Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators
UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these equations despite numerous works involving vario...
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Institute of Mathematics, NAS of Ukraine
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| author | Ali, Z. I. Sango, M. Ali, Z. I. Sango, M. |
| author_facet | Ali, Z. I. Sango, M. Ali, Z. I. Sango, M. |
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We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these equations despite numerous works involving various generalizations of the monotonicity condition. Key to our work is a sign result for the Ito differential of an approximate solution that we establish, as well as several compactness results of the analytic and probabilistic nature, and a characterization of pseudomonotone operators due to F. E. Browder. |
| doi_str_mv | 10.37863/umzh.v74i7.2286 |
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DOI: 10.37863/umzh.v74i7.2286
UDC 519.21
Z. I. Ali, M. Sango (Univ. Pretoria, South Africa)
PROBABILISTIC WEAK SOLUTIONS
FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS
INVOLVING PSEUDOMONOTONE OPERATORS*
IМОВIРНIСНI СЛАБКI РОЗВ’ЯЗКИ
НЕЛIНIЙНИХ СТОХАСТИЧНИХ ЕВОЛЮЦIЙНИХ ЗАДАЧ,
ЩО МIСТЯТЬ ПСЕВДОМОНОТОННI ОПЕРАТОРИ
We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish
the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these
equations despite numerous works involving various generalizations of the monotonicity condition. Key to our work is a
sign result for the Itô differential of an approximate solution that we establish, as well as several compactness results of
the analytic and probabilistic nature, and a characterization of pseudomonotone operators due to F. E. Browder.
Вивчається важливий клас стохастичних нелiнiйних еволюцiйних задач iз псевдомонотонними елiптичними части-
нами. Встановлено iснування ймовiрнiсних слабких (або мартингальних) розв’язкiв. На сьогоднi не iснує теорiї
розв’язностi, розробленої для таких рiвнянь, незважаючи на те, що є багато робiт, в яких вивчаються рiзнi уза-
гальнення умов монотонностi. Ключем до нашої роботи є знаковий результат для диференцiала Iто наближеного
розв’язку, який ми встановлюємо, а також кiлька результатiв щодо компактностi аналiтичної та ймовiрнiсної природи
i характеризацiя псевдомонотонних операторiв по Ф. E. Браудеру.
1. Introductory background. Stochastic partial differential equations (SPDEs) have become one of
the main areas of research in mathematics and in applied sciences due to their crucial relevance in the
modelling of important processes either subjected or generating random excitations or fluctuations
such as turbulence in fluids, filtering theory, random media, finance; just to cite a few.
Their investigation may be traced back to the pioneering work of Bensoussan and Temam [4],
[3] followed by the theses of Pardoux [27] and Viot [39]. These works generalized the deterministic
results of Lions [25], Browder [12], Vishik [40] to their stochastic counterparts and had a huge
influence on the field, as witnessed by the numerous important works that followed; for instance,
[1, 20, 19, 31, 28]; just to cite a few. The monotonicity and compactness methods were key in the
progress made. The weakening of the monotonicity condition by local monotonicity for SPDEs was
undertaken in recent years in works by Liu, Röckner and their coworkers [22 – 24, 14]. It is not an
exaggeration to say that these latest works have revived the interest for the investigation of existence
of solutions for nonlinear SPDEs which can’t be handled by the popular method of semigroup theory.
Despite these impressive advances made in the field of SPDEs, several important classes of
equations have up to date not been studied by experts. Among them the fundamental class of
evolution SPDEs involving pseudomonotone operators. Their deterministic counterparts have been
the object of investigation by two of the most influential mathematicians of our time, namely Brezis
* Dedicated to the loving memory of Academician Igor Volodymyrovych Skrypnyk (on his 80th birthday).
