Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data
We introduce new sets of functions different both from the space introduced in [Ph. Bénilan, L. Boccardo, T. Gallou?t, R. Gariepy, M. Pierre, and J. L. Vazquez, Ann. Scuola Norm. Super. Pisa, 22, No. 2, 241–273 (1995)] and from the Rakotoson T -set introduced in [J. M. Rakotoson, Different. Integr....
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| author | Li, Fengquan Лі, Фенґяуан |
| author_facet | Li, Fengquan Лі, Фенґяуан |
| author_sort | Li, Fengquan |
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| datestamp_date | 2019-12-05T09:49:28Z |
| description | We introduce new sets of functions different both from the space introduced in [Ph. Bénilan, L. Boccardo, T. Gallou?t, R. Gariepy, M. Pierre, and J. L. Vazquez, Ann. Scuola Norm. Super. Pisa, 22, No. 2, 241–273 (1995)] and from the Rakotoson T -set introduced in [J. M. Rakotoson, Different. Integr. Equat., 6, No. 1, 27–36 (1993); J. Different. Equat., 111, No. 2, 458–471 (1994)]. In the new framework of sets, we present some summability results for the entropy solutions of nonlinear parabolic equations. |
| first_indexed | 2026-03-24T02:17:45Z |
| format | Article |
| fulltext |
UDC 517.9
Fengquan Li (School Math. Sci., Dalian Univ. Techology, China)
SOME REGULARITY OF ENTROPY SOLUTIONS
FOR NONLINEAR PARABOLIC EQUATIONS WITH IRREGULAR DATA
РЕГУЛЯРНIСТЬ ЕНТРОПIЙНИХ РОЗВ’ЯЗКIВ
НЕЛIНIЙНИХ ПАРАБОЛIЧНИХ РIВНЯНЬ З НЕРЕГУЛЯРНИМИ ДАНИМИ
We introduce new sets of functions which are different to the space introduced in [Bénilan Ph., Boccardo L., Gallouët T.,
Gariepy R., Pierre M., Vazquez J. L. An L1-theory of existence and uniqueness of solutions of non-linear elliptic equations
// Ann. Scuola norm. super. Pisa. – 1995. – 22, № 2. – P. 241 – 273] and Rakotoson’s T -set in [Rakotoson J. M. Generalized
solutions in a new type of sets for problems with measures as data // Different. and Integr. Equat. – 1993. – 6, № 1. –
P. 27 – 36; T-sets and relaxed solutions for parabolic equations // J. Different. Equat. – 1994. – 111, № 2. – P. 458 – 471].
In the new framework of sets, we give some summability results of entropy solutions for nonlinear parabolic equations.
Введено новi множини функцiй, що вiдрiзняються як вiд простору, що був введений у [Bénilan Ph., Boccardo L.,
Gallouët T., Gariepy R., Pierre M., Vazquez J. L. An L1-theory of existence and uniqueness of solutions of non-linear
elliptic equations // Ann. Scuola norm. super. Pisa. – 1995. – 22, № 2. – P. 241 – 273], так i вiд T -множини Ракотосона,
що була введена в [Rakotoson J. M. Generalized solutions in a new type of sets for problems with measures as data
// Different. and Integr. Equat. – 1993. – 6, № 1. – P. 27 – 36; T-sets and relaxed solutions for parabolic equations //
J. Different. Equat. – 1994. – 111, № 2. – P. 458 – 471]. В рамках нової колекцiї множин отримано новi результати про
сумовнiсть ентропiйних розв’язкiв нелiнiйних параболiчних рiвнянь.
1. Introduction. In this paper we will consider the following problem:
∂u
∂t
− div(a(x, t, u,Du)) = f in Q,
u = 0 on Σ,
u(x, 0) = u0 in Ω,
(P)
where Ω is a bounded open subset of RN , N ≥ 2, and T > 0, Q = Ω× (0, T ), Σ denotes the lateral
surface of Q, f ∈Mb(Q), u0 ∈Mb(Ω). HereMb(Q),Mb(Ω) denote the spaces of bounded Radon
measures on Q and Ω respectively.
There are many works contributing to nonlinear elliptic equations and parabolic equations with
measure data (see [1, 3 – 7, 9 – 21]).
If f ∈ Lp′
(
0, T ;W−1,p′(Ω)
)
, u0 ∈ L2(Ω), the existence of solutions for this problem is a classical
result due to Lions’s methods in [8].
If f ∈ Mb(Q), u0 ∈ Mb(Ω) and p > 2 − 1
N + 1
, Boccardo and Gallouët proved the existence
of distributional solution to problem (P) in [6] (also see [5, 7, 10, 13 – 15]). If 1 < p ≤ 2− 1
N + 1
,
one can’t expect the solution to be in the classical Sobolev space. In order to deal with this case,
Rakotoson has introduced T -sets and the notion of relaxed solutions in [12] (also see [11]). To study
elliptic equations, T 1,p
0 (Ω) has been introduced in [3] too.
