Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations
The L∞-estimates of weak solutions are established for a quasilinear nondiagonal parabolic system of singular equations whose matrix of coefficients satisfies special structural conditions. A procedure based on the estimation of linear combinations of the unknowns is used.
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irk-123456789-1653852020-02-14T01:27:47Z Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations Portnyagin, D.V. Статті The L∞-estimates of weak solutions are established for a quasilinear nondiagonal parabolic system of singular equations whose matrix of coefficients satisfies special structural conditions. A procedure based on the estimation of linear combinations of the unknowns is used. L∞ - оцінки слабких розв'язків встановлено для квазілінійної недіагональної параболічної системи сингулярних рівнянь, матриця коефіцієнтів якої задовольняє спеціальні структурні умови. Використовується техніка, що базується на оцінці лінійних комбінацій невідомих. 2006 Article Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations / D.V. Portnyagin // Український математичний журнал. — 2006. — Т. 58, № 8. — С. 1084–1096. — Бібліогр.: 11 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165385 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Portnyagin, D.V. Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations Український математичний журнал |
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The L∞-estimates of weak solutions are established for a quasilinear nondiagonal parabolic system of singular equations whose matrix of coefficients satisfies special structural conditions. A procedure based on the estimation of linear combinations of the unknowns is used. |
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Article |
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Portnyagin, D.V. |
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Portnyagin, D.V. |
| author_sort |
Portnyagin, D.V. |
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Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations |
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Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations |
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Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations |
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Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations |
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Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations |
| title_sort |
boundedness of weak solutions of a nondiagonal singular parabolic system of three equations |
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Інститут математики НАН України |
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2006 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165385 |
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Boundedness of weak solutions of a nondiagonal singular parabolic system of three equations / D.V. Portnyagin // Український математичний журнал. — 2006. — Т. 58, № 8. — С. 1084–1096. — Бібліогр.: 11 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT portnyagindv boundednessofweaksolutionsofanondiagonalsingularparabolicsystemofthreeequations |
| first_indexed |
2025-07-14T18:23:10Z |
| last_indexed |
2025-07-14T18:23:10Z |
| _version_ |
1837647676025339904 |
| fulltext |
UDC 517.9
D. V. Portnyagin (Inst. Condensed Matter Physics Nat. Acad. Sci. Ukraine, Lviv)
BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL
SINGULAR PARABOLIC SYSTEM OF THREE EQUATIONS
OBMEÛENIST| SLABKYX ROZV’QZKIV NEDIAHONAL|NO}
SYNHULQRNO} PARABOLIÇNO} SYSTEMY
TR|OX RIVNQN|
L
∞
-estimates of weak solutions are established for a quasilinear nondiagonal parabolic system of singular
equations with matrix of coefficients satisfying special structure conditions. A technique based on
estimating the linear combinations of unknowns is employed to this end.
L
∞
-ocinky slabkyx rozv’qzkiv vstanovleno dlq kvazilinijno] nediahonal\no] paraboliçno]
systemy synhulqrnyx rivnqn\, matrycq koefici[ntiv qko] zadovol\nq[ special\ni strukturni
umovy. Vykorystovu[t\sq texnika, wo bazu[t\sq na ocinci linijnyx kombinacij nevidomyx.
1. Introduction. In the present paper we study the boundedness of weak solutions to
the quasilinear nondiagonal parabolic system of three singular equations in divergence
form under special assumptions upon its structure.
It is well-known that the De Giorgi – Nash – Moser estimates are no longer valid in
general for an elliptic system, the latter can be regarded as a special case of the
parabolic version. An example of an unbounded solution to the linear elliptic system
with bounded coefficients was built up by De Giorgi in [1]. There is yet another
example due to J. Ne c∨as and J. Sou c∨ek of nonlinear elliptic system with the
coefficients sufficiently smooth, but the weak solution not belonging to W
2,
2
. These
two and many other examples illustrate that the regularity problem for elliptic systems
proves to be far more complicated than that for second order elliptic equations and that
the smooth properties of solutions are not only determined by the smoothness of data,
but strongly depend upon the structure of system.
Until now a priori estimates of De Giorgi type have been extended only to a special
class of parabolic systems of equations, the so-called weakly coupled systems. The
system is said to be weakly coupled if it is coupled only through the terms which are
not differentiated, each equation containing derivatives of just one component.
There exists yet another approach to a priori estimates for a parabolic system of
second order differential equations [2]. This applies for diagonal systems which on
freezing the leading coefficients and discarding the right-hand sides and lower order
terms reduce to just one single equation rewritten several times in turn for all the
unknown functions; see also [3, p. 27; 4, p. 32, 33; 5].
The technique we are utilizing has been employed earlier in [6] for semilinear
systems (see also [7 – 9]), and consists in switching to new functions, for each of which
the estimate is established in a conventional way, whence the final conclusion about
each component of the vector function solution follows. This technique allows for
extension to nondiagonal systems with nonlinearities in the spatial derivatives also.