c\bigcirc Z. I. ALI, M. SANGO, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7 871
872 Z. I. ALI, M. SANGO
and Browder in the seminal works [5, 9] (Chapt. 17) and [10]. Their results expanded the frontiers of
study of many important classes of nonlinear partial differential equations, among them the so-called
class of strongly nonlinear elliptic and parabolic equations with zeroth-order nonlinear perturbation
terms introduced and studied in [6, 7, 13, 11]. The survey papers by Dubinskii and Skrypnik [16,
17, 36, 37] give authoritative accounts of further developments with extensive references. Due to
the lack of a relevant theory for pseudomonotone SPDEs, the work on stochastic strongly nonlinear
parabolic equations has so far been out of reach. To make a breakthrough in that direction of research,
the genuine pseudomonotone case that is being investigated and settled in our work is unavoidable. It
should be noted that a sign condition is key in the application of the main result on pseudomonotone
operators. In the deterministic case, an efficient intermediary tool used to that effect is a sign condition
involving the derivative of an approximation of the solution of the problem at hand (see, for instance,
Lemma 7.4 in [30, p. 192], and [21]). The stochastic version of that result is subtle and its proof more
delicate than in the deterministic case. The key to our work is the successful establishment of such a
sign condition involving Itô’s differential of an approximation of our required solution. Our notion of
solution is that of martingale solution or weak probabilistic. One initial main challenge in this work in
the fact that Galerkin’s method seems hopeless, since the corresponding system of stochastic ordinary
differential equations satisfied by the Fourier coefficients of the Galerkin approximation lacks the
counterpart of Carathéodory existence theorem for deterministic ordinary differential equations which
was key in the deterministic theory (see, for instance, [30]). We rely instead on a numerical scheme
introduced by Gyöngy and Millet in [18] for the case of strongly monotone nonlinear stochastic
parabolic equations. Our main result can be seen as a generalization of the corresponding results of
[18, 20, 22 – 24, 27, 39] and many others to pseudomonotone stochastic parabolic equations. From the
probabilistic methodological side, we rely on the still unavoidable fundamental compactness results
of Prokhorov [29] and Skorokhod [35] which are crucial in establishing probabilistic weak solutions.
Specifically, we consider stochastic nonlinear evolution problem
(P )
\left\{ du+At(u)dt = f(t)dt+G(t, u)dW (t),
u(0) = 0,
for t \in [0, T ], where u = u(t) is the unknown process, the contributors f and G to the forcing are
given, W is a d -dimensional Wiener process and At = A(t, u) is a pseudomonotone operator acting
from a reflexive and separable Banach space V to its dual V \prime . Namely for almost everywhere (a.e.)
t \in [0, T ], At is bounded and if uj converges to u weakly in V and \mathrm{l}\mathrm{i}\mathrm{m}j\rightarrow \infty \langle At(uj), uj - u\rangle V \prime ,
V \leq 0, then
At(uj) - \rightarrow At(u) weakly in V \prime and \langle At(uj), uj\rangle V \prime ,V - \rightarrow \langle At(u), u\rangle V \prime ,V .
The theory of pseudomonotone operators was pioneered by Brezis in [5] and actively studied
by Browder [9]. Our characterization of pseudomonotonicity is due to Browder [10] (see also [37],
Chapt. 1, \S 1).
This paper is organized as follows. In Section 2, we state the assumptions on the investigated
problem and formulate our main result. In Section 3, we introduce a suitable numerical scheme for
an approximation of the solution of our original problem and establish some crucial compactness
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 873
results for a sequence of probability measures generated by the approximating solutions. Section 4
is devoted to the proof of convergence of the approximating solutions to the genuine one leading
to the proof of the main result on the existence of a probabilistic weak (martingale) solution. In
Section 5, we provide an example of application of the main result to a stochastic evolution problem
involving higher-order nonlinear partial differential operators. The last section is devoted to some
closing remarks on the comparison of our work with those of Liu and Röckner [22 – 24] dealing with
local monotonicity.
2. Assumptions and formulation of the main result. Let V be a reflexive and separable
Banach space compactly embedded into the separable Hilbert space H; H is identifiable with its
dual H \prime , and we denote by \langle \cdot , \cdot \rangle the duality pairing of V \prime and V . Then we have the Gelfand triple
V \subset H \simeq H \prime \subset V \prime .
