Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems
We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the regularity of solutions of nonlinear differential equations in the case where the classical Cauchy-Liouville-Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the boundedne...
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| author | Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. |
| author_facet | Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. |
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| description | We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the regularity of solutions of nonlinear differential equations in the case where the classical Cauchy-Liouville-Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the boundedness and continuity of variations with respect to initial data and discuss their applications to the dynamics of unbounded lattice Gibbs models. |
| first_indexed | 2026-03-24T02:43:18Z |
| format | Article |
| fulltext |
UDC 517.9
A. Val. Antoniouk, A. Vict. Antoniouk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
NONLINEAR-ESTIMATE APPROACH
TO THE REGULARITY OF INFINITE-DIMENSIONAL
PARABOLIC PROBLEMS*
PIDXID DO REHULQRNOSTI NESKINÇENNOVYMIRNYX
PARABOLIÇNYX ZADAÇ, WO ÌRUNTU{T|SQ
NA NELINIJNYX OCINKAX
We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the
regularity of solutions of nonlinear differential equations in the case where the classical Cauchy –
Liouville – Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the
boundedness and continuity of variations with respect to initial data and discuss their applications to the
dynamics of unbounded lattice Gibbs models.
Pokazano, qkym çynom zastosuvannq nelinijnyx symetrij poxidnyx vysokoho porqdku dozvolq[
vyvçaty rehulqrnist\ rozv’qzkiv nelinijnyx dyferencial\nyx rivnqn\ u vypadku, koly klasyçnu
sxemu Koßi – Liuvillq – Pikara nemoΩlyvo zastosuvaty. Zokrema, otrymano nelinijni ocinky na
obmeΩenist\ ta neperervnist\ variacij za poçatkovymy umovamy i rozhlqnuto ]x zastosuvannq do
dynamiky neobmeΩenyx ©ratkovyx hibbsivs\kyx system.
1. Statement of problem: symmetries of variational equations and regularity
schemes in the non-Lipschitz case. The problem of correct definition of differential
operators is naturally linked with the construction of the functional spaces of their
action and study of the associated regularity problems. There is one classical regularity
scheme, usually attributed to Cauchy, Liouville, and Picard, that demonstrates how the
Lipschitz assumptions on coefficients of equation lead to the regular dependence of
solutions with respect to the initial data and parameters of different kinds [1 – 4].
Consider the first order differential equation
yt
(
x
) = x –
0
t
sF y x ds∫ ( )( ) (1)
with nonlinear drift F such that
∃ Kn :
sup ( )( )
x
nF x
∈R
1
≤ Kn
,
in particular, F is Lipschitz continuous, i.e., | F
(
x
) – F
( y
) | ≤ K1
| x – y |. In adaptation
to (1), the Cauchy – Liouville – Picard scheme applies the fixed point arguments to the
construction of the unique solution to (1) via the following iteration formula:
y xt
n+1( ) = x –
0
t
s
nF y ds∫ ( ) , n ≥ 1.
In a similar way, the implicit function arguments
Φ( , )x y t{ } = yt
– x +
0
t
sF y ds∫ ( ) = 0
* Research is partially supported by theme 24 Institute of Mathematics NAS Ukraine.
© A. Val. ANTONIOUK, A. Vict. ANTONIOUK, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, #5 579
580 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
and the estimate ′ −Φy 1 ≤ t
K1 ≤ ε for small t lead to the first order differentiability
with respect to the initial data: ∃
∂
∂x
y xt ( ) = – ′[ ] ′Φ Φy x .
The higher-order differentiability of yt
(
x
) on initial data x for small t > 0 is
derived from the theorem on the differentiability of implicit function. Finally, due to
the semigroup property of flow yt + s
(
x
) = yt
( ys
(
x
)
), solution yt
(
x
) is regular with
respect to the initial data for all t ≥ 0.
In application to the associated semigroup (
Pt f )
(
x
) = f ( yt (
x
)
), the Cauchy –
Liouville – Picard scheme leads to the quasicontractive regularity estimates of any
order:
Pt L Cb
n( ) ≤ exp , ,tMK Kn1 …( ) (2)
in the standard spaces of continuously differentiable functions with bounded
derivatives Cb
n . This scheme admits natural generalizations to the infinite-dimensional
Banach space X and, in fact, inspired to a great degree the development of modern
functional analysis. In particular, the concept of full metric space and Browder index
theorems were closely related with the fixed point results [1 – 4]. In a similar way, the
development of infinite-dimensional analogies of implicit function theorems inspired
the study of differentiability in spaces of Frechet differentiable functions and the
interpretation of variations as Frechet derivatives y xt
n( )( ) ∈ Bn = L
(
X, Bn – 1
), B0 = X.
However, it is still a question what happens, when the map F is essentially
nonlinear, i.e., does not have bounded derivatives and, therefore, is no more globally
Lipschitz. In this case, the fixed point arguments and implicit function techniques
could not be applied because of unbounded derivatives F
(
j
)
. One can also construct a
counter-example [5] (Chapter 1.2), which demonstrates that spaces Cb
n of
continuously differentiable functions with bounded derivatives are not preserved by
non-Lipschitz semigroups.
What techniques can be introduced in the essentially nonlinear case and how to
work with the C
∞
regularity problems in this case is the main question of this article.
The principal idea lies in the backgrounds of differentiable calculus. Because
nonlinearity means nonlinear responses to linear operations, like derivative, let us
consider the n th derivative of nonlinear function
F y n( ) ( )[ ] = ′[ ] −
F y y
n
( ) ( ) ( )1 1
=
= ′F y y n( ) ( ) +
j j n s n
s j j
s
sF y y y
1
1
2 1+…+ = = −
∑ …
, ,
( ) ( ) ( )( ) + F y yn n( ) ( )( ) 1[ ] . (3)
The main observation is that the right-hand side of (3) contains simultaneously n th
order derivative y
(
n
) and first derivative in n th degree y
n( )1[ ] . Similar symmetry is
also reflected in the intermediate terms
y yj js( ) ( )1 … ∼ y
j( )1 1[ ] … y
js( )1[ ] ∼ y
n( )1[ ] because j1 + … + js = n. (4)
Symmetry (4) has immediate consequences for the nonlinear evolutional equations
like (1). Consider the higher-order derivative of process yt
x with respect to the initial
data
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 581
yt
n( ) =
∂
∂
n
n t
x
x
y = i. d. –
0
t n
n t
x
x
F y dt∫ ( )∂
∂
=
= i. d. –
0 11
1
t
j j n s
s
t
x
t
j
t
j
s
sF y y y dt∫ ∑
+…+ = ≥
( ) …[ ]
,
( ) ( ) ( ), , , (5)
where H x h ha
a
( )( ) , ,1 …[ ] means a th order directional derivative at point x ∈ R
n.
For equation (5), variation y n( ) on the left-hand side is proportional to the first
variation y
(
1
) in n th power on the right-hand side, or, after taking the n th root,
y
(
1
) ≈ y n n( )[ ] /1
. (6)
This property becomes fundamental. It appears that the knowledge of symmetry (6)
is sufficient for the study of regularity in the essentially non-Lipschitz case. Let us
introduce the expression which is homogeneous with respect to the symmetry (6):
ρn
( y, t
) =
j
n
j t
x
t
j m j
p y y
=
∑ ( ) /
1
2 ( ) . (7)
Due to yt
j( ) = ∂x
j
t
xy( ) , this expression reflects the regularity of solution yt
(
x
) with
respect to the initial data x. The main statement is that there is a hierarchy of weights
{
p
}, determined by nonlinearity parameters of map F, that leads to quasicontractive a
priori estimate on regularity.