The main idea of our approach is as follows: instead of trying to establish estimates
for each component of solution ( u, v, w ) rather to introduce some linear combinations
of components of the solution:
H1 = α1 u + β2 v + w,
H2 = α2 u + β2 v + w, (1.1)
H3 = α3 u + β3 v + w,
in general some functions H of t, x, u, v , w, for each of which the estimates hold
© D. V. PORTNYAGIN, 2006
1084 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL SINGULAR … 1085
and from whose estimates we shall be able to derive the estimates for the components
of solution ( u, v, w ).
In the present paper although restricting ourselves to systems of second order
equations in divergence form possessing special structure, we demonstrate
boundedness of solution to quasilinear singular parabolic system of three equations in
which coupling occurs in the leading derivatives and whose leading coefficients depend
on x, u, v, w and ux , vx , wx .
2. Basic notations and hypotheses. We shall be concerned with a system of three
equations of the form:
ut –
∂
∂ ( )
x
A x u w u w
i
i x x x
( )( , , , , , , )1 v v = B
(
1
)
( x, u, v, w, ux , vx , wx ),
vt –
∂
∂ ( )
x
A x u w u w
i
i x x x
( )( , , , , , , )2 v v = B
(
2
)
( x, u, v, w, ux , vx , wx ), (2.1)
wt –
∂
∂ ( )
x
A x u w u w
i
i x x x
( )( , , , , , , )3 v v = B
(
3
)
( x, u, v, w, ux , vx , wx ), x ∈ Q.
The boundary conditions of the Dirichlet type are assigned:
( x – g1 , v – g2 , w – g3 ) ( x, t ) ∈ W p
0
1, ( )Ω a.e. t ∈ ( 0, T ),
(2.2)
( u, v, w ) ( x, 0 ) = ( u0 , v0 , w0 ) ( x ).
A solution to system (2.1) with Dirichlet data (2.2) is understood in the weak sense,
as in [10].
Definition 2.1. A measurable vector function ( u1, u2, u3
) = ( u, v, w ) is called a
weak solution of problem (2.1), (2.2) if
u
j ∈ C T L0 2, ; ( )Ω( ) ∩ L T Wp p0 1, ; ( ), Ω( )
and for all t ∈ ( 0, T ]
Ω
∫ u x t dxj
jϕ ( , ) +
Ω×( ]
∫∫ − +{ }
0, t
j
jt i
j
j xu A d x d
i
ϕ ϕ τ =
=
Ω
∫ u x dxj
j0 0ϕ ( , ) +
Ω×( ]
∫∫
0, t
j
jB d x dϕ τ
for all testing functions
ϕ ∈ W T L1 2 20, , ; ( )Ω( ) ∩ L T Wp p0 0
1, ; ( ), Ω( ) , ϕ ≥ 0.
The boundary condition in (2.2) is meant in the weak sense.
Let us also define the boundary norms of functions that will come useful in the
follow-up considerations.
Definition 2.2. Let Ω be a domain in R
n
( here n is any natural number)
and ∂ Ω a portion of its boundary; W ( Ω ) be any Sobolev space. For a function
u ( x ) specified on ∂ Ω we define
u W( )∂Ω = inf ( )ψ
ψ W Ω ,
where the infimum is taken in all functions ψ ∈ W ( Ω ) such that ψ ( x ) = u ( x ) a. e.
on ∂ Ω . We shall denote by W ( ∂ Ω ) a functional space for which the
aforementioned norm is finite.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1086 D. V. PORTNYAGIN
Let us describe the notions, quantities and functions that will appear in this paper.
Here and onward we accept the following notations: Q = ( 0, T ]; S = ∂ Ω × ( 0, T ];
∂ Q ≡ Ω × { }{ }0 ∪ ∂ × ( ]{ }Ω 0, T ; Ω is a bounded domain in R
n
with piecewise
smooth boundary; x ∈ Ω; T > 0; t ∈ ( 0, T ]; 1 < p < 2; p < n; i = 1, … , n; j = 1, 2, 3
and summation convention over repeated indices is assumed; u , v ,
w ∈ C T L0 2, ; ( )Ω( ) ∩ L T Wp p0 1, ; ( ), Ω( ) ; W p
0
1, ( )Ω is a space of functions in
W p1, ( )Ω vanishing on ∂ Ω in the sense of traces for a.e. t ∈ ( 0, T ]. Throughout the
paper, for the sake of brevity, by | s | and | si | is denoted the distance in 3 n-
dimensional and n-dimensional Euclidean space respectively, i.e.,
| s | =
j i
n
i
js
= =
∑ ∑ ( )
1
3
1
2
, | si | =
i
n
i
js
=
∑ ( )
1
2
,
where si
j stands for a 3 n-component vector.