Dall’ Aglio, Orsina have discussed the existence and regularity of solutions under three kinds of
conditions about f and u0 in a framework of Sobolev spaces as p > 2 − 1
N + 1
in [7] (also see
c© FENGQUAN LI, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8 1107
1108 FENGQUAN LI
[6, 9]). Similar problem has been discussed in [5], but in which the summability of solutions with
respect to space and time was considered separately. In [17] Segura de León and Toledo have given a
precise summability result of entropy solutions to problem (P) and its gradient with respect to space
and time in a framework of Lebesgue and Marcinkiewicz space under the assumptions of f ∈ L1(Q),
u0 ∈ L1(Ω) or f = 0, u0 ∈ L1(Ω) (also see [16]).
Up to now, I haven’t found any regular results of solutions in the case of 1 < p ≤ 2 − 1
N + 1
,
f ∈ L1(0, T ;L1 logL1(Ω)). Since in the case of f ∈ L1(Q), u0 ∈ L1(Ω), one can prove the solution
to problem (P) in a unified framework of Marcinkiewicz space Mq(Q)
(
here q =
N(p− 1) + p
N + 1
)
for all p > 1, and deduce that the solution belongs to the classical Sobolev space Lq(0, T ;W 1,q
0 (Ω)),
q <
N(p− 1) + p
N + 1
, in the case of p > 2 − 1
N + 1
(see [1, 17]). Can we prove the solution in a
unified framework of Marcinkiewicz space in the case of f ∈ L1(0, T ;L1 logL1(Ω)) for all p > 1
as that of [1, 17]? The answer is negative because we have known in the case of p > 2 − 1
N + 1
,
f ∈ L1(0, T ;L1 logL1(Ω)), the solution to problem (P) belongs to the limit case Lq̄
(
0, T ;W 1,q̄
0 (Ω)
)
,
q̄ =
N(p− 1) + p
N + 1
, from [7] and [5], but we can’t deduce that the limit case from the Marcinkiewicz
space. Furthermore it’s impossible to prove the solution in the framework of Rakotoson’s T -set
because it’s difficult to obtain Φ(u) ∈ Lp
(
0, T ;W 1,p
0 (Ω)
)
(see [12]). Similar question also appears
in the case of f ∈ Lm(Q), 1 < m <
(N + 2)p
(N + 2)p−N
.
To solve the above questions, here I will introduce a new type of functional sets which are
different both from the ones introduced by Rakotoson’s (T -set), T 1,p
0 (Q) and the Marcinkiewicz
space (see [4]). I will discuss the summability results of entropy solutions to problem (P) with respect
to time and space in the framework of new functional sets. Similar case to elliptic equations has been
discussed by the author in [20].
The author has also studied how the growth of a(x, t, s, ξ) with respect to s affected the regularity
of entropy solutions in [21].
The paper is organized as follows. In Section 2 we will give a new type of functional sets and
specify the link with the classical Sobolev spaces. In Section 3 assumptions and statements of the
main results will be given. In Section 4 we will complete the proof of the main results.
2. A new type of functional sets. In this paper we need to define four new types of functional
sets.
For k > 0, we set Tk(σ) = max{−k,min{k, σ}} ∀σ ∈ R.
For 1 < p < +∞, the definition of T 1,p
0 (Ω) can be found in [3]. u ∈ Lp
(
0, T ; T 1,p
0 (Ω)
)
if and
only if Tk(u) ∈ Lp
(
0, T ;W 1,p
0 (Ω)
)
for every k > 0.
Let
T 1,p
0T (Q) =
{
u ∈ Lp
(
0, T ; T 1,p
0 (Ω)
) ∣∣∣∣∣ sup
k>0
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)1+δ
dxdt < +∞ ∀δ > 0
}
,
T 1,p
0T1
(Ω) =
{
u ∈ T 1,p
0T (Q)
∣∣∣∣∣ ∃C > 0 and increasing function Θ such that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1109
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)
dxdt ≤ C[Θ(‖Tk(u)‖L∞(0,T ;L1(Ω))) + 1] ∀k > 0
}
.
For any given 1 < m < +∞, let
T 1,p
0Tm
(Q) =
u ∈ T 1,p
0T (Q)
∣∣∣∣∣ ∃C > 0 such that
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)λ
dxdt ≤ C
∫
Q
(1 + |Tk(u)|)
(1−λ)m
m−1 dxdt
1− 1
m
∀k > 0 and 0 < λ < 1
.