In the following sections, we consider an example of infinite dimensional model,
for which it could be done a sufficiently detail research of regularity properties. In
particular, we obtain a set of nonlinear estimates on the a priori boundedness and
continuity of variations for dynamics of such models and derive applications to the
C
∞
-properties of associated parabolic evolutions. More motivational details and
peculiarities of research of these models are discussed in [5 – 8].
2. Influence of nonlinearity on the properties of infinite-dimensional systems.
Notations and preliminary results. We discuss below an important infinite-
dimensional model, that describes the lattice approximations of multidimensional
Euclidean :P
(
ϕ
):d -field theory [9, 10]. It corresponds to the formal Gibbs measure
d
µ
(
x
) =
1 1
2
0
Z
b x x e d x
j k j k r
k j k j
k
F x
k
d
kexp
, :
( )−
− ≤
−
∈
−∑ ∏
Z
and its kinetic energy operator H, defined by integration by parts via
( , )
,
Hu u
L
d
2 R
Z µ( ) =
k kd d x
u x d x
∈
∑ ∫ ∂
∂
Z
R
Z
( ) ( )
2
µ ,
has the form
H =
k
k k k
kd
F x Bx
x∈
∑ − + ′ +[ ] ∂
∂
Z
∆ ( ) ( ) .
The corresponding semigroup Pt f (
x
(
0
)
) = E f xtξ( ) ( )0 0( )( ) is defined in terms of
solutions to the infinite system of stochastic differential equations
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
582 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
d t xkξ( ) ( ),0 0( ) = 2 dW tk ( ) – F t x B t x dtk k
ξ ξ( ) ( ) ( ) ( ), ,0 0 0 0( )( ) + ( )( ){ } ,
(8)
ξk x( ) ( ),0 00( ) = xk
( )0 , k ∈ Z
d,
where, for the probability space (
Ω, F, P
) with filtration Ft
, the process W
(
t
) =
= W tk k d( ){ } ∈Z
, t ≥ 0, is an Ft-adapted Wiener process defined on Ω with values in
l2
(
a
) = l2
(
a, Z
d
),
k kd a∈∑ Z
= 1, and identity covariance operator. The linear finite-
diagonal map B : RZ
d
→ RZ
d
is defined by
B x =
j j k r
j
k
b k j x
d:
( )
− ≤ ∈
∑ −
0 Z
and the nonlinear map F : RZ
d
� x → F
(
x
) = F xk k d( ){ } ∈Z
∈ R
Z
d
is generated by
the C
∞
monotone function F, F
(
0
) = 0, which satisfies the condition of polynomial
growth at the infinity:
∃ k ≥ – 1 ∀ i ≥ 1 :
F x F yi i( ) ( )( ) ( )− ≤ C x y x yi − + +( )1 k . (9)
To study the regular properties of semigroup Pt
, we need to write the
representation for derivatives of semigroup ∂τ
Pt f : for τ = { j1
, … , jm
} and ∂τ =
= ∂ …∂/τ ∂x xj jm1
,
∂τ
(
Pt f )
(
x
(
0
)
) =
s
m
s
s
s
f
= … =
∑ ∑ ∂ ( ) ⊗…⊗
1
0
1
1
γ γ τ
γ γξ ξ ξ
∪ ∪
E ( ) ( ) , , (10)
where ∂( )s f = ∂γ γ
f
s{ } =
denotes the set of s th order partial derivatives of
function. We also use the notation
∂ ( ) ⊗…⊗( ) ( ) ,s f
s
ξ ξ ξγ γ
0
1
=
j j
j j j j
s
d
s s s
f
1
1 1 1
0
, ,
, ,
( )
, ,
… ∈
…{ }∑ ∂( )( ) …
Z
ξ ξ ξγ γ .
The equation on variational process
ξτ = ξ
ξ
∂ ∂τ
τ
k
k
j j k
t x
x x
n d
,
( ) ( )
( ) ( )
( , )
=
∂
…
∈
0 0
0 0
1 Z
is derived by the formal successive differentiation of (8) with respect to x( )0 :
d
dt
kξ τ, = – ′( )F k kξ ξ τ
( )
,
0 –
j j k r
jb k j
:
,( )
− ≤
∑ −
0
ξ τ – ϕk, τ
,
(11)
ξk, τ
(0) =
δ τ τ
τ
k j
dj, , , ,
, ,
= = { } ⊂
>
1
0 1
Z
where ϕk, τ = ϕk, τ
(
ξ
(
0
), ξ⋅, γ
, γ ⊂ τ, γ ≠ τ
) equals to
ϕk, τ =
γ γ τ
γ γξ ξ ξ
1
1
2
0
∪ ∪… = ≥
∑ ( ) …
s
s
s
s
k kF
,
( ) ( )
, , . (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 583
In (12), the summation
γ γ τ1 2∪ ∪… = ≥
∑
s s,
is made over all possible subdivisions of the
set τ = { j1
, … , jn }, ji ∈ Z
d, on the nonintersecting subsets γ1
, … , γs ⊂ τ, with
| γ1 | + … + | γs | = | τ |, s ≥ 2, | γi | ≥ 1.
Let us introduce nonlinear expression, which reflects the symmetries of variations
for lattice models:
ρτ
(
ξ; t
) = E
s
n
s t
s
l c
m
p z
m= ⊂ =
∑ ∑
1
( )
,
( )
γ τ γ
γξ
γ γ
γ , (13)
where τ = { j1
, … , jn }, ji ∈ Z
d, p s are polynomial functions depending on zt =
= ξ( ) ( )
( )
,0 0 2
2
t x
l a( ) and mγ = m1 / | γ |, | γ | is a number of points in the set γ ∈ Z
d.
The set of all vectors c = ck k d{ } ∈Z
such that δc = sup k j k jc c− = /1 < ∞ is denoted
by P.
Theorem 1. Let F satisfy (9), x
(
0
) ∈ l a
2 1 2( )
( )
k+ ,
k kd a∈∑ Z
= 1, a ∈ P,
xγ ∈ l dcmγ γ( ) , γ ⊂ τ, d ≥ a m− + /( )k 1 21 , mγ = m1 / | γ |, m 1 ≥ | τ | and ξ
(
0
), {
ξ
γ
}γ ⊂ τ
form the strong solutions to systems (8), (11). Suppose that functions pi
(
z
), i =
= 1, … , n, and vectors {
c
γ
}γ ⊂ τ ⊂ P in (13) satisfy the following conditions:
1) ∃ ε > 0, ∃ K > 0 such that
∀ z ∈ R+ : pi (
z
) ≥ ε, ( ) ( ) ( )1 + ′ + ′′( )z p z p zi i ≤ K
pi
(
z
); (14)
2) ∃ Kp ∀ j = 2, … , n ∀ i1
, … , is
, i1 + … + is = j, s ≥ 2:
p z zj
j m( ) ( )( )[ ] + + /1 1 21k ≤ K p z p zp i
i
i
i
s
s
1
1( ) ( )[ ] …[ ] ; (15)
3) for any subdivision of the set γ = α1 ∪ … ∪ α s
, γ ⊂ τ on nonempty
nonintersecting subsets α1
, … , αs
, s ≥ 2, there is R
sγ α α, , ,1 … such that
∀ k ∈ Z
d :
c ak k
m
,
( )
γ
γ[ ] − + /k 1 21 ≤ R c c
s s
s
k kγ α α α
α
α
α
, , , , ,1 1
1
… [ ] …[ ] . (16)
Then there is a constant M ∈ R
1 such that the nonlinear quasicontractive
estimate
ρτ
(
ξ; t
) ≤ e
M
t
ρτ
(
ξ; 0
) (17)
holds.
Proof of this result may be found in [5, 6] . In Theorem 3, we give a more general
result, which implies, in particular, estimate (17).