For r \in [1,\infty ], T > 0 and X a Banach space, Lr(0, T ;X) denotes the usual Lebesgue –
Bochner space of functions defined on [0, T ] with values in X endowed with the usual norm. Given a
complete probability space
\bigl(
\Omega ,\scrF , (\scrF t)t\in I ,\BbbP
\bigr)
with a filtration (\scrF t)t\in I (I is the time interval [0, T ]),
and the numbers 1 \leq r \leq \infty , 1 \leq q < \infty , Lq
\bigl(
\Omega , Lr(0, T ;X)
\bigr)
denotes the space of progressively
measurable processes endowed with the corresponding norm.
We now formulate assumptions on our problem (P ). Let 2 \leq p <\infty .
(i) For a.e. t \in [0, T ], At : V \rightarrow V \prime is pseudomonotone.
(ii) There exist a constant c1 > 0 and a nonnegative function h1 \in L1([0, T ]) such that
\langle At(u), u\rangle \geq c1
\bigm\| \bigm\| u\bigm\| \bigm\| p
V
- h1(t) for all u \in V and a.e. t \in [0, T ].
(iii) There exist a constant c2 > 0 and a function h2 \in Lp\prime ([0, T ]) such that\bigm\| \bigm\| At(u)
\bigm\| \bigm\|
V \prime \leq c1
\bigm\| \bigm\| u\bigm\| \bigm\| p - 1
V
+ h2(t) for all u \in V and a.e. t \in [0, T ].
(iv) The nonlinear operator G(t, u) : [0, T ]\times H \rightarrow Hd is continuous in (t, u) and there exists a
positive constant C such that \bigm\| \bigm\| G(t, u)\bigm\| \bigm\|
Hd \leq C
\bigl(
1 + \| u(t)\| H
\bigr)
;
Hd denotes the product of d copies of H .
(v) f(t) is a deterministic functional on V, measurable and there exists a positive constant C
such that
T\int
0
\bigm\| \bigm\| f(t)\bigm\| \bigm\| p\prime
V \prime dt \leq C.
Next, we define the concept of probabilistic weak or martingale solution for the problem (P ).
Definition 1. A probabilistic weak solution of the problem (P ) is a system\bigl(
\Omega ,\scrF , (\scrF t)0\leq t\leq T ,\BbbP ,W, u
\bigr)
,
where
(1) (\Omega ,\scrF ,\BbbP ) is a probability space, (\scrF t) a filtration on it,
(2) W is a d-dimensional \scrF t-standard Wiener process,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
874 Z. I. ALI, M. SANGO
(3) (\omega , t) \rightarrow u(\omega , t) is progressively measurable,
(4) u \in L2(\Omega , L\infty (0, T ;H)) \cap Lp
\bigl(
\Omega , Lp(0, T ;V )
\bigr)
, and, for all t \in [0, T ], u(t) satisfies the
integral identity
\bigl(
u(t), v
\bigr)
-
t\int
0
\langle As(u), v\rangle ds =
t\int
0
\langle f(s), v\rangle ds+
\left( t\int
0
G(s, u(s))dW (s), v
\right) \forall v \in V, \BbbP -a.s. (1)
Note that the last equation implies that almost surely (a.s.)
u(\cdot ) \in C(0, T ;V \prime )
and since u(\cdot ) is also bounded in H, then it is almost surely in C (0, T ;H weak), the space of
H -valued weakly continuous functions on [0, T ]; that, for any v \in H, the function
[0, T ] \rightarrow [0,\infty ) : t \mapsto \rightarrow
\bigl(
u(t), v
\bigr)
is continuous. This follows by arguing as in [38] (Chapt. 3, \S 3).
The definition means that the probability space (\Omega ,\scrF ,\BbbP ) and the Wiener process W are unknown
alongside the process u.