By parabolicity of system (2.1) it is meant that the part without derivatives with
respect to time is elliptic. The notion of ellipticity of a system of differential equations
is understood in the following sense, as it is introduced in [11]:
∃ λ > 0 0 < F = F ( x ) ∈ Lp p/ −( )1 ( Q ) | ∀ sj
i ∈ R
3
n
(2.3)
∀ rj ∈ R
3
∀ x ∈ R
n
: A x r s si
j
j
i( , , ) ≥ λ | s | p – F.
It should be emphasized that we impose neither the Legendre nor the Legendre –
Hadamard condition. The Legendre condition stems from the calculus of variations,
the problem of the minimization of functional, as a sufficient conditions for the
existence of extremal. Since it is to be calculated on the extremal it bears no relation to
the set-up of a problem. Its usage as the ellipticity condition in the theory of systems of
equations is entirely technically motivated. The Legendre – Hadamard condition,
being a weakened version of the Legendre one, has been regarded by many authors as a
more natural ellipticity condition for systems. Both conditions produce an obstacle
from the technical point of view in the approach used by ours, and that is why we
dispense with them and accept the ellipticity condition (2.3) for quasilinear system as
the most appropriate to our ends.
About A x r si
j ( , , ) it is assumed that they are measurable Ω × R
3 × R
3
n → R
functions that satisfy the ellipticity conditions and are subject to the growth conditions:
∃ Λ 2 > 0 ∀ si
j ∈ R
3
n
∀ r
j ∈ R
3
∀ x ∈ R
n
: A x r si
j ( , , ) ≤ Λ 2 | s | p
–
1
, (2.4)
and to the following structure conditions:
∃ α j , β j ∈ R; Det
α α α
β β β
1 2 3
1 2 3
1 1 1
≠ 0 such that
∀ si
j ∈ R
3
n
∀ r
j ∈ R
3
∀ x ∈ R
n
:
α β λ α β1
1
1
2 3
1 1
1
1
2 3A x r s A x r s A x r s x r s s s si i i i i i
( ) ( ) ( )( , , ) ( , , ) ( , , ) ( , , )+ + − + +( ) ≤
≤ ξ 1 ( x, r, s ) + F1 , (2.5a)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL SINGULAR … 1087
α β λ α β2
1
2
2 3
2 2
1
2
2 3A x r s A x r s A x r s x r s s s si i i i i i
( ) ( ) ( )( , , ) ( , , ) ( , , ) ( , , )+ + − + +( ) ≤
≤ ξ 2 ( x, r, s ) + F2 , (2.5b)
α β λ α β3
1
3
2 3
3 3
1
3
2 3A x r s A x r s A x r s x r s s s si i i i i i
( ) ( ) ( )( , , ) ( , , ) ( , , ) ( , , )+ + − + +( ) ≤
≤ ξ 3 ( x, r, s ) + F3 , (2.5c)
λ j = λ j ( x, r, s ) > 0, ξ j = ξ j ( x, r, s ) > 0 are some measurable Ω × R
3
× R
3
n
→ R
functions of x, u, v , w , ux , vx , wx on which the following growth conditions are
imposed:
∃ Λ 1 , Λ 2 > 0 ∀ si
j ∈ R
3
n
∀ r
j ∈ R
3
∀ x ∈ R
n
:
0 < Λ1
1 2 3 2
α βj i j i i
p
s s s+ +
−
≤ λ j ( x, r, s ) ≤ Λ2
1 2 3 2
α βj i j i i
p
s s s+ +
−
, (2.6)
ξ j ( x, r, s ) ≤ ξ 0 | s | ν, 0 < ν =
p p
n p
( )( )− −
+
1 1 1κ ; (2.7)
ξ 0 is a positive number,
Fj ( x, t ) ∈ L
θ
( Q ), θ =
p n
p
+
− −( )( )1 1 1κ
, κ1 ∈ ( 0, 1 ), (2.8)
moreover
α1 , β2 > 1, (2.9)
α2 , α3 , β1
β3 < 1, (2.10)
3
1
2 1
1
2
1
3 3max , max , , ,
p
Λ
[ ]− −α β α β ≤
Λ1
2 p p
, (2.11)
6 ξ0 ≤
Λ1
12 p p+ . (2.12)
Remark 2.1. It is not difficult to check by direct calculation, taking into account
the fact that Fj ∈ L p n p( ) ( )( )+ − −( )/ 1 1 1κ , that structure conditions (2.5a) – (2.5c) along
with (2.6) and (2.12) imply the ellipticity condition (2.3) with λ =
Λ1
12 p p+ and F ≡
≡ C F F F p p
1 1 2 3
1+ +( ) / −( ) + C2
, C1, 2 are numbers depending only on the data.
About right-hand sides B
j
( x, r, s ) it is assumed that they are measurable Ω ×
× R
3
× R
3
n
→ R functions satisfying:
∃ ε ∈ 0
12
1,
( )p
n p
−
+
κ
Λ 3 > 0 ∀ si
j ∈ R
3
n
∀ r
j ∈ R
3
∀ x ∈ R
n
:
(2.13)
B x r sj ( , , ) ≤ Λ 3 | s | ε.