For any given 0 < m < 1, let
T 1,p
0Tm
(Q) =
u ∈ T 1,p
0T (Q)
∣∣∣∣∣ sup
k>0
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)m
dxdt < +∞
.
Remark 2.1. Here T 1,p
0T (Q) is different both from T 1,p
0 (Q) introduced by Benilan, Boccardo,
Gallouët, Gariepy, Pierre and Vazquez in [3] and Rakotoson’s T -set in [12]. The above four types of
functional sets are new. Similar functional sets are defined in [19].
The relations between the above four kind of functional spaces and the classical Sobolev spaces
are stated as follows:
Proposition 2.1. If p > 2− 1
N + 1
, then
L∞(0, T ;L1(Ω)) ∩ T 1,p
0T (Q) ⊂ Lr(0, T ;W 1,q
0 (Ω)),
where r and q satisfy the following inequalities:
1 ≤ q < min
{
N(p− 1)
N − 1
, p
}
, 1 ≤ r < p,
and
N(p− 2) + p
r
+
N
q
> N + 1.
Furthermore, as p > 2, we can take
r = p and q <
p
2
.
Proof. Working as in the proof of Lemma 2.2 of [5] we can prove this propositionby by replacing
u with Tk(u) and taking the limit as k goes to +∞.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1110 FENGQUAN LI
Proposition 2.2. If p > 2− 1
N + 1
, then
L∞(0, T ;L1(Ω)) ∩ T 1,p
0T1
(Q) ⊂ Lr(0, T ;W 1,q
0 (Ω)),
where r and q satisfy
1 ≤ q ≤ N(p− 1)
N − 1
, if p < N,
1 ≤ q < p, if p ≥ N,
1 ≤ r < p,
and
N(p− 2) + p
r
+
N
q
= N + 1.
Furthermore, as p > 2, we can take
r = p and q =
p
2
.
Proof. For any given k > 0 and u ∈ L∞(0, T ;L1(Ω)) ∩ T 1,p
0T1
(Q), Hölder’s inequality implies
that ∫
Ω
|DTk(u)|qdx ≤
≤
∫
Ω
|DTk(u)|p
(1 + |Tk(u)|)
dx
q
p
∫
Ω
(1 + |Tk(u)|)
q
p−q dx
1− q
p
.
For any 1 ≤ r < p, using Hölder’s inequality again, we get
T∫
0
‖DTk(u)‖rLq(Ω)dt ≤
≤
T∫
0
∫
Ω
|DTk(u)|p
(1 + |Tk(u)|)
dxdt
r
p
T∫
0
∫
Ω
(1 + |Tk(u)|)
q
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
≤
≤ C
[
Θ
(
‖Tk(u)‖L∞(0,T ;L1(Ω))
)
+ 1
] r
p
T∫
0
∫
Ω
(1 + |Tk(u)|)
q
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
≤
≤ C1
1 +
T∫
0
‖Tk(u)‖
r
p−r
L
q
p−q (Ω)
dt
1− r
p
,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1111
where C1 denotes the positive constant independent of u and k. From then on, Ci denote analogous
constants, which can vary from line to line. Applying Gagliardo – Nirenberg embedding inequality
(also see [2]), we have
‖Tk(u(t))‖
L
q
p−q (Ω)
≤ C2‖DTk(u(t))‖θLq(Ω)‖Tk(u(t))‖1−θ
L1(Ω)
≤
≤ C3‖DTk(u(t))‖θLq(Ω), (2.1)
where θ satisfies
p− q
q
= θ
(
1
q
− 1
N
)
+
1− θ
1
. (2.2)
Thus, from (2.1) and (2.2) it follows that
T∫
0
‖DTk(u)‖
r
θ
L
q
p−q (Ω)
dt ≤ C4
T∫
0
‖DTk(u)‖rLq(Ω)dt.
Let
r
θ
=
r
p− r
, (2.3)
then
T∫
0
‖DTk(u)‖rLq(Ω)dt ≤ C5
1 +
T∫
0
‖DTk(u)‖rLq(Ω)dt
1− r
p
≤
≤ C6 + ε
T∫
0
‖DTk(u)‖rLq(Ω)dt. (2.4)
Taking ε =
1
2
in (2.4), then
T∫
0
‖DTk(u)‖rLq(Ω)dt ≤ C7. (2.5)
Taking k → ∞ in (2.5) and using Fatou’s lemma, it is easy to see that u ∈ Lr(0, T ;W 1,q
0 (Ω)).
Combining (2.2) with (2.3), we can deduce that
1 ≤ q ≤ N(p− 1)
N − 1
, if p < N,
1 ≤ q < p, if p ≥ N,
1 ≤ r < p,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1112 FENGQUAN LI
and
N(p− 2) + p
r
+
N
q
= N + 1.