The main result of this paper is the following theorem.
Theorem 1. Assume that the conditions (i) – (v) are satisfied and V is compactly embedded
into H . Then problem (P) has a probabilistic weak (martingale) solution in the sense of the above
Definition 1.
The remaining part of the paper is devoted to the proof of this theorem.
3. Numerical approximation of (\bfitP ) and compactness results. In this section, we introduce
a suitable numerical scheme for Problem (P ) and derive needed compactness results for probability
measures generated by the approximating solutions.
From the onset, the pseudomonotonicity of At limits the methodological options in the proof
of Theorem 1, since due to lack of a convenient stochastic version of Carathéodory’s determinis-
tic theorem on the existence of solutions to ordinary differential equations, we are unable to use
Galerkin’s method. We rely instead on a semidiscretized version of (P ), following [18]. We set
f = 0, since the presence of f does not add any complication. Let \{ ti\} be a regular partition of the
interval [0, T ] given by ti = i
T
M
, i = 0, 1, . . . ,M and set \tau =
T
M
. On an intermediary probability
space (\=\Omega , \=\scrF , \=\BbbP ) with a prescribed d-dimensional standard Wiener process \=W (t), we consider the
approximation
\bigl(
uM (t)
\bigr)
of the presumed solution u(t) of problem (P ), which is required to satisfy
the following conditions: uM (0) = 0,
uM (ti+1) = uM (ti) - \tau AM
ti
\bigl(
uM (ti+1)
\bigr)
+ \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta \=Wi, (2)
where
AM
ti (\cdot ) =
1
\tau
ti+1\int
ti
A(s, \cdot )ds, \~GM
ti (ti, \cdot ) =
1
\tau
ti+1\int
ti
G(s, \cdot )ds, \Delta \=Wi = \=W (ti+1) - \=W (ti),
and we define the piecewise functions uM (t) and AM.
t (\cdot ) by setting
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 875
uM (t) = uM (ti+1) for t \in (ti, ti+1], i = 0, . . . ,M - 1, (3)
and
AM.
t (\cdot ) = AM
ti+1
(\cdot ) for t \in (ti, ti+1], i = 0, . . . ,M - 1.
Before proceeding further, let us estimate the H -norm of \~GM
ti (ti, \cdot ). We have, by using Hölder’s
inequality and Fubini’s theorem, that
\bigm\| \bigm\| \bigm\| \~GM
ti (ti, \cdot )
\bigm\| \bigm\| \bigm\| 2
H
\leq 1
\tau
ti+1\int
ti
\bigm\| \bigm\| G(s, uM (s))
\bigm\| \bigm\| 2
H
ds. (4)
We deal next with the a priori estimates of the sequence (uM ).
3.1. Estimates for \bfitu \bfitM . For sufficiently small \tau , denoting by Id the identity operator, it is
known that the operator Id + \tau AM
ti is pseudomonotone (see [30, p. 203]). Therefore owing to the
results of [5] (see also [37] (Chapt. 1, \S 4), we have that for almost all \=\omega \in \=\Omega , (2) has at least a
weak solution uM (ti+1) \in V, given uM (ti) \in V .
From relation (2), we obtain \bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 2
H
-
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
=
= \tau 2
\bigm\| \bigm\| AM
ti
\bigl(
uM (ti+1)
\bigr) \bigm\| \bigm\| 2
H
+
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 2
H
+
+2
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr)
- 2\tau
\bigl\langle
AM
ti
\bigl(
uM (ti+1)
\bigr)
, uM (ti)
\bigr\rangle
-
- 2\tau
\Bigl\langle
AM
ti
\bigl(
uM (ti+1)
\bigr)
, \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr\rangle
.