In what follows for the sake of conciseness we shall use the notations:
ũ0 =
u x x t
g x t x t T
0
1
0
0
( ), , ,
( , ), , , ,
∈ =
∈∂ ∈( )
Ω
Ω
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1088 D. V. PORTNYAGIN
ṽ0 =
v0
2
0
0
( ), , ,
( , ), , , ,
x x t
g x t x t T
∈ =
∈∂ ∈( )
Ω
Ω
w̃0 =
w x x t
g x t x t T
0
3
0
0
( ), , ,
( , ), , , .
∈ =
∈∂ ∈( )
Ω
Ω
Let us introduce in addition the following functional space:
Definition 2.3.
˜ ( )W Q = L W Tp p′ ′( )1 0, ( , );Ω ∩ L T Wp p0 1, ; ( ), Ω( ) , p
′ =
p
p − 1
,
i.e., the function u belongs to ˜ ( )W Q if the integral
0
T
t
p p p pu u u u∫ ∫ ′ ′+ ∇ + +( )
Ω
is finite.
On the functions gj ( x, t ), ( u0 , v0 , w0 ) ( x ) in boundary data (2.2) we assume to be
fulfilled the following assumptions:
ũ0 ∈ ˜ ( )W Q∂ , ̃v0 ∈ ˜ ( )W Q∂ , w̃0 ∈ ˜ ( )W Q∂ ;
and, in addition:
gj ( x, t ) ∈ L
∞
( S ), ( u0 , v0 , w0 ) ( x ) ∈ L∞ × { }( )Ω 0 .
3. Estimate for the sum of squares. For the ongoing considerations we need to
estimate the integral of the sum of squares of the spacial derivatives of the components
of solution of problem (2.1), (2.2).
Our goal in this section is to prove the following statement.
Theorem 3.1. Let ( u , v , w ) be a solution to problem (2.1), (2.2) and the
hypotheses (2.5a) – (2.5c), (2.6), (2.7) – (2.12) and (2.13) are satisfied, then there
hold the estimates:
sup ˜
0
0
2
< <
∫ −
t T
u u
Ω
+ sup ˜
0
0
2
< <
∫ −
t T Ω
v v + sup ˜
0
0
2
< <
∫ −
t T
w w
Ω
+
+
0
0 0 0
T
p p pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )
Ω
˜ ˜ ˜v v ≤ C
and
0
T
p p pu w∫ ∫ ∇ + ∇ + ∇( )
Ω
v ≤ C
with constant C depending only on the data: F j
, ˜ ˜ ( )u W Q0 ∂ ,
˜ ˜ ( )v0 W Q∂ ,
˜ ˜ ( )w W Q0 ∂ , p, n, Λ 1 , Λ 2 , ξ0 , κ1 , α j , β j , ε, mes Q, and independent of u , v
and w.
Remark 3.1. By ũ0 , ṽ0 and w̃0 in the formulation of the theorem and in the
follow-up proof is meant any function from ˜ ( )W Q assuming the values of either ũ0
or ṽ0 , w̃0 on the parabolic boundary. Therefore the final statement remains valid
with the boundary norms.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL SINGULAR … 1089
Proof. Multiply the first equation of (2.1) by u u−( )˜0 , the second one by
v v−( )˜ 0 , and the third by w w−( )˜ 0 . After adding all three together and integrating
over the domain Ω × ( 0, t ) this results in:
Ω( )
˜
t
u u∫ −( )1
2 0
2 +
Ω( )
˜
t
∫ −( )1
2 0
2v v +
Ω( )
˜
t
w w∫ −( )1
2 0
2 +
+
0
1
0
t
A u u∫ ∫ ∇ −( )
Ω
�( ) ˜ +
0
2
0
t
A∫ ∫ ∇ −( )
Ω
�( ) ˜v v +
0
3
0
t
A w w∫ ∫ ∇ −( )
Ω
�( ) ˜ ≤
≤
0
1
0
t
B u u∫ ∫ −
Ω
( ) ˜ +
0
2
0
t
B∫ ∫ −
Ω
( ) ˜v v +
0
3
0
t
B w w∫ ∫ −
Ω
( ) ˜ , (3.1)
where the integration by parts with respect to time variable in the first two terms and
the initial condition is taken into account. On the strength of ellipticity condition (2.3)
and growth conditions on A
(
1
),
(
2
),
(
3
)
(2.4), the second group of terms on the left
admits the estimation:
0
1
0
2
0
3
0
t
A u u A A w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )
Ω
� � �( ) ( ) ( )˜ ˜ ˜v v =
=
0
1 2 3 1
0
2
0
3
0
t
A u A A w A u A A w∫ ∫ ∇ + ∇ + ∇ − ∇ − ∇ − ∇( )
Ω
� � � � � �( ) ( ) ( ) ( ) ( ) ( )˜ ˜ ˜v v ≥
≥
0
t
p p pu w∫ ∫ ∇ + ∇ + ∇( )λ
Ω
v –
0
1 1 1
t
p p pu w∫ ∫ ∇ + ∇ + ∇( )− − −λ
Ω
v ×
× ∇ + ∇ + ∇( )u w0 0 0v –
0
1 2 3
t
F F F∫ ∫ + +
Ω
≥
≥
0
1
2
t
p p pu w∫ ∫ ∇ + ∇ + ∇( )λ
Ω
v –
– C p u w
t
p p p( , )λ
0
0 0 0∫ ∫ ∇ + ∇ + ∇( )
Ω
v – C ≥
≥
0
0 0 0
1
2
t
p p pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )λ
Ω
˜ ˜ ˜v v –
–
0
0 0 0
t
p p pC p u w∫ ∫ ∇ + ∇ + ∇( )˜ ( , )λ
Ω
v – C.