By checking the above estimate, it is possible to choose r = p, q =
p
2
in the case of p ≥ 2.
Proposition 2.2 is proved.
Proposition 2.3. If 1 < m <
(N + 2)p
(N + 2)p−N
, p > 2− 1
N + 1
, then
L∞(0, T ;L
[(2−m)−(1−m)p]N
N+p−pm (Ω)) ∩ T 1,p
0Tm
(Q) ⊂ Lr(0, T ;W 1,q
0 (Ω)),
for every pair (r, q) satisfying
1 ≤ q ≤ NNm(p− 1)− p(p− 2)(m− 1)
2p(m− 1) +N(N −m)
, if p < N,
1 ≤ q < p, if p ≥ N,
1 ≤ r ≤ p,
and
N(p− 2) +mp
r
+
N [1 + (p− 1)(m− 1)]
q
= N + 2−m.
Proof. For any given u ∈ L∞
(
0, T ;L
[(2−m)−(1−m)p]N
N+p−pm (Ω)
)
∩ T 1,p
0Tm
(Q), 0 < λ < 1, 1 ≤ q < p,
by Hölder’s inequality we get ∫
Ω
|DTk(u)|qdx ≤
≤
∫
Ω
|DTk(u)|p
(1 + |Tk(u)|)λ
dx
q
p
∫
Ω
(1 + |Tk(u)|)
λq
p−q dx
1− q
p
. (2.6)
For any 1 ≤ r < p, Hölder’s inequality and (2.6) yield
T∫
0
‖DTk(u)‖rLq(Ω)dt ≤
≤
T∫
0
∫
Ω
|DTk(u)|p
(1 + |Tk(u)|)λ
dxdt
r
p
T∫
0
∫
Ω
(1 + |Tk(u)|)
λq
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
≤
≤ C8
∫
Q
(1 + |Tk(u)|)(1−λ)m′
r
pm′
T∫
0
∫
Ω
(1 + |Tk(u)|)
λq
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1113
≤ C9
1 +
∫
Q
|Tk(u)|(1−λ)m′dxdt
r
pm′
T∫
0
∫
Ω
|Tk(u)|
λq
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
. (2.7)
Taking
λ =
Nm− p(m− 1)(N + 2)
N + p− pm
(2.8)
and working as in the proof of Lemma 2.4 in [5] we can prove∫
Q
|Tk(u)|(1−λ)m′dxdt ≤ C10. (2.9)
Using Gagliardo – Nirenberg embedding inequality, then
‖Tk(u(t))‖
L
λq
p−q (Ω)
≤ C11‖DTk(u(t))‖θLq(Ω)‖Tk(u(t))‖1−θ
L2−λ(Ω)
≤
≤ C12‖DTk(u(t))‖θLq(Ω), (2.10)
where 0 ≤ θ ≤ 1 satisfies
p− q
λq
= θ
(
1
q
− 1
N
)
+
1− θ
2− λ
. (2.11)
Applying (2.9) and (2.10) to (2.7), we get
T∫
0
‖DTk(u(t))‖rLq(Ω)dt ≤ C13 + C14
T∫
0
‖DTk(u(t))‖
θλr
p−r
Lq(Ω)dt
1− r
p
.
Choosing
θλr
p− r
= r, (2.12)
then we have
T∫
0
‖DTk(u(t))‖rLq(Ω)dt ≤ C15. (2.13)
Taking k → ∞ in (2.13) and using Fatou’s lemma, we obtain u ∈ Lr(0, T ;W 1,q
0 (Ω)). From (2.8),
(2.11) and (2.12) it follows that
N(p− 2) +mp
r
+
N [1 + (p− 1)(m− 1)]
q
= N + 2−m. (2.14)
By (2.8), (2.12), (2.14) and 0 ≤ θ ≤ 1, we can deduce that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1114 FENGQUAN LI
q ≤ NNm(p− 1)− p(p− 2)(m− 1)
2p(m− 1) +N(N −m)
.
This inequality and q < p yield
1 ≤ q ≤ NNm(p− 1)− p(p− 2)(m− 1)
2p(m− 1) +N(N −m)
, if p < N,
1 ≤ q < p, if p ≥ N.
Moreover, by checking the above proof, (2.14) still hold for the case of r = p.