At this point, we substitute uM (ti) by uM (ti+1) + \tau AM
ti
\bigl(
uM (ti+1)
\bigr)
- \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi in the
pairing of AM
ti and uM (ti) and get after some cancellations\bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 2
H
=
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
- \tau 2
\bigm\| \bigm\| AM
ti
\bigl(
uM (ti+1)
\bigr) \bigm\| \bigm\| 2
H
+
+
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 2
H
- 2\tau
\bigl\langle
AM
ti
\bigl(
uM (ti+1)
\bigr)
, uM (ti+1)
\bigr\rangle
+
+2
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr)
.
We deduce that \bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 2
H
+ 2\tau
\bigl\langle
AM
ti
\bigl(
uM (ti+1)
\bigr)
, uM (ti+1)
\bigr\rangle
\leq
\leq
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
+
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 2
H
+ 2
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr)
. (5)
In particular, \bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 2
H
\leq
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
+
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 2
H
+
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
876 Z. I. ALI, M. SANGO
+2
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr)
+ 2
ti+1\int
ti
| h1(s)| ds (6)
in view of condition (ii) applied to the second term in the left-hand side of (5).
Due to the vanishing of the expectation of the last term in (5), the conditions on At and the
relation (4), we deduce from (5) that
\=\BbbE
\bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 2
H
+ \=\BbbE
ti+1\int
ti
\bigm\| \bigm\| uM (s)
\bigm\| \bigm\| p
V
ds \leq
\leq \=\BbbE
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
+ \=\BbbE
ti+1\int
ti
\bigm\| \bigm\| G \bigl( s, uM (s)
\bigr) \bigm\| \bigm\| 2
H
ds+
+2
ti+1\int
ti
| h1(s)| ds for i = 0, 1, . . . ,M - 1. (7)
Of interest for our purpose are higher order moments for
\bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\|
H
; the fourth moment will do.
For that, we square both sides of (6) and get
\=\BbbE
\bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 4
H
\leq \=\BbbE
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 4
H
+
5\sum
l=1
Il, (8)
where
I1 = \=\BbbE
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 4
H
, I2 = 4\=\BbbE
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr) 2
,
I3 = 2\=\BbbE
\biggl[ \bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 2
H
\biggr]
,
I4 = 4\=\BbbE
\Bigl[ \bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr) \Bigr]
,
I5 = 4\=\BbbE
\biggl[ \bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| 2
H
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\Bigr) \biggr]
.
We now estimate each term in the right-hand side of (8). We have
I1 \leq C \=\BbbE
\biggl( \bigm\| \bigm\| \Delta Wi
\bigm\| \bigm\| 4 \bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr) \bigm\| \bigm\| \bigm\| 4
H
\biggr)
.
But \~GM
ti being
-
\scrF ti -measurable, it is independent of \Delta Wi, and hence
I1 \leq C \=\BbbE
\Bigl( \bigm\| \bigm\| \Delta Wi
\bigm\| \bigm\| 4\Bigr) C \=\BbbE
\biggl( \bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr) \bigm\| \bigm\| \bigm\| 4
H
\biggr)
\leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 877
\leq C\tau 2\=\BbbE
\biggl( \bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr) \bigm\| \bigm\| \bigm\| 4
H
\biggr)
. (9)
Noting that
\Bigl(
uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr) \Bigr)