Here the use is also made of Young’s inequality and the inequality
| a + b | p ≤ C ( p ) a bp p+( ) ∀ a, b ∈ R. (3.2)
The first three terms on the right of (3.1) in virtue of Young’s inequality, the Sobolev
inequality and growth condition (2.13) can be estimated like that:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1090 D. V. PORTNYAGIN
0
1
0
t
B u u∫ ∫ −
Ω
( ) ˜ +
0
2
0
t
B∫ ∫ −
Ω
( ) ˜v v +
0
3
0
t
B w w∫ ∫ −
Ω
( ) ˜ ≤
≤
0
0 0 0
t
u w u u w w∫ ∫ ∇ + ∇ + ∇( ) − + − + −( )
Ω
v v vε ˜ ˜ ˜ ≤
≤ δ1 C1 ( ε, p )
0
0 0 0
t
pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )
Ω
˜ ˜ ˜v v +
+ δ2 C2 ( p )
0
0 0 0
t
pu u w w∫ ∫ − + − + −( )
Ω
˜ ˜ ˜v v +
+ C C u w Q1 2 1 2 0 0 0, ,, , ˜ , ˜ , ˜ ,δ v mes( ) ≤
≤
δ3
0
0 0 0
t
pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )
Ω
˜ ˜ ˜v v + C3
.
Here it has been taken into account that
ε
p
+
1
p
<
1
2
+
1
p
≤ 1. Collecting the above
estimates, from (3.1) we get:
1
2 0
2
0
2
0
2
Ω( )
˜ ˜ ˜
t
u u w w∫ −( ) + −( ) + −( )[ ]v v +
+
0
0 0 0
1
2
t
p p pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )λ
Ω
˜ ˜ ˜v v ≤
≤
δ5
0
0 0 0
t
p p pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )
Ω
˜ ˜ ˜v v +
+ C Q F u w5 5 0 0 0mes , , , ˜ , ˜ , ˜δ v( ) .
Let us choose δ5 =
1
4
λ , which yields the inequality
1
2 0
2
0
2
0
2
Ω( )
˜ ˜ ˜
t
u u w w∫ −( ) + −( ) + −( )[ ]v v +
+
0
0 0 0
1
4
t
p p pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )λ
Ω
˜ ˜ ˜v v ≤
≤ C Q F u w4 5 0 0 0mes , , , ˜ , ˜ , ˜δ v( ). (3.3)
On this step we take the supremum in t in the left-hand side of (3.3) and obtain the
estimate
sup ˜
0
0
2
< <
∫ −
t T
u u
Ω
+ sup ˜
0
0
2
< <
∫ −
t T Ω
v v + sup ˜
0
0
2
< <
∫ −
t T
w w
Ω
+
+
0
0 0 0
T
p p pu u w w∫ ∫ ∇ −( ) + ∇ −( ) + ∇ −( )( )
Ω
˜ ˜ ˜v v ≤ C5
with constant C5 depending on n, p, ε, λ, Fj
, p, n, Λ
1
, Λ
2
, ξ0
, κ
1
, α j , β j , ε,
mes Q, and, on the strength of Remark 2.1, the boundary norms ˜ ˜ ( )u W Q0 ∂ ,
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BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL SINGULAR … 1091
˜ ˜ ( )v0 W Q∂ , and ˜ ˜ ( )w W Q0 ∂ of functions in the boundary conditions only. Hence the
second statement of the theorem in self-evident.
4. Estimates of L
∞∞∞∞-norms. Let us now turn our attention to the question of
boundedness of weak solutions to a system whose coefficients satisfy assumptions
(2.5a) – (2.5c). Our main result is the following theorem.
Theorem 4.1. Let ( u, v, w ) be a solution to system (2.1). If there exist such
numbers αj , βj ,
α α α
β β β
1 2 3
1 2 3
1 1 1
≠ 0
satisfying assumptions (2.5a) – (2.5c), then for the three linearly independent
functions H1 , H2 , and H3 defined by (1.1) the following estimates hold:
H L Q1 ∞ ( ) ≤ C1 ,
H L Q2 ∞ ( ) ≤ C2 ,
H L Q3 ∞ ( ) ≤ C3 .