As a consequence of nonlinear estimate, we can obtain regular properties of
semigroup Pt in the spaces of continuously differentiable functions.
Denote by Lipr
(
l
2
(
a
)
) the space of continuous functions on the space
l
2
(
a
) ⊂
⊂ RZ
d
, equipped with norm
f
rLip = sup
( )
( ) ( )x l a
l a
r
f x
x∈
+
+( )2
2
1
1 +
+ sup
( ) ( )
, ( ) ( ) ( ) ( )x y l a
l a l a l a
r
f x f y
x y x y∈
−
− + +( )2
2 2 2
1
< ∞. (18)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
584 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
For some m ∈ N, denote by Θ
m the array of pairs
p p m, : ,G G( ) ( ) ∈{ }Θ , where
G = G
1 ⊗ … ⊗ G
m is m-tensor constructed by vectors G
i ∈ P , i = 1, … , m, and p
is a smooth function of polynomial behavior (14).
Definition 1. The array Θ = Θ
1 ∪ … ∪ Θ
n, n ∈ N, is quasicontractive with
parameter k iff for any m = 2, … , n, for any (
p, G
) ∈ Θ
m , and all i, j ∈
∈ {
1, … , m
}, i ≠ j, there exists a pair
˜, ˜p G( ) ∈ Θ
m
–
1 such that
∃ K ∀ z ∈ R+
1
: ( ) ˜( )( )1 1 2+ + /z p zk ≤ K
p
(
z
),
ˆ ,G i j l{ }( ) ≤ K G̃l , l = 1, … , m – 1. (19)
Above (
m – 1
)-tensor ̂
,G i j{ } is constructed from m-tensor G by the rule
̂
,G i j{ } =
G A G G G
i j
i j m1 1⊗ ⊗ ⊗ ⊗− +
↑
… …
ˆ
( )( ) k x.
The notation
G G
i
s1 ⊗ ⊗…̂ means that the i th vector is omitted in tensor product and
G B G
j
s1 ⊗ ⊗ ⊗ ⊗
↑
… … means that the vector B is inserted on j th place in tensor
product. Inequality (19) is understood as a coordinate inequality between two vectors.
For multifunction of m th order u
(
m
)
(
x
) = u x k k km i
d
τ τ( ), , , ,= …{ } ∈{ }1 Z ,
x ∈ l
2
(
a
), we introduce the seminorm
u m
m
( )
Θ
=
sup sup
( )
( ) ,
( )
( )x l a p
m
m l am
m m
mu x
p x∈ ( )∈ ( )2 2
2
G
G
Θ
(20)
with
u xm
m
( )( )
G
2
= τ τ= …{ }⊂∑ …
j j j j
m
m
d m
G G u x
1 1
1 2
, ,
( )
Z
for G
m = G
1 ⊗ … ⊗ G
m.
Let r ≥ 0, n ≥ 1, and let Θ = Θ
1 ∪ … ∪ Θ
n be a quasicontractive array with
parameter k. We say that a function f belongs to the space CΘ, r
(
l
2
(
a
)
) iff
f ∈ Lipr
(
l
2
(
a
)
) and the following statement are true:
1) there is a set of partial derivatives ∂ ∂( ) ( ), ,1 f fn…{ } such that for any m ∈
∈ {1, … , n
} the coordinates of multifunctions ∂( ) ( )m
j j
f x
m
{ } …1
= ∂τ f x( ) , τ =
= { j1
, … , jm
} are continuous: ∂τ f ∈ C
(
l
2
(
a
), R
1
), and for all x
(0) ∈ l
2
(
a
), h ∈
∈ X∞
(
[
a, b
]
) the equalities
f x h
a
b( ) ( )0 + ⋅( ) =
a
b
k
k kds f x h s h s
d
∫ ∑
∈
∂ +( ) ′
Z
( ) ( ) ( )0 (21)
and for τ = { j1
, … , jl }, | τ | = l ≤ n – 1:
∂ + ⋅( )τ f x h
a
b( ) ( )0 =
a
b
k
k kds f x h s h s
d
∫ ∑
∈
{ }∂ +( ) ′
Z
τ∪
( ) ( ) ( )0 , (22)
hold;
2) the norm is finite:
f C rΘ,
= f
rLip + max
,
( )
m n
m f m
=1
∂
Θ
< ∞. (23)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 585
The space X
∞
(
[
a, b
]
) is defined as
X
∞
(
[
a, b
]
) =
p c
pAC a b l c
≥ ∈
∞ [ ]( )
1,
, , ( )
P
∩ (24)
and A
C
∞
(
[
a, b
], X
) = h C a b X h L a b X∈ [ ]( ) ∃ ′ ∈ [ ]( ){ }∞, , : , , for the Banach space X.
The following theorem states the regular properties of semigroup Pt in the scale
CΘ, r
.
Theorem 2. Let F satisfy (9) and let Θ be a quasicontractive array with
parameter k. Then, for any t ≥ 0, P t : CΘ, r → CΘ , r and there exist constants
KΘ, r
, MΘ, r such that
∀f ∈ CΘ, r : P ft C rΘ,
≤ K e fr
M t
C
r
rΘ
Θ
Θ,
,
,
.
Proof of this result is quite complicated and may be found in [5, 6].
The relation between variations and derivatives of a semigroup
∂( ) ( )n
tP f x =
j j n s
s
t x t x
j
t x
j
s
sf
1
1
1
0
+…+ = ≥
∑ ( ) …
,
( )
, ,
( )
,
( )E ξ ξ ξ (25)
is applied as a main tool to obtain the quasicontractive estimates on the regularity of
semigroup from estimates (17).
3. Main result: non-Lipschitz gap between boundedness and continuity. In this
section, we discuss a principally infinite-dimensional effect inherent for the infinite-
dimensional nonlinear evolutions.
First, recall that the classical Cauchy – Liouville – Picard scheme for equations with
coefficients with bounded derivatives implies C
∞-properties of semigroups in the
spaces of smooth functions with the same topology of boundedness and continuity of
derivatives: for a function f such that
∀ i = 1, … , n ∀ x, y ∈ B0 :
∂( ) ( )i f x
iB
≤ K,
∂ − ∂( ) ( )( ) ( )i if x f y
iB
≤ K x y− B0
,
the consideration of the semigroup ( )( )P f xt = f y xt ( )( ) , generated by Lipschitz
differential flow y xt ( ) = x –
0
t
sF y x ds∫ ( )( ) , gives no more that exponential growth of
constants in the recurrently defined sequence of spaces Bi = L
(
B0
, B i – 1
) over the
Banach space B0
:
∃ M ∀ t ≥ 0 ∀ i = 1, … , n ∀ x, y ∈ B0 :
∂( ) ( )( )i
tP f x
iB
≤ K
e
M
t,
∂ − ∂( ) ( )( )( ) ( )( )i
t
i
tP f x P f y
iB
≤ Ke x yMt − B0
.
In other words, in terms of the norm
f nC = max sup ( ) , sup
( ) ( )
, ,
( )
,
( ) ( )
i n x
i
x y
i i
f x
f x f y
x yi
i
= … ∈ ∈
∂
∂ − ∂
−
0
0 0 0B B B
B
B
,
there is a quasicontractive property:
∃ M ∀ f ∈ C
n
: P ft nC ≤ e
M
t f nC .