Proposition 2.4. (i) If 1 < γ ≤ N
N − 1
, 1 +
(2− γ)(N − 1)
N
< p, then
L∞(0, T ;Lγ(Ω)) ∩ T 1,p
0T2−γ
(Q) ⊂ Lr(0, T ;W 1,q
0 (Ω)), (2.15)
for every pair (r, q) satisfies
1 ≤ q ≤ N(p− 2 + γ)
N − 2 + γ
, if p < N,
1 ≤ q < p, if p ≥ N,
1 ≤ r < p,
and
Nγ
q
+
(N + γ)p− 2N
r
= N + γ. (2.16)
(ii) If
N
N − 1
< γ < 2, 1+
(2− γ)(N − 1)
N
< p <
2N
N + 1
, then (2.15) holds and r and q satisfy
N(p− 2 + γ)
N − 2 + γ
< q ≤ pγ
2
,
1 ≤ r < p,
and (2.16). Furthermore, as p ≥ 2
γ
, we can take
r = p and q =
pγ
2
.
Proof. For any given u ∈ L∞(0, T ;Lγ(Ω)) ∩ T 1,p
0T2−γ
(Q), k > 0, 1 ≤ q < p, by Hölder’s
inequality we get ∫
Ω
|DTk(u)|qdx ≤
≤
∫
Ω
|DTk(u)|p
(1 + |Tk(u)|)2−γ dx
q
p
∫
Ω
(1 + |Tk(u)|)
(2−γ)q
p−q dx
1− q
p
. (2.17)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1115
For 1 ≤ r < p, Hölder’s inequality and inequality (2.17) imply that
T∫
0
‖DTk(u)‖rLq(Ω)dt ≤
≤
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)2−γ dxdt
r
p
T∫
0
∫
Ω
(1 + |Tk(u)|)
(2−γ)q
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
≤
≤ C16
T∫
0
∫
Ω
(1 + |Tk(u)|)
(2−γ)q
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
≤
≤ C17
1 +
T∫
0
∫
Ω
|Tk(u)|
(2−γ)q
p−q dx
(p−q)r
(p−r)q
dt
1− r
p
. (2.18)
Applying Gagliardo – Nirenberg embedding inequality to Tk(u(t)), we have
‖Tk(u(t))‖
L
(2−γ)q
p−q (Ω)
≤ C18‖DTk(u(t))‖θLq(Ω)‖Tk(u(t))‖1−θLγ(Ω) ≤
≤ C19‖DTk(u(t))‖θLq(Ω), (2.19)
where 0 ≤ θ ≤ 1 and satisfies
p− q
(2− γ)q
= θ
(
1
q
− 1
N
)
+
1− θ
γ
. (2.20)
(2.18), (2.19) yield
T∫
0
‖DTk(u(t))‖rLq(Ω)dt ≤ C20 + C21
T∫
0
‖DTk(u(t))‖
θ(2−γ)r
p−r
Lq(Ω) dt
1− r
p
.
Let
θ(2− γ)r
p− r
= r. (2.21)
Thus we obtain
T∫
0
‖DTk(u(t))‖rLq(Ω)dt ≤ C22. (2.22)
Taking k → ∞ in (2.22) and by Fatou’s lemma, it is easy to see that u ∈ Lr(0, T ;W 1,q
0 (Ω)). From
(2.20) and (2.21) it follows that
(N + γ)p− 2N
r
+
Nγ
q
= N + γ. (2.23)
By (2.20), (2.21), (2.23) and 0 ≤ θ ≤ 1, we can deduce that the conditions of q and r in (i) and (ii).
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1116 FENGQUAN LI
3. Assumptions and statements of the main results. Let a : Q × R × RN → RN be a
Carathéodory function satisfying for almost every (x, t) ∈ Q and every (s, ξ) ∈ RN+1, ξ ∈ RN ,
ξ′ ∈ RN , ξ 6= ξ′,
a(x, t, s, ξ)ξ ≥ α|ξ|p, (3.1)
|a(x, t, s, ξ)| ≤ β(a0(x, t) + |s|p−1 + |ξ|p−1), (3.2)
[a(x, t, s, ξ)− a(x, t, s, ξ′)][ξ − ξ′] > 0, (3.3)
where α, β are two positive constants, a0 is a nonnegative function belonging to Lp
′
(Q), p′ =
p
p− 1
.
Definition 3.1. A measurable function u ∈ L∞(0, T ;L1(Ω)) will be called an entropy solution
to problem (P) if Tk(u) ∈ Lp(0, T ;W 1,p
0 (Ω)), Sk(u(·, t)) ∈ L1(Ω) ∀k > 0 ∀t ∈ [0, T ], and u
satisfies
∫
Ω
Sk(u(t)− φ(t))dx+
t∫
0
〈φτ , Tk(u− φ)〉 dτ+
+
∫
Q
a(x, τ, u,Du)DTk(u− φ)dxdτ ≤
≤
∫
Ω
Sk(u0 − φ(0))dx+
∫
Q
fTk(u− φ)dxdτ
∀k > 0 ∀φ ∈ Lp(0, T ;W 1,p
0 (Ω)) ∩ L∞(Q) such that φt ∈ Lp
′
(0, T ;W−1,p′(Ω)) + L1(Q).