Hence it is easily seen that the same estimates take place for the components of the
solution themselves:
u L Q∞ ( ) ≤ C1 , v L Q∞ ( ) ≤ C2 , w L Q∞ ( ) ≤ C3 ,
where constants C1, 2, 3 depend only on the data: n, p, ε, λ, F
j
, p, n, Λ
1
, Λ
2
,
ξ0
, κ
1
, α j , β j , ε , mes Q , g
S1 2 3, , ,( )∞ ,
u w0 0 0, , ,( )v ∞ Ω ; constants in the
embedding theorems and is independent of u, v, and w.
To prove the theorem we need the well-known Stampacchia’s lemma:
Lemma 4.1. Let ψ ( y ) be a nonnegative nondecreasing function defined on [
l0
,
∞
) which satisfies:
ψ ( m ) ≤
C
m l
l
( )
( )
−
{ }ϑ ψ δ for m > l ≥ l0
,
with ϑ > 0 and δ > 1. Then
ψ ( l0 + d ) = 0,
where d = C l1
0
1 12/ / /{ } − −ϑ ϑψ δ δ δ( ) ( ) ( ) .
For proof see [11, p. 8] (Lemma 4.1). We make also use of the following lemma
(see [10, p. 7] (Proposition 3.1)).
Lemma 4.2. If u ∈ L T L∞( )0 2, ; ( )Ω ∩ L T Wp p0 0
1, ; ( ), Ω( ) then there holds the
inequality
0
T
qu∫ ∫
Ω
≤ C u u
T
p
t T
p n
0 0
2∫ ∫ ∫∇
< <
/
Ω Ω
ess sup
with q = p
n
n
+ 2
and constant C depending only on p and n.
Proof of Theorem 4.1. Let α1 , β1 be from hypotheses (2.5a). Multiply the first
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1092 D. V. PORTNYAGIN
equation of (2.1) by α1 , add the second one multiplied by β 1 , and the third one.
Choose H1 ≡ sign ( α1 u + β1 v + w ) α β1 1u w l+ + −( )+v as a testing function with
l ≥ l0 =
max ,( )α β α β1 1 1 2 3 1 0 1 0 0 0g g g u wL S L+ + + +[ ]∞ ∞ × { }( )v Ω . After
integrating in t from 0 to t, t ≤ T, and in x over the domain Ω, this results in
1
2 1
2
Ω( )t
H∫ +
0
1
1
1
2 3
1
t
A A A H∫ ∫ + + ∇
Ω
α β
� � �( ) ( ) ( ), =
=
0
1
1
1
2 3
1
t
B B B H∫ ∫ + +( )
Ω
α β( ) ( ) ( ) .
Making use hypotheses (2.5a) – (2.5c) we have
0
1
1
1
2 3
1
t
A A A H∫ ∫ + + ∇
Ω
α β
� � �( ) ( ) ( ), =
=
0
1
1
1
2 3
1 1 1
t
A A A u w∫ ∫ + + − ∇ + ∇ + ∇[ ]
Ω
α β λ α β
� � �( ) ( ) ( ) ( )v +
+
λ α β1 1 1 1( ),∇ + ∇ + ∇ ∇u w Hv ≥
0
1 1 1 1
t
u w H∫ ∫ ∇ + ∇ + ∇ ∇
Ω
λ α β( ),v –
–
0
1
1
1
2 3
1 1 1 1
t
A A A u w H∫ ∫ + + − ∇ + ∇ + ∇ ∇
Ω
α β λ α β
� � �( ) ( ) ( ) ( )v ≥
≥
0
1 1 1 1
t
u w H∫ ∫ ∇ + ∇ + ∇ ∇
Ω
λ α β v , –
0
1 1 1 1
t
H F H∫ ∫ ∇ + ∇( )
Ω
ξ ;
which hence yields the inequality
1
2 1
2
Ω( )t
H∫ +
0
1 1 1 1
t
u w H∫ ∫ ∇ + ∇ + ∇ ∇
Ω
λ α β v , ≤
≤
0
1 1 1
t
F H∫ ∫ +( ) ∇
Ω
ξ + C B H
t
0
1∫ ∫
Ω
,
where it is denoted B = α1 B
1
+ β1 B
2
+ B
3
; λ and ξ are functions from (2.5a) –
(2.5c); C = C ( α1 , β1 , p, n ) is a constant. Since t ∈ ( 0, T ] is arbitrary, then taking
the supremum we have
sup
0
1
2
< <
∫
t T
H
Ω
+ C H
T
p
1
0
1∫ ∫ ∇
Ω
≤
0
1 1 1
T
F H∫ ∫ +( ) ∇
Ω
ξ + C B H
T
0
1∫ ∫
Ω
, (4.1)
where C = C ( Λ1 , α1 , β1 , p, n ) and the use has been made of the assumption upon
function λ1 (2.6). Applying generalized Holder’s inequality consequently to the terms
on the right yields
