Root Elements . . . Small Motions of Viscous Relaxing Fluid
References
[1] D. Zakora, On the Spectrum of Rotating Viscous Relaxing Fluid, Zh. Mat. Fiz.
Anal. Geom. 12 (2016), No. 4, 338–358.
[2] D. Zakora, A Symmetric Model of Viscous Relaxing Fluid. An Evolution Problem,
Zh. Mat. Fiz. Anal. Geom. 8 (2012), No. 2, 190–206.
[3] A.S. Marcus, Introduction to Spectral Theory of Polinomial Operator Pencils, Shti-
inca, Kishenev, 1986 (Russian).
[4] D.A. Zakora, Operator Approach to Ilushin’s Model of Viscoelastic Body of
Parabolic Type, Sovrem. Mat. Fundam. Napravl. 57 (2015), 31–64 (Russian); Engl.
transl.: J. Math. Sci. (N.Y.) 225 (2015), No. 2, 345–380.
[5] T.Ya. Azizov and I.S. Iohvidov, Basic Operator Theory in Spaces with Indefinite
Metrics, Nauka, Moscow, 1986 (Russian).
[6] N.D. Kopachevsky and S.G. Krein, Operator Approach to Linear Problems of Hy-
drodynamics. Vol. 2: Nonself-Adjoint Problems for Viscous Fluids, Birkhäuser Ver-
lag, Basel–Boston–Berlin, 2003.
Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 4 413
|