We denote by L1 logL1(Ω) Orlicz space on Ω. Now, we state the main results of this paper.
Theorem 3.1. Let u be an entropy solution to problem (P) and f ∈ L1(Q), u0 ∈ L1(Ω). Then
under the hypothesis (3.1), u ∈ T 1,p
0T (Q).
Theorem 3.2. Let u be an entropy solution to problem (P) and f ∈ L1(0, T ;L1 logL1(Ω)),
u0 ∈ L1 logL1(Ω). Then under the hypothesis (3.1), u ∈ T 1,p
0T1
(Q).
Theorem 3.3. Let u be an entropy solution to problem (P) and f ∈ Lm(Q), 1 < m <
<
(N + 2)p
(N + 2)p−N
, u0 = 0. Then under the hypothesis (3.1), u ∈ T 1,p
0Tm
(Q).
Theorem 3.4. Let u be an entropy solution to problem (P) and f = 0, u0 ∈ Lγ(Ω), 1 < γ < 2.
Then under the hypothesis (3.1), u ∈ T 1,p
0T2−γ
(Q).
Remark 3.1. In Theorems 3.1 – 3.4, we obtain the precise summability results of entropy solu-
tions to problem (P) by using four types of functional spaces.
Remark 3.2. If f ∈ Lm(Q), u0 = 0, the existence and regular results of solutions to problem (P)
were obtained in Theorem 1.9 and Remark 2.5 of [5] as p ≥ 2. However, here Theorem 3.3 and
Proposition 2.3 get rid of the condition of p ≥ 2 and give a precise summability result of entropy
solutions to problem (P).
Remark 3.3. Theorem 3.4 and Proposition 2.4 improve those results obtained in [9, 16] and
Theorem 5.5 (1) in [17] (in the case of bounded domain).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1117
4. Proofs of the main results. In order to prove the main results of this paper, we need the
following lemmas.
Lemma 4.1. If f ∈ L1(Q), u0 ∈ L1(Ω) and suppose that u is an entropy solution to problem (P),
then
∫
{h≤|u|<h+k}
|Du|pdxdt ≤ k
∫
{|u|≥h}
|f |dxdt+
∫
{|u0|≥h}
|u0|dx
∀k, h > 0. (4.1)
Proof. The proof can be seen in [9].
For convenience, let
Am =
{
(x, t) ∈ Q : m ≤ |u(x, t)| < m+ 1
}
and
A0m =
{
x ∈ Ω : m ≤ |u(x)| < m+ 1
}
.
For any given k > 0, let K = [k] denote the maximal integer not beyond k.
Lemma 4.2. Let u be an entropy solution to problem (P), then for any fixed 0 < τ < 1 and
large enough positive integer l > 1, we have
l∑
m=1
∫
{|u|≥m−1}
|f |m−τdxdt ≤ 1
1− τ
∫
Q
|f |(1 + |Tl(u)|)1−τdxdt, (4.2)
l∑
m=1
∫
{|u0|≥m−1}
|u0|m−τdx ≤
1
1− τ
∫
Ω
|u0|(1 + |Tl(u0)|)1−τdx (4.3)
and
l∑
m=1
m−1
∫
{|u|≥m−1}
|f |dxdt+
∫
{|u0| ≥m−1}
|u0|dx
≤
≤
∫
Q
|f |(1 + ln(1 + |Tl(u)|))dxdt+
∫
Ω
|u0|(1 + ln(1 + |Tl(u0)|))dx. (4.4)
Proof. From the formula of Abel’s summation it follows that
l∑
m=1
∫
{|u|≥m−1}
|f |m−τdxdt =
l∑
m=1
∞∑
h=m−1
∫
Ah
|f |m−τdxdt =
=
∫
{|u|≥l−1}
|f |
l∑
m=1
m−τdxdt+
l−1∑
m=1
∫
Am−1
|f |
m∑
h=1
h−τdxdt ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1118 FENGQUAN LI
≤ 1
1− τ
∫
{|u|≥l−1}
|f |l1−τdxdt+
1
1− τ
l−1∑
m=1
∫
Am−1
|f |m1−τdxdt ≤
≤ 1
1− τ
∫
{|u|≥l−1}
|f |(1 + |Tl(u)|)1−τdxdt+
+
1
1− τ
l−1∑
m=1
∫
Am−1
|f |(|1 + Tm(u)|)1−τdxdt =
=
1
1− τ
∫
{|u|≥l−1}
|f |(1 + |Tl(u)|)1−τdxdt+
+
1
1− τ
l−1∑
m=1
∫
Am−1
|f |(|1 + Tl(u)|)1−τdxdt =
=
1
1− τ
∫
Q
|f |(1 + |Tl(u)|)1−τdxdt.