0
1 1
T
F H∫ ∫ ∇
Ω
≤ ∇
∫ ∫
− −/ /
H Fp Q Q
T
A l
p
1 1
0
1 1 1
, , ( )θ
θ
χ
Ω
, (4.2a)
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BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL SINGULAR … 1093
0
1 1
T
H∫ ∫ ∇
Ω
ξ ≤ ∇
/
/ /
∫ ∫
− −
H p Q p Q
T
A l
p p
1 1
0
1 1
, , ( )ξ χν
ν
Ω
, (4.2b)
0
1
T
B H∫ ∫
Ω
≤ H Bq Q p Q
T
A l
q p
1
0
1 1
, , ( )/
/ /
∫ ∫
− −
ε
ε
χ
Ω
, (4.2c)
where χ A ( l ) is a characteristic function of the set A ( l ) = x H l t∈ ≥{ }Ω 1 ( ) . From
conditions (2.8), (2.7), (2.13) and Theorem 3.1 it follows that
|| F1 ||θ, Q ≤ C2 , || ξ1 ||p / ν, Q ≤ C3 , || B || p / ε, Q ≤ C4 . (4.3)
Collecting (4.2a) – (4.2c) and taking account of (4.3) from (4.1) we obtain the
inequality
sup
0
1
2
< <
∫
t T
H
Ω
+
0
1
T
pH∫ ∫ ∇
Ω
≤
≤ C H lp Q
p
1 1
1 1 1∇ { } − −/ /
, ( )ψ θ + C H lp Q
p p
2 1
1 1∇ { } − −/ /
, ( )ψ ν +
+ C H lq Q
q p
3 1
1 1
, ( )ψ ε{ } − −/ / , (4.4)
here we denoted:
ψ ( l ) =
0
1
T
A H l l t dt∫ ≥{ }mes ( , ) .
From Lemma 4.2 it follows:
H q Q1 , ≤ sup
( )
0
1
2
0
1
< <
+
∫ ∫ ∫+ ∇
/
t T
T
p
p n qn
H H
Ω Ω
. (4.5)
From relation (4.4) and this inequality we get
sup
0
1
2
< <
∫
t T
H
Ω
+
0
1
T
pH∫ ∫ ∇
Ω
≤
≤ C H H l l
t T
T
p
p
p p p
1
0
1
2
0
1
1
1 1 1 1 1sup ( ) ( )
< <
− − − −∫ ∫ ∫+ ∇
{ } + { }( )
/
/ / / /
Ω Ω
ψ ψθ ν +
+ C H H l
t T
T
p
n p nq
q p
2
0
1
2
0
1
1 1sup ( )
( )
< <
+
− −∫ ∫ ∫+ ∇
{ }
/
/ /
Ω Ω
ψ ε . (4.6)
Applying Young’s inequality to the right-hand side of (4.6) gives
sup
0
1
2
< <
∫
t T
H
Ω
+
0
1
T
pH∫ ∫ ∇
Ω
≤ C l p p p
1
1 1 1 1ψ θ( ) ( ){ } − −( ) −( )/ / / +
+ C l p p p p
2
1 1 1ψ ν( ) ( ){ } − −( ) −( )/ / / + C l q p nq n p
3
1 1ψ ε( ) ( ) #
{ } − −( ) +( )/ / / ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 8
1094 D. V. PORTNYAGIN
with C1, 2, 3 = C u v w F p n1 2 3 0 0 0 1 2, , ˜ , ˜ , ˜ , , , , , , , ,ν θ τ Λ Λ( ) and
nq
n p+
#
such that
nq
n p+
−# 1
+
nq
n p+
= 1. Resorting again to (4.5) implies
H q Q
nq n p
1 ,
( )( ) / +
≤ C l p p p
1
1 1 1 1ψ θ( ) ( ){ } − −( ) −( )/ / / +
+ C l p p p p
2
1 1 1ψ ν( ) ( ){ } − −( ) −( )/ / / + C l q p nq n p
3
1 1ψ ε( ) ( ) #
{ } − −( ) +( )/ / / . (4.7)
Let us estimate:
( ) ( )m l m q− { } /ψ 1 = ( ) ( )m l
T
A m
q
−
∫ ∫
/
0
1
Ω
χ <
0
1
1T
q
A m
q
H∫ ∫
/
Ω
χ ( ) <
< H q Q1 , ,
where m > l ≥ l
0
. Substituting this into (4.7) we come down to
( ) ( )m l mq− ψ ≤ C l p p n p n p
1
1 1 1 1ψ θ( ) ( ) ( ){ } − −( ) + −( )/ / / +
+ C l p p p n p n p
2
1 1 1ψ ν( ) ( ) ( ){ } − −( ) + −( )/ / / + C l q p nq n p n p n
3
1 1ψ ε( ) ( ) ( )#
{ } − −( ) +( ) +( )/ / / /
or, succinctly
ψ ( m ) ≤
C
m l
lq
1 1
( )
( )
−
{ }ψ δ +
C
m l
lq
2 2
( )
( )
−
{ }ψ δ +
C
m l
lq
3 3
( )
( )
−
{ }ψ δ (4.8)
with
δ1 = 1
1 1
1
− −
+
−
p
p n p
n pθ
( )
( )
, δ2 = 1
1
1
− −
+
−
p p
p n p
n p
ν ( )
( )
,
and
δ3 = 1
2 2
−
+
−
+
−
+
n
p n p
n
n p
n
p n( ) ( )
ε
.