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586 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
Below we show that the non-Lipschitz coefficients in equation cause the
depending on the nonlinearity parameters gap between the topologies of the
boundedness and continuity of semigroup derivatives. We study the action of
semigroup Pt in spaces equipped with topology
f nE =
max sup
( )
, sup
( ) ( )
, ,
( )
,
( ) ( )
˜
j n x
j
j x y
j j
j
f x
q x
f x f y
x y p x y
j j
= …
∂
( )
∂ − ∂
− +( )
0
B B
. (26)
We show that the study of smooth properties of non-Lipschitz semigroup, in particular,
the quasicontractive estimates
∃ M =
M nE ∀ f ∈ E
n
: P ft nE ≤ e
M
t
f nE
requires to introduce a gap between the topologies of boundedness and continuity
pj
(
z
) = Polk
(
z
) ⋅ qj
(
z
), B̃ j = Ck Bj
depending on the non-Lipschitz order k of F. Note that the operator gap Ck displays
an essentially infinite-dimensional effect, because the finite-dimensional norms are
equivalent.
Let us shortly discuss the key idea. Due to representation (25), the continuity of
variations, i.e., estimates on ξt x
j
,
( ) – ξt y
j
,
( ) for solutions of (11), is necessary in order to
control the continuity of semigroup derivatives. The principal part of equation in
ξt x
j
,
( ) – ξt y
j
,
( ):
d
dt t x
j
t y
jξ ξ,
( )
,
( )−( ) = –
′( ) + ′( )[ ] −( )F Ft x t y
t x
j
t y
jξ ξ
ξ ξ, ,
,
( )
,
( )
0 0
2
–
–
′( ) − ′( )[ ] +( )F Ft x t y
t x
j
t y
jξ ξ
ξ ξ, ,
,
( )
,
( )
0 0
2
+ …
points on the similarity of behaviour,
ξ ξ
ξ ξ
t x
j
t y
j
Y
t x t y
j
,
( )
,
( )
, ,
−
−0 0 ∼ ξt x
j
X j
,
( ) + ξt y
j
X j
,
( ) , (27)
due to the relation ′( ) − ′( )( )F Ft x t yξ ξ, ,
0 0 ∼ ξ ξt x t y, ,
0 0−( ) up to accuracy of some
polynomial ξ ξt x t y, ,,0 0( ) factor.
Let pγ
, qγ ∈ C
∞
(
R+
), γ ⊂ τ, be positive monotone functions of polynomial
behaviour, i.e., such that
∃ ε > 0 ∀ z ∈ R+ : pγ
(
z
) ≥ ε, ′p zγ ( ) ≥ ε,
∃ C > 0 : ( ) ( )1 + ′′z p zγ ≤ C ′p zγ ( ),
(
1 + z
) ′p zγ ( ) ≤ C
pγ
(
z
). (28)
Let us introduce a generalization of expression (13), reflecting the higher-order
symmetries of variations
ρ ξ ξτ
b x y t( , ; ) =
γ τ
δ
γ γ γξ ξ ξ ξ
γ γγ
γ
γ
γ
⊂
∅ ∅∑ − ( ) +
E x y
l a t
x y x
l c
m y
l c
m
p n
m m2 ( )
,
( ) ( )
, (29)
where we use the notation nt
x y, for the sum of norms,
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NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 587
nt
x y, = ξ∅
x
l a
t( )
( )2
2
+ ξ∅
y
l a
t( )
( )2
2
,
with corresponding sense of nx y
0
, = x l a2
2
( ) + y l a2
2
( ) .
Then we write the continuity part
ρ ξ ξτ
c x y t( , ; ) =
γ τ
δ
γ γ γξ ξ ξ ξγ
γ
γ
γ
γ
⊂
∅ ∅
−
( )∑ − ( ) − + /E k
x y
l a
m
t
x y x y
l a c
m
q n
m
m
2
1 2( )
,
( ) (30)
with the multiple ξ ξ
δ γ
∅ ∅
−
−x y m
, that reflects the similarity of behavior (27).
Introduce nonlinear expression
ρ ξ ξτ( , ; )x y t = ρ ξ ξτ
b x y t( , ; ) + ρ ξ ξτ
c x y t( , ; )
for the joint boundedness and continuity of variations.
In the following theorem, we obtain the estimates on the continuous dependence of
solutions ξγ
x of variational equations (11) with respect to the initial data x in terms of
quasicontractive behavior of expression ρ ξ ξτ( , ; )x y t .
Theorem 3. Let F satisfy (9), δ ≥ m1 ≥ | τ |, and let ξ∅
x , ξ∅
y , ξγ
x , ξγ
y , γ ⊂ τ,
be generalized solutions to (8) and (11) with initial data x, y ∈ l
2
(
a
) and x γ
,
yγ ∈ l dcmγ γ( ) , dk ≥ ak
m− + /( )k 1 21 , correspondingly.
Suppose that
{
cγ
, γ ⊂ τ
} satisfy hierarchy (16);
functions {
pγ
, γ ⊂ τ
} of polynomial behaviour satisfy the relations: ∀α1 ∪ …
… ∪ αs = γ ⊂ τ, s ≥ 2 ∃ Kp
:
p z z m
γ
γ
( ) ( )( )[ ] + + /1 1 21k ≤ K p z p zp s
s
α
α
α
α
1
1( ) ( )[ ] …[ ] , z ∈ R+
; (31)
functions of polynomial behaviour {
qγ
, γ ⊂ τ
} are such that
q z z
m
γ
γ( )( )1
2+ /k
= pγ
(
z
).
Then there exists Mγ such that
ρ ξ ξτ( , ; )x y t = eM t x yτ ρ ξ ξτ( , ; )0 . (32)
Proof. This theorem is proved by induction on the number of points in the set τ.
First, we suppose that the initial data x, y ∈ l a
2 1 2( )
( )
k+ , i.e., ξ∅
x , ξ∅
y and ξγ
x , ξγ
y
form strong solutions to (8) and (11).
Introduce notation for i = 1, … , | τ |:
h ti x y
τ ξ ξ( , ; ) =
0 0
1
, ,
( , ; ) ( , ; ) , ,
,
i
g t g t i
i
b x y c x y
=
+[ ] ≥
⊂ ≤
∑
γ τ γ
γ γξ ξ ξ ξ
where
g tb
γ ( ) = E ξ ξ ξ ξ
δ
γ γ γ
γ γ
γ
γ γ
γ
∅ ∅− ( ) +
x y
l a t
x y x
l c
m y
l c
m
p n
m m2 ( )
,
( ) ( )
,
g tc
γ ( ) = E kξ ξ ξ ξ
δ
γ γ γ
γ
γ γ
γ
γ∅ ∅
−
( )− ( ) − + /
x y
l a
m
t
x y x y
l a c
m
q n
m
m
2
1 2( )
,
( ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
588 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
Note that h tn x y
τ ξ ξ( , ; ) = ρ ξ ξτ( , ; )x y t at n = | τ | and
h ti x y
τ ξ ξ( , ; ) = h ti x y
τ ξ ξ−1( , ; ) +
γ τ γ
γ
⊂ =
∑
,
( )
i
g t (33)
with gγ
(
t
) = g tb
γ ( ) + g tc
γ ( ) .
If the prove by induction that
∀ i ≤ n ∃ Mi : h ti x y
τ ξ ξ( , ; ) ≤ e hM t i x yi
τ ξ ξ( , ; )0 , (34)
then we obtain the statement of theorem at i = n.
The inductive assumption at i = 0 is trivial. Suppose that inequality
gγ
(
t
) ≤ e gC t1 0γ ( ) + C e h s ds
t
C t s i x y
2
0
11∫ − −( ) ( , ; )τ ξ ξ (35)
is already proved for i = {
0, … , n0 – 1
}.