Working as in the proof of (4.2), we can prove (4.3) and (4.4).
Proof of Theorem 3.1. Let u be an entropy solution to problem (P), then Tk(u) ∈ Lp(0, T ;
W 1,p
0 (Ω)) ∀k > 0, thus u ∈ Lp
(
0, T ; T 1,p
0 (Ω)
)
.
By (4.1) (here h = m, k = 1), we get∫
Q
|DTk(u)|p
(1 + |Tk(u)|)1+δ
dxdt =
K∑
m=0
∫
Am
|DTk(u)|p
(1 + |Tk(u)|)1+δ
dxdt ≤
≤
K∑
m=0
1
(1 +m)1+δ
∫
Am
|Du|pdxdt ≤
≤
∞∑
m=0
1
(1 +m)1+δ
(‖f‖L1(Q) + ‖u0‖L1(Ω)) ≤ C(δ),
where C(δ) is a positive constant independent of k. Thus it easy to see that u ∈ T 1,p
0T (Q).
Proof of Theorem 3.2. If f ∈ L1(0, T ;L1 logL1(Ω)), u0 ∈ L1 logL1(Ω), let u be an entropy
solution to problem (P). By virtue of L1(0, T ;L1 logL1(Ω)) ⊂ L1(Q), L1 logL1(Ω) ⊂ L1(Ω), we
can deduce u ∈ T 1,p
0T (Q) from Theorem 3.1.
For any given k > 0, if 0 < k ≤ 1, it’s obvious. If k > 1, by (4.1) (here h = m, k = 1) and
(4.4) (here l = K), we get∫
Q
|DTk(u)|p
1 + |Tk(u)|
dxdt =
K∑
m=0
∫
Am
|DTk(u)|p
(1 + |Tk(u)|)
dxdt ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1119
≤
K∑
m=0
1
1 +m
∫
Am
|Du|pdxdt ≤
≤
K∑
m=0
1
1 +m
∫
{|u|≥m}
|f |dxdt+
∫
{|u0|≥m}
|u0|dx
=
=
K∑
m=1
1
m
∫
{|u|≥m−1}
|f |dxdt+
∫
{|u0|≥m−1}
|u0|dx
+
+
1
K + 1
∫
{|u|≥K}
|f |dxdt+
∫
{|u0|≥K}
|u0|dx
≤
≤ 2
∫
Q
|f |(1 + ln(1 + |TK(u)|))dxdt+
+2
∫
Ω
|u0|(1 + ln(1 + |TK(u0)|))dx ≤
≤ 2
∫
Q
|f |(1 + ln(1 + |Tk(u)|))dxdt+
+2
∫
Ω
|u0|(1 + ln(1 + |Tk(u0)|))dx ≤
≤ 2
∫
Q
|f |(1 + ln(1 + |Tk(u)|))dxdt+
+2
∫
Ω
|u0|(1 + ln(1 + |u0|))dx ≤
≤ C
[
Θ(‖Tk(u)‖L∞(0,T ;L1(Ω))) + 1
]
, (4.5)
where C is a positive constant independent of k. The final inequality in (4.5) is due to f ∈
∈ L1(0, T ;L1 logL1(Ω)), u0 ∈ L1 logL1(Ω). The details can be seen in [7]. Thus we get
u ∈ T 1,p
0T1
(Q).
Proof of Theorem 3.3. In the following, we only need to prove that for any given k > 0,
0 < λ < 1, there exists a positive constant C independent of k such that
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)λ
dx ≤ C
∫
Q
(1 + |Tk(u)|)
(1−λ)m
m−1 dxdt
1− 1
m
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1120 FENGQUAN LI
In fact, by (4.1) (here h = n, k = 1, u0 = 0), (4.2) (here l = K, τ = λ) and Hölder’s inequality, we
obtain
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)λ
dxdt =
K∑
n=0
∫
An
|DTk(u)|p
(1 + |Tk(u)|)λ
dxdt ≤
≤
K∑
n=0
1
(n+ 1)λ
∫
An
|Du|pdxdt ≤
≤
K∑
n=0
1
(n+ 1)λ
∫
{|u|≥n}
|f |dxdt =
=
K∑
n=1
1
nλ
∫
{|u|≥n−1}
|f |dxdt+
1
(K + 1)λ
∫
{|u|≥K}
|f |dxdt ≤
≤ 2
1− λ
∫
Q
|f |(1 + |TK(u)|)1−λdxdt ≤
≤ 2
1− λ
∫
Q
|f |(1 + |Tk(u)|)1−λdxdt ≤
≤ 2
1− λ
‖f‖Lm(Q)
∫
Q
(1 + |Tk(u)|)
(1−λ)m
m−1 dxdt
1− 1
m
.