From the assumption upon Fj
, (2.8), it follows that
1 –
1
p
–
1
θ
>
n p
p n p
( )
( )
−
+
1
; and thus δ1 > 1;
from the hypotheses on ξj (2.7)
0 < ν <
p p
n p
( )−
+
1
,
it follows that
1 –
1
p
–
ν
p
>
n p
p n p
( )
( )
−
+
1
; and thus δ2 > 1;
from the hypotheses on B
j
(2.13)
0 < ε <
p
n p
2
+
,
hence
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BOUNDEDNESS OF WEAK SOLUTIONS TO NONDIAGONAL SINGULAR … 1095
1 –
n
p n( )+ 2
–
ε
p
>
n
n p+
–
n
p n( )+ 2
; and thus δ3 > 1.
Without loss of generality we may assume that ψ ( l ) < 1. In fact, from the first
statement of Theorem 3.1 and (4.5) it follows that
( ) ( )l l l q− { } /
0
1ψ = ( ) ( )l l
T
A l
q
−
∫ ∫
/
0
0
1
Ω
χ <
0
1 0
1T
A l
q
H l∫ ∫ −( )
/
Ω
χ ( ) <
< H l q Q1 0− , ≤ sup ( ) ( )
( )
0
1 0
2
0
1 0
< <
+
∫ ∫ ∫− + ∇ −
/
t T
T
p
p n qn
H l H l
Ω Ω
≤ C̃ ,
where l ≥ l0
; and hence
ψ ( l ) ≤
˜
( )
C
l l
q
q− 0
; and it’s easy to see that ψ ( l ) < 1 whenever l > C̃ + l0
.
Since ψ ( l ) is nonincreasing function, ψ ( l ) < 1 is true for all l > C̃ + l0
. Due to this,
(4.8) yields
ψ ( m ) ≤
C
m l
lq( )
( )
−
{ }ψ δ , (4.9)
with δ = min [ δ1 , δ2 , δ3 ] and C = max [ C1 , C2 , C3 ]. On the strength of Lemma 4.1
from relation (4.9) we can conclude that
ψ( )l d0 + = 0
for some d sufficiently large, but finite, depending only on the data: n, p, ε, λ , F
j
,
p, n, Λ 1 , Λ 2 , ξ0 , κ1 , αj , βj , ε, mesQ g
S1 2 3, , ,( )∞ , u w0 0 0, , ,( )v ∞ Ω ; constants in
the embedding theorems and is independent of u, v, and w. Analogously is done for
H2 = α2 u + β2 v + w and H3 = α3 u + β3 v + w, where α2, 3 and β2, 3 are from
(2.5b) – (2.5c).
It is not difficult to see from the previous considerations that the same estimates
hold for the components ( u, v, w ) of solution themselves. In fact,
|| u || ∞ =
u∆
∆
∞ =
=
( )( ) ( )( ) ( )( )α β β β α β β β α β β β1 1 2 3 2 2 1 3 3 3 1 2u w u w u w+ + − − + + − + + + − ∞v v v
∆
=
=
( ) ( ) ( )β β β β β β2 3 1 1 3 2 1 2 3− − − + − ∞H H H
∆
≤
≤
β β β β β β2 3 1 1 3 2 1 2 3− + − + −C C C
∆
,
|| v || ∞ =
v∆
∆
∞ =
=
( )( ) ( )( ) ( )( )α β α α α β α α α β α α1 1 2 3 2 2 1 3 3 3 1 2u w u w u w+ + − − + + − + + + − ∞v v v
∆
=
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1096 D. V. PORTNYAGIN
=
( ) ( ) ( )α α α α α α2 3 1 1 3 2 1 2 3− − − + − ∞H H H
∆
≤
≤
α α α α α α2 3 1 1 3 2 1 2 3− + − + −C C C
∆
,
|| w || ∞ = || ( α1 u + β1 v + w ) – α1 u – β1 v || ∞ ≤
≤ || H1 – α1 u – β1 v || ∞ ≤ || H1 || ∞ + | α1 | || u || ∞ + | β1 | || v || ∞,
where || ⋅ || ∞ stands for L∞ norm. Hence follows the statement.
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Received 23.02.2005
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