Then representation (33) and inductive assumption (34) imply estimate (32) for
x, y ∈ l a
2 1 2( )
( )
k+ :
h ti x y
τ ξ ξ( , ; ) ≤ e hM t i x yi− −1 1 0τ ξ ξ( , ; ) +
+
γ τ γ
γ τ ξ ξ
⊂ =
− −∑ ∫+
−
,
( )( ) ( , ; )
i
C t
t
C t s M s i x ye g C e e h dsi1 1 10 02
0
1 ≤
≤ e C t hM C t i x yi( ) ( , ; )− + +( )1 1 1 2 02
τ
τ ξ ξ ≤ e h
M C C t i x yi− + +( )1 1 22
0
τ
τ ξ ξ( , ; ).
We now prove inequality (35). First, we derive the estimate
g tb
γ ( ) ≤ gb
γ ( )0 + A g s ds
t
b
1
0
∫ γ ( ) + A h s ds
t
i
2
0
1∫ −
τ ( ) (36)
for boundedness part gb
γ of nonlinear expression and then, using the special symmetry
(27) between gb
γ and gc
γ , obtain inequality
g tc
γ ( ) ≤ gc
γ ( )0 + B g ds
t
c
1
0
∫ γ + B g ds
t
b
2
0
∫ γ + B h ds
t
i
3
0
1∫ −
τ (37)
for continuity part gc
γ . Together with (36), this finally implies (35).
Estimate (36). The Ito formula for gb
γ = E I I p nt
x y
1 2
δ
γ
,( ) with finite variation
processes
I1 = ξ ξ∅ ∅−x y
l a2 ( )
, I2 = ξγ
γ γ
γx
l c
m
m ( )
+ ξγ
γ γ
γy
l c
m
m ( )
(38)
implies
g tb
γ ( ) ≤ gb
γ ( )0 + E
0
1 2
t
t
x yI I dp n∫ ( )δ
γ
, + p n I d It
x y
γ
δ,( ) 2 1 + I p n d It
x y
1 2
δ
γ
,( ) . (39)
To estimate first integral in (39), we write the stochastic differential of p nt
x y
γ
,( )
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 589
d p nt
x y
γ
,( ) = – L p n dtx y
t
x y, ,
γ ( ) + 2 ′ ( )p nt
x y
γ
, ξ ξ∅ ∅+x y
t l a
dW,
( )2
, (40)
where the second order differential operator Lx y, acts by the rule
L p nx y x y, ,
0( ) = – 2 0′( )
∈
∑p n ax y
k
k
d
,
Z
– 2 0
2 2′′( ) +
∈
∑p n a x yx y
k
k k k
d
, ( )
Z
+
+ 2 0 2 2
′( ) + + +{ }p n x F x Bx y F y B yx y
l a l a
,
( ) ( ), ( ) , ( ) . (41)
Using (40) and the estimate
L p nx y x y, ,
0( ) ≥ – M p np
x y
0
,( ) (42)
which is analogous to [7] (Hint 9), we have
E
0
1 2
t
I I dp∫ δ
γ = –
0
1 2
t
x y
t
x yI I L p n dt∫ ( )E δ
γ
, , ≤ M g dtp
t
b
γ γ
0
∫ . (43)
The monotonicity of map F : ξ ξ ξ ξ∅ ∅ ∅ ∅− ( ) − ( )x y x y
l a
F F,
( )2
≥ 0 and the Ito formula
for the stochastic differential of I1
δ
d I1
δ = δ ξ ξ ξ ξ ξ ξ
δ
∅ ∅
−
∅ ∅ ∅ ∅− − −x y
l a
x y x y
l a
d d
2 2
2
( ) ( )
,
applied to the second integral in (39) give
E
0
2 1
t
t
x yI p n d I∫ ( )γ
δ, =
= – F I p n F F B dt
t
t
x y x y
l a
x y x y x y
l a
δ ξ ξ ξ ξ ξ ξ ξ ξγ
δ
0
2
2
2 2
∫ ( ) − − ( ) − ( ) + −( )∅ ∅
−
∅ ∅ ∅ ∅ ∅ ∅E ,
( ) ( )
, ≤
≤ δ γB g t dtl a
t
b
L 2
0
( ) ( )( ) ∫ . (44)
To estimate the third term in (39), we use the Ito formula for stochastic differential
of I2
d I2 = m d d dtx x
l c
y y
l cm m
γ γ γ γ γξ ξ ξ ξ
γ γγ γ
[ ] + [ ]
#
( )
#
( )
, ,
for
ξ#
( )
, y
l cp
=
k
k k k
p
k
d
c y
∈
−∑
Z
ξ ξ 2
and inequality
ξ ϕ#,
lm
≤
1
m l
m
m
ϕ +
m
m l
m
m
− 1 ξ . (45)
Therefore, by F ′ ≥ 0, we have
E
0
1 2
t
t
x yI p n d I∫ ( )δ
γ
, = – m I p n
t
t
x y
γ
δ
γ
0
1∫ ( )E , ×
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
590 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
× ξ ξ ξ ξ ϕ ξ ξ ξ ξ ϕγ γ γ γ γ γ γ γ
γ γγ γ
x x x x x
l c
y y y y y
l c
F B F B dt
m m
[ ] ′( ) + + + [ ] ′( ) + +
∅ ∅
#
( )
#
( )
, , ≤
≤
m B m K g t dtl c
t
b
mγ γ γ γ
γ γL ( ) ( ) ( )( ) + −
∫1
0
+ E
0
1
t
t
x yI p n∫ ( )δ
γ
, ×
×
α α γ
α α α αξ ξ ξ ξ ξ ξ
γ γγ
γ
γ
γ
1
1 1
2∪ ∪… = ≥
∅ ∅∑ ( ) … + ( ) …
s
s m s ms
s x x x
l c
m s y y y
l c
m
F F dt
,
( )
( )
( )
( )
, (46)
where K γ is a number of all possible subdivisions of the set γ on sets α1 , … , αs ,
s ≥ 2.
Both terms in (46) will be estimated analogously. Using condition (9) on map F
F s
k
x m( )
,ξ γ
∅( ) ≤ C a a a an
F m
k
m
k k k
x
k k
y
m( ) + +( )− +
∅ ∅
+/ /γ γ γ
ξ ξ( )
, ,
( )k k1 2 2 2 1 2
≤
≤ C a nn
F m
k
m
t
x y m( ) +( )− + +/ /γ γ γ( ) , ( )k k1 2 1 2
1
and applying hierarchy (16) of weights cγ{ } and representation ξα
γm =
= ξα
α γ
αm( ) /
, we estimate first term in (46) by
E
0
1
21
1
t
t
x y
s
s x x x
l c
m
I p n F
s
s m
∫ ∑( ) ( ) …
… = ≥
∅
δ
γ
α α γ
α αξ ξ ξ
γ γ
γ,
,
( )
( )
∪ ∪
=
=
α α γ
δ
γ γ α αξ ξ ξγ γ γ
1
1
2 0
1
∪ ∪… = ≥ ∈
∅∑ ∫ ∑( ) ( ) …
s
d
s
s
t
t
x y
k
k
s
k
x m
k
x m
k
x m
I p n c F dt
,
,
,
( )
, , ,E
Z
≤
≤ C Rn
F m
ss
( )
… = ≥
∑ /γ
α α γ
γ α
γ
1 2
1
∪ ∪ ,
, ×
×
0
1
1 2
1
1
t
t
x y
t
x y m
k l
s
k k
x m
I p n n c dt
d
l l
l
l
∫ ∑ ∏( ) +( ) ( )+
∈ =
/ /
E
kδ
γ α α
α γ
γ αξ, , ( )
, ,
Z
. (47)
Using hierarchy (31) applied to polynomials { pγ } and inequality | x1 … xn | ≤
≤ x qq
1 1
1 / + … + x qn
q
n
n / with ql = | γ | / | αl | we continue (47) by
C Rn
F m
ss
( )
… = ≥
∑ /γ
α α γ
γ α
γ
1 2
1
∪ ∪ ,
, ×
×
0
1
1 2
1
1
t
t
x y
t
x y m
k l
s
k k
x m
I p n n c dt
d
l l
l
l
∫ ∑ ∏( ) +( ) ( )+
∈ =
/ /
E
kδ
γ α α
α γ
γ αξ, , ( )
, ,
Z
≤
≤
K C Rp n
F m
ss
1
2
1
1
/ /( )
… = ≥
∑γ
α α γ
γ α
γγ
∪ ∪ ,
, ×
×
l
s
l
t
t
x y x
l c
m y
l c
m
I p n dt
l l m l m
l ll
l
l
l
=
( ) ( )∑ ∫ ( ) +
1 0
1
α
γ
ξ ξδ
α α α
α αα
α
α
αE , ≤
≤
K C R h s dsp n
F m
s
t
i
s
1
2
1
0
1
1
/ /( )
… = ≥
−∑ ∫γ
α α γ
γ α
γ
τ
γ
∪ ∪ ,
, ( ) . (48)
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NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 591
Finally, (44), (48), and analogous estimates for second term in (46) prove
inequality (36).