Proof of Theorem 3.4. For any given k > 0, taking λ = 2 − γ, (4.1) (here h = m, k = 1,
f = 0), (4.3) (here l = K, τ = λ) and Hölder’s inequality imply that
∫
Q
|DTk(u)|p
(1 + |Tk(u)|)λ
dxdt =
K∑
m=0
∫
Am
|DTk(u)|p
(1 + |Tk(u)|)λ
dxdt ≤
≤
K∑
m=0
1
(1 +m)λ
∫
Am
|Du|pdxdt ≤
≤
K∑
m=0
1
(1 +m)λ
∫
{|u0|≥m}
|u0|dx =
=
K∑
m=1
1
mλ
∫
{|u0|≥m−1}
|u0|dx+
1
(K + 1)λ
∫
{|u0|≥K}
|u0|dx ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
SOME REGULARITY OF ENTROPY SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS . . . 1121
≤ 2
1− λ
∫
Ω
|u0|(1 + |TK(u0)|)1−λdx ≤ 2
1− λ
∫
Ω
|u0|(1 + |Tk(u0)|)1−λdx ≤
≤ 2
1− λ
∫
Ω
|u0|(1 + |u0|)1−λdx ≤ 2
γ − 1
∫
Ω
|u0|dx+
∫
Ω
|u0|γdxdt
.
Hence u ∈ T 1,p
0T2−γ
(Q).
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Received 02.10.12,
after revision — 26.06.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
|
| id | umjimathkievua-article-2049 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:17:45Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e3/631254540e6b2b758076c7691e5b70e3.pdf |
| spelling | umjimathkievua-article-20492019-12-05T09:49:28Z Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data Регулярність ентропійних розв'язків нелінійних параболічних рiвнянь з нерегулярними даними Li, Fengquan Лі, Фенґяуан We introduce new sets of functions different both from the space introduced in [Ph. Bénilan, L. Boccardo, T. Gallou?t, R. Gariepy, M. Pierre, and J. L. Vazquez, Ann. Scuola Norm. Super. Pisa, 22, No. 2, 241–273 (1995)] and from the Rakotoson T -set introduced in [J. M. Rakotoson, Different. Integr. Equat., 6, No. 1, 27–36 (1993); J. Different. Equat., 111, No. 2, 458–471 (1994)]. In the new framework of sets, we present some summability results for the entropy solutions of nonlinear parabolic equations. Введено нові множини Функцій, що відрізняються як від простору, що був введений у [Benilan Ph., Boccardo L., Gallouet T., Gariepy R., Pierre M., Vazquez J. L. An L1-theory of existence and uniqueness of solutions of non-linear elliptic equations // Ann. Scuola norm. super. Pisa. - 1995. - 22, № 2. - P. 241-273], так i від $T$-множини Ракотосона, що була введена в [Rakotoson J.M.Generalized solutions in a new type of sets for problems with measures as data // Different. and Integr. Equat. - 1993. - 6, № 1. - P. 27-36; T-sets and relaxed solutions for parabolic equations // J. Different. Equat. - 1994. - 111, № 2. - P. 458-471]. В рамках нової колекції множин отримано нові результати про сумовність ентропійних розв'язків нелінійних параболічних рівнянь. Institute of Mathematics, NAS of Ukraine 2015-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2049 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 8 (2015); 1107-1121 Український математичний журнал; Том 67 № 8 (2015); 1107-1121 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2049/1115 https://umj.imath.kiev.ua/index.php/umj/article/view/2049/1116 Copyright (c) 2015 Li Fengquan |
| spellingShingle | Li, Fengquan Лі, Фенґяуан Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data |
| title | Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data |
| title_alt | Регулярність ентропійних розв'язків нелінійних параболічних рiвнянь з нерегулярними даними |
| title_full | Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data |
| title_fullStr | Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data |
| title_full_unstemmed | Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data |
| title_short | Certain Regularity of the Entropy Solutions for Nonlinear Parabolic Equations with Irregular Data |
| title_sort | certain regularity of the entropy solutions for nonlinear parabolic equations with irregular data |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2049 |
| work_keys_str_mv | AT lifengquan certainregularityoftheentropysolutionsfornonlinearparabolicequationswithirregulardata AT lífengâuan certainregularityoftheentropysolutionsfornonlinearparabolicequationswithirregulardata AT lifengquan regulârnístʹentropíjnihrozv039âzkívnelíníjnihparabolíčnihrivnânʹzneregulârnimidanimi AT lífengâuan regulârnístʹentropíjnihrozv039âzkívnelíníjnihparabolíčnihrivnânʹzneregulârnimidanimi |