Estimate (37). Similar to (36), g tc
γ ( ) = E I I q n
m
t
x y
1 3
δ
γ
γ− ( ), with I 1 introduced in
(38) and
I3 = ξ ξγ γ
γ
γ
γ
γ
x y
l a c
m
m k
m− + /( )( )k 1 2 .
Because I
m
1
δ γ−
and I3 are finite variation processes, by applying the Ito formula to
g tc
γ ( ) , we obtain
g tc
γ ( ) = gc
γ ( )0 +
+ E
0
1 3 3 1 1 3
t
m
t
x y
t
x y m m
t
x yI I dq n q n I d I I q n d I∫ − − −( ) + ( ) + ( ){ }δ
γ γ
δ δ
γ
γ γ γ, , , . (49)
Representation (40) of stochastic differential of q nt
x y
γ
,( ), inequality (42), and the
monotonicity of map F imply the following estimates for the first and second terms in
(49):
E
0
1 3
t
m
t
x yI I dq n∫ − ( )δ
γ
γ , = –
0
1 3
t
m x y
t
x yI I L q n dt∫ − ( )E
δ
γ
γ , , ≤ M g t dtq
t
c
γ γ
0
∫ ( ) , (50)
E
0
3 1
t
t
x y m
q n I d I∫ ( ) −
γ
δ γ, = – ( ) ,
( )
δ ξ ξγ γ
δ γ− ( ) −∫ ∅ ∅
− −
m q n I
t
t
x y x y
l a
m
0
3
2
2
E ×
× ξ ξ ξ ξ ξ ξ∅ ∅ ∅ ∅ ∅ ∅− ( ) − ( ) + −( )x y x y x y
l a
F F B dt,
( )2
≤
≤ ( ) ( )( )δ γ γ− ( ) ∫m B q s dsl a
t
c
L 2
0
. (51)
The estimation of the third term in (49) reflects the similarity of behaviour (27).
Using the Ito formula for I3 with ˜ ,ck γ = a ck
m
k
( )
,
k+ /1 2γ
γ ,
d I3 = − −[ ] ′( ) − ′( ) + −( ) + −∅ ∅ ( )
m F F B dtx y x x y y x y x y
l cm
γ γ γ γ γ γ γ γ γξ ξ ξ ξ ξ ξ ξ ξ ϕ ϕ
γ γ
#
˜
, ,
representation (12) of ϕγ
, the monotonicity of map F ( F ′ ≥ 0 ) and, where necessary,
adding and subtracting new terms, we have
E
0
1 3
t
m
t
x yI q n d I∫ − ( )δ
γ
γ , = − ( )∫ −
m I q n
t
m
t
x y
γ
δ
γ
γ
0
1E , ×
× ξ ξ ξ ξ ξ ξ ξ ξ ϕ ϕγ γ γ γ γ γ γ γ
γ γ
x y x x y y x y x y
l c
F F B dt
m
−[ ] ′( ) − ′( ) + −( ) + −∅ ∅ ( )
#
˜
, ≤
≤ m B m K q dtl c
t
c
mγ γ γ γ
γ γ
γL ˜ ( )( )( ) + − +( )
∫1 1
0
+ (52)
+
0
1
t
m
t
x y x y y
l c
m
I q n F F dt
m
∫ −
∅ ∅ ( )( ) ′( ) − ′( )( )E
δ
γ
γ
γ
γ
ξ ξ ξ
γ
γ,
˜
+ (53)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
592 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
+
α α γ
δ
γ
γ
1 2 0
1
∪ ∪… = ≥
−∑ ∫ ( )
s s
t
m
t
x yI q n
,
,E ×
× F F dts x s y y y
l c
m
s m
( ) ( )
˜
ξ ξ ξ ξα α
γ
γ
γ
∅ ∅ ( )( ) − ( )[ ] …
1
+ (54)
+
α α γ
δ
γ
γ
1 2 1 0
1
∪ ∪… = ≥ =
−∑ ∑ ∫ ( )
s s j
s t
m
t
x yI q n
,
,E ×
× F dts x y y x y x x
l c
m
j j j j s
m
( )
˜
ξ ξ ξ ξ ξ ξ ξα α α α α α
γ
γ
γ
∅ ( )( ) … −( ) …
− +1 1 1
. (55)
Above inequality (45) and the boundedness of finite diagonal map B in any space
lp
(
c
), c ∈ P, were applied.
We use the relation q n nt
x y
t
x y m
γ
γ, ,( ) +( ) /
1
2k
= p nt
x y
γ
,( ) and assumption (9) on
map F,
′( ) − ′( )[ ]∅ ∅ ( )+ /F Fx y y
l a c
m
m
mξ ξ ξγ
γ
γ
γ
γ
( )k 1 2 ≤
≤ C c an
F m
k
k k k
x
k
y
m
d
( ) −( )
∈
∅ ∅∑
/γ
γ ξ ξ
γ
Z
, , ,
2 2
×
× a a ak k k
x
k k
y
m
k
y m
+ +( )∅ ∅
/
ξ ξ ξ
γ γ
γ, , ,
2 2 2k
≤
≤ C nn
F m x y
l a
m
t
x y m y
l c
m
m
( ) − +( )∅ ∅
/γ γ
γ
ξ ξ ξγ
γ
γ
γ
2
1
2
( )
,
( )
k
, (56)
to estimate term (53):
0
1
t
m
t
x y x y y
l c
m
I q n F F dt
m
∫ −
∅ ∅ ( )( ) ′( ) − ′( )( )E
δ
γ γ
γ
γ
ξ ξ ξ
γ
γ,
˜
≤ C g t dtn
F m
t
b( ) ∫γ
γ
0
( ) . (57)
To estimate term (54), we first use 1 ≤ ak
m− + /( )k 1 2γ and hierarchy (16) to get the
following estimate:
F Fs x s y y y
l c
m
s m
( ) ( )
˜
ξ ξ ξ ξα α
γ
γ
γ
∅ ∅ ( )( ) − ( )[ ] …
1
≤
≤ C c an
F m
k
k k
m
k
x
k
y m
k
x
k
y
m
k
y
k
y m
d
s
( ) − + +( ) …
∈
+ /
∅ ∅ ∅ ∅∑
/γ γ γ
γ α αξ ξ ξ ξ ξ ξ
γ γ
Z
,
( )
, , , , , ,
k k1 2 2 2 2
1
1
≤
≤ C n cn
F m x y
l a
m
t
x y m
k
k k
y
k
y m
d
s
( ) − +( ) …∅ ∅
/
∈
∑γ γξ ξ ξ ξγ γ
γ α α
2 1
1
2
( )
,
, , ,
k
Z
≤
≤ C n Rn
F m x y
l a
m
t
x y m( ) − +( )∅ ∅
/ /γ γξ ξ γ
γ α
γ
2
1
2 1
( )
,
,
k
k j
s
k k
y m
d
j j
j j
c
∈ =
∑ ∏
/
Z 1
, ,α α
α γ
ξ α
≤
≤ C n Rn
F m x y
l a
m
t
x y m( ) − +( )∅ ∅
/ /γ γξ ξ γ
γ α
γ
2
1
2 1
( )
,
,
k
j
s
y
l c
m
j
m j
j
j
j
= ( )∏
/
1
ξα
α γ
α α
α
. (58)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 593
Hence, by using the inequality | x1 … xn | ≤ x qq
1 1
1 / + … + x qn
q
n
n / with qj =
= | γ | / | αj |, the relation q z z m
γ
γ( ) 1 2+( ) /k = p zγ ( ) , and hierarchy (31) of
polynomials { pγ }, we obtain the estimate for term (54):
α α γ
δ
γ
γ
1 2 0
1
∪ ∪… = ≥
−∑ ∫ ( )
s s
t
m
t
x yI q n
,
,E F F dts x s y y y
l c
m
s m
( ) ( )
˜
ξ ξ ξ ξα α
γ
γ
γ
∅ ∅ ( )( ) − ( )[ ] …
1
≤
≤
C R I q n I nn
F m
s
t
m
t
x y m
t
x y m
s
( ) ( ) +( )
… = ≥
−∑ ∫/ /γ γ γ
α α γ
γ α
γ δ
γ
γ
1 2
1
0
1 1
2
1
∪ ∪ ,
,
, ,E
k
×
×
j
s
y
l c
m
j
m
j
j j
j
=
∏
/
1
ξα
α γ
α α
α
( )
≤
≤
C K R I p nn
F m
p
s
t
j
s
t
x y y
l c
m
s
j j
m
j
j j
j( ) ( )
/ /
/
… = ≥ =
∑ ∫ ∏γ
α
αγ
α α γ
γ α
γ δ
α α
α γ
ξ
α
1
2
1
0
1
11 ∪ ∪ ,
,
,
( )
E ≤
≤ C K R g t dtn
F m
p
s j
s
j
t
b
s
j( ) / /
… = ≥ =
∑ ∑ ∫γ γ
α α γ
γ α
γ
α
α
γ
1
2
1
1 01 ∪ ∪ ,
, ( ) . (59)
It remains to estimate (55). Assumption (9) on map F gives
F xs( )( ) ≤ C a x yn
F
k + +( ) + /2 2 1 2( )k
and, by taking ˜ ,ck γ = ak
m( )k+ /1 2γ ck, γ
, we arrive at the estimate
F s x y y x y x x
l c
m
j j j j s
m
( )
˜
ξ ξ ξ ξ ξ ξ ξα α α α α α
γ γ
γ
∅ ( )( ) … −( ) …
− +1 1 1
≤
≤ C a cn
F m
k
k
m
k k
x
k
y
m
d
( ) + +( )
∈
+ /
∅ ∅
+ /
∑γ γ γ
γ ξ ξ
Z
( )
, , ,
( )k k1 2 2 2 1 2
1 ×
× ξ ξ ξ ξα α α α
γ γ γ
k
y m
k
x
k
y m
k
x m
j j i, , , ,1
… − … ≤
≤ C n Rn
F m
t
x y m x y
l c
m
j j
m
j
j j
j( ) +( ) −
+
( )
/ / /γ
α
αγ
α
γ α
γ
α α
α γ
ξ ξ1
1 2 1, ( )
, ˜
k
×
×
l l j
s
x
l c
m y
l c
m
l m l m
l
l l l l
l l
= ≠
( ) ( )∏ + +
/
1
1
,
ξ ξα α
α γ
α αα
α
α
α , (60)
where, on the last step, we use
ck, γ = ck, γ
ak
m− + /( )k 1 2γ ak
m( )k+ /1 2γ ≤
≤ R c c ak k k
m
s
s
γ α
γ
α
α γ
α
α γ γ
, , ,
( )1 1 2
1
1/ / /[ ] …[ ] + /k
=
= R c a ck k
m
l l j
s
kj
j
l
lj
γ α
γ
α
α γ
α
α γα
, ,
( )
,
,
1 1 2
1
/ / /+ /
= ≠
[ ] [ ]∏k
.
To estimate (55), we note that hierarchy (31) of weights pγ and the relation
q z z m
γ
γ( ) 1 2+( ) /k = p zγ ( ) imply
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
594 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
q z z m
γ
γ( ) ( )1 1 2+( ) + /k ≤ 1 2 1
1
1+( ) …− / / / /z K p pm
p s
sk γ γ
α
α γ
α
α γ =
= K q pp
p
l l j
s
j
j
l
l1
1
/ / /
= ≠
∏α
α γ
α
α γ
,
.
Finally, substituting (60) in (55) and representing δ – mγ as
δ – mγ = δ
α
γα−( )m
j
j +
l l j
s
l
= ≠
∑
1,
δ α
γ
,
we have the estimate for term (55):
α α γ
δ
γ
γ
1 2 1 0
1
∪ ∪… = ≥ =
−∑ ∑ ∫ ( )
s s j
s t
m
t
x yI q n
,
,E ×
× F dts x y y x y x x
l c
m
j j j j s
m
( )
˜
ξ ξ ξ ξ ξ ξ ξα α α α α α
γ
γ
γ
∅ ( )( ) … −( ) …
− +1 1 1
≤
≤ C R Kn
F m
s
p
s
( )
… = ≥
∑ / /γ
α α γ
γ α
γ γ
1 2
1 1
∪ ∪ ,
, ×
×
j
s t
x y
l a
m
t
x y x y
l c
m
j
j j j
m
j
q n
j j
j
=
∅ ∅
−
( )∑ ∫ − ( ) −
/
1 0
2
E ξ ξ ξ ξ
δ
α α α
α γ
α
α
α
α
( )
,
˜
×
×
l l j
s
x y
l a t
x y x
l c
m y
l c
m
p n dt
l l m
l
l m
l
l
l ll l= ≠
∅ ∅ ( ) ( )∏ − ( ) + +
/
1 2
1
,
( )
,ξ ξ ξ ξ
δ
α α α
α γ
α αα
α
α
α ≤
≤
C R K g g dtn
F m
s
p
j
s t
j c
l l j
s
l b
s
j l( ) +
… = ≥ = = ≠
∑ ∑ ∫ ∑/ /γ
α α γ
γ α
γ γ
α α
α
γ
α
γ
1 2
1 1
1 0 1∪ ∪ ,
,
,
≤
≤ C R K h dtn
F m
s
p
t
i
s
( )
… = ≥
−∑ ∫/ /γ
α α γ
γ α
γ γ
τ
1 2
1 1
0
1
∪ ∪ ,
, . (61)
Collecting together (50) – (52), (57), (59), and (61), we conclude that inequality (37) is
proved.
The possibility to close nonlinear estimate (32) from x, y ∈ l a
2 1 2( )
( )
k+ to x ,
y ∈ l2
(
a
) follows from [6] (Theorems 3.4 and 3.11).
4. Conclusion: application to the regularity of heat semigroups. As a
consequence of the nonlinear estimate for the joint boundedness and continuity of
variations (32), one can estimate the gap between the boundedness and continuity of
semigroup derivatives in the corresponding scales of functional spaces. Next theorem
announces this result.
The Banach space E
Θ, r
(
l
2
(
a
)
), Θ = Θb ∪ Θ c
, consists of functions
f ∈ Lipr
(
l
2
(
a
)
) which have partial derivatives up to n th order ∂ … ∂{ }( ) ( ), ,1 f fn ,
∂{ } …
( )
, ,
m
k k
f
m1
= ∂ …{ }k km
f x
1, , ( ) and whose norm is finite:
f
rEΘ,
= f
rLip + max ,
, ,
( ) ( )
m n
m mf f
b
m
c
m
= …
∂ ∂( )
1 Θ Θ
< ∞,
where
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
NONLINEAR-ESTIMATE APPROACH TO THE REGULARITY … 595
∂( )m f
b
mΘ
=
max sup
( )
, ( )
( )
( )q x l a
m
m l am
m
b
m
mf x
q xG
G
( ) ∈ ∈
∂
( )Θ
2 2
2 , (62)
∂( )m f
c
mΘ
=
= max sup
( ) ( )
, , ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
q x y l a
m m
l a m l a l a l a l am
m
c
m
mf x f y
x y q x y x yH
H
( ) ∈ ∈
∂ − ∂
− +( ) + +( ) /Θ
2
2 2 2 2 2
2 2 2 2 2
1
k
(63)
with
∂( )m f mG
2
=
τ
τ
= …{ }⊂
∑ … ∂
j j
j j
m
m
d
m
G G f x
1
1
1 2
, ,
( )
Z
for G
m
= G
1
⊗
…
⊗ G
m
, G
i
∈
P, and similar expression for H
m
= H
1
⊗
…
⊗ H
m
.
The partial derivatives ∂ … ∂{ }( ) ( ), ,1 f fn of function f ∈
E
Θ, r
are understood in
the sense of identities (21), (22).
Henceforth, we demand that array Θb in (62) be generated by the array Θc (63)
by the law
∀
m
=
1,
…
, n : Θb
m = { qm j
m
j
m
,G( ) =1
such that
G j
m = H
1
⊗
…
⊗ A
–
(
k
+
1
)
H
j
⊗
…
⊗ H
m
(64)
for
q H Hm
m m
c
m, H = ⊗…⊗( ) ∈ }1 Θ .
Note that for quasicontractive array Θc with parameter k, the array Θb = Θb
1 ∪
…
…
∪ Θb
n generated by (64) is also a quasicontractive one, which could be directly
checked.
Theorem 4. Let F satisfy (9), let Θ = Θc ∪ Θ b for Θ c be a quasi-
contractive array with parameter k (Definition 2), and Θ b be generated by Θc
by rule (64).
Then, for all t ≥ 0, Pt : E
Θ, r
→ E
Θ, r
and there exist K
Θ, r
, M
Θ, r
such that
∀
f ∈
E
Θ, r : P ft rEΘ,
≤ K e fr
M tr
rΘ
Θ
Θ,
,
,E . (65)
Proof. The detailed proof will appear in [8], here we only sketch its main steps.
First, one can derive estimates
Eqn t x t y t x
n
t y
n
Y
m n
n
ξ ξ ξ ξ, , ,
( )
,
( )0 0+( ) − /
≤
≤ e x y p x yM t m n
x X
m
y X
m
n
˜
,
( )
,
( )− +( ) +{ }/
1 0
1
0
1
1 1
ξ ξ , (66)
which follow by formal appeal to (30), (32) at δ = m / n < m, if n th term of ρn
c is
remained on the left-hand side of (32) and the special initial data from (11), are
substituted therefore ρn
c
t=0 = 0 and summands on j ≥ 2 in ρn
b
t=0 disappear on the
right-hand side of (32).
However, estimate (32) holds only in the domain δ ≥ m, otherwise one would face
singular terms like 1 0 0/ −
−
ξ ξ
δ
t x t y
m
, , in expression (30). Having guessed in
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
596 A. Val. ANTONIOUK, A. Vict. ANTONIOUK
Theorem 3 the precise form of weights { pj , q j } and topologies { Xj , Y j } for
boundedness and continuity, we apply in [8] (Theorem 5) the evolutionary equation
techniques to the nonautonomous inhomogeneous equation (11) with its special initial
data to reach the value δ = m / n and obtain estimate (66), important in the proof of
Theorem 4.
1. Hadamard J. Lectures on Cauchy’s problem in linear partial differential equations. – Yale Univ.
Press, New Haven, 1923. Reprinted by Dover, New York, 1952.
2. Kamke E. Differentialgleichungen reeler functionen // Math. in Monogr. und Lehrbüchern. –
Leipzig: Acad. Verlag, 1930. – 7.
3. Klein F. Vorlesungen über die Entwicklung der Mathematik in 19 Jahrhundert // Grundlehren
Math. Wiss. – Berlin: Springer, 1926.
4. Simmons G. F. Differential equations with applications and historical notes // Int. Ser. in Pure and
Appl. Math. – McGraw-Hill Book Comp., 1972.
5. Antoniouk A. Val., Antoniouk A. Vict. Nonlinear effects in C
∞
-smooth properties of infinite
dimensional evolutions. Classical Gibbs models (in Russian). – Kiev: Naukova Dumka, 2006. –
187 p.
6. Antoniouk A. Val., Antoniouk A. Vict. Nonlinear effects in the regularity problems for infinite
dimensional evolutions of classical lattice Gibbs systems // Condensed Matter Phys. (volume
dedicated to 80th birthday of I. R. Yuchnovskii). – 2006. – 32 p.
7. Antoniouk A. Val., Antoniouk A. Vict. How the unbounded drift shapes the Dirichlet semigroups
behaviour of non-Gaussian Gibbs measures // J. Funct. Anal. – 1996. – 135. – P. 488 – 518.
8. Antoniouk A. Val., Antoniouk A. Vict. Nonlinear estimates approach to the non-Lipschitz gap
between boundedness and continuity in C
∞
-properties of infinite dimensional semigroups //
Nonlinear Boundary Problems. – 2006. – 18 p.
9. Glimm J., Jaffe A. Quantum physics. A functional integral point of view. – Springer Verlag, 1987.
10. Simon B. The : P ( ϕ ) : 2 Euclidean (quantum) field theory. – Princeton Univ. Press, 1974.
Received 11.10.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
|
| id | umjimathkievua-article-3477 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:43:18Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/493973aa37e01c3d9dab2ebed3c751b0.pdf |
| spelling | umjimathkievua-article-34772020-03-18T19:55:42Z Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems Підхід до регулярності нескінченновимірних параболічних задач, що грунтуються на нелінійних оцінках Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the regularity of solutions of nonlinear differential equations in the case where the classical Cauchy-Liouville-Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the boundedness and continuity of variations with respect to initial data and discuss their applications to the dynamics of unbounded lattice Gibbs models. Показано, яким чином застосування нелінійних симетрій похідних високого порядку дозволяє вивчати регулярність розв'язків нелінійних диференціальних рівнянь у випадку, коли класичну схему Коші - Ліувілля - Пікара неможливо застосувати. Зокрема, отримано нелінійні оцінки на обмеженість та неперервність варіацій за початковими умовами і розглянуто їх застосування до динаміки необмежених ґраткових гіббсівських систем. Institute of Mathematics, NAS of Ukraine 2006-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3477 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 5 (2006); 579–596 Український математичний журнал; Том 58 № 5 (2006); 579–596 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3477/3691 https://umj.imath.kiev.ua/index.php/umj/article/view/3477/3692 Copyright (c) 2006 Antoniouk A. Val.; Antoniouk A. Vict. |
| spellingShingle | Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| title | Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| title_alt | Підхід до регулярності нескінченновимірних параболічних задач, що грунтуються на нелінійних оцінках |
| title_full | Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| title_fullStr | Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| title_full_unstemmed | Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| title_short | Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| title_sort | nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3477 |
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