is independent of \Delta Wi and using Young’s inequality, we get
I2 \leq C \=\BbbE
\bigm\| \bigm\| \Delta Wi
\bigm\| \bigm\| 2\=\BbbE \Bigl( uM (ti), \~G
M
ti
\bigl(
ti, u
M (ti)
\bigr) \Bigr)
\leq
\leq C\tau
\biggl(
\=\BbbE
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 4
H
+ \=\BbbE
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr) \bigm\| \bigm\| \bigm\| 4
H
\biggr)
. (10)
Similarly
I3 \leq C\tau
\biggl(
\=\BbbE
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 4
H
+ \=\BbbE
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr) \bigm\| \bigm\| \bigm\| 4
H
\biggr)
. (11)
It is clear that
I4 = 0. (12)
Using Hölder’s inequality, we easily show that
I5 \leq C\tau
\biggl(
\=\BbbE
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 4
H
+ \=\BbbE
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr) \bigm\| \bigm\| \bigm\| 4
H
\biggr)
. (13)
Combining the estimates (8) – (13) and (4), it follows that
\=\BbbE
\bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| 4
H
\leq \=\BbbE
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 4
H
+ C \=\BbbE
ti+1\int
ti
\bigm\| \bigm\| uM (s)
\bigm\| \bigm\| 4
H
ds+
+\=\BbbE
ti+1\int
ti
\bigm\| \bigm\| G\bigl( s, uM (s)
\bigr) \bigm\| \bigm\| 4
H
ds for i = 0, . . . ,M - 1. (14)
Summing up the estimates (7) for i = 0, . . . , l - 1, l = 1, . . . ,M, and using assumption (iv) on the
nonlinear function G, we have
\=\BbbE
\bigm\| \bigm\| uM (tl)
\bigm\| \bigm\| 2
H
+ \=\BbbE
tl\int
0
\bigm\| \bigm\| uM (s)
\bigm\| \bigm\| p
V
ds \leq
\leq \=\BbbE \| u0\| 2H + \=\BbbE
tl\int
0
\Bigl(
1 +
\bigm\| \bigm\| uM (s)
\bigm\| \bigm\| 2
H
\Bigr)
ds, l = 1, . . . ,M,
from which we deduce, owing to Gronwall’s lemma, that
\mathrm{s}\mathrm{u}\mathrm{p}
M
\mathrm{s}\mathrm{u}\mathrm{p}
l=1,...,M
\=\BbbE
\bigm\| \bigm\| uM (tl)
\bigm\| \bigm\| 2
H
\leq C, \mathrm{s}\mathrm{u}\mathrm{p}
M
\mathrm{s}\mathrm{u}\mathrm{p}
l=1,...,M
\=\BbbE
tl\int
0
\bigm\| \bigm\| uM (s)
\bigm\| \bigm\| p
V
ds \leq C,
and subsequently, we get
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878 Z. I. ALI, M. SANGO
\mathrm{s}\mathrm{u}\mathrm{p}
M
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\=\BbbE
\bigm\| \bigm\| uM (t)
\bigm\| \bigm\| 2
H
\leq C, \mathrm{s}\mathrm{u}\mathrm{p}
M
\=\BbbE
T\int
0
\bigm\| \bigm\| uM (s)
\bigm\| \bigm\| p
V
ds \leq C. (15)
We analogously also have
\mathrm{s}\mathrm{u}\mathrm{p}
M
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\=\BbbE
\bigm\| \bigm\| uM (t)
\bigm\| \bigm\| 4
H
\leq C. (16)
From condition (iii) on our data, we easily show that\bigm\| \bigm\| As
\bigl(
uM (ti+1)
\bigr) \bigm\| \bigm\|
V \prime \leq
\leq C
\biggl( \bigm\| \bigm\| uM (ti+1)
\bigm\| \bigm\| p
p\prime
V + \| h2(s)\| Lp\prime (D)
\biggr)
, i = 0, . . . ,M - 1, (17)
and thanks to the second estimate in (15) and the condition on h2, we deduce that
E
T\int
0
\bigm\| \bigm\| As
\bigl(
uM (s)
\bigr) \bigm\| \bigm\| p\prime
V \prime ds <\infty . (18)
Similarly
E
T\int
0
\bigm\| \bigm\| Gs
\bigl(
uM (s)
\bigr) \bigm\| \bigm\| 4
H
ds <\infty . (19)
Our next task is to estimate the incremental variation of uM on the interval [ti, ti+1] in the dual
space V \prime . This estimate will be crucial for the proof of needed compactness results.
From the relations (2), we obtain\bigm\| \bigm\| uM (ti+1) - uM (ti)
\bigm\| \bigm\| p\prime
V \prime \leq C\tau p
\prime \bigm\| \bigm\| AM
ti
\bigl(
uM (ti+1)
\bigr) \bigm\| \bigm\| p\prime
V \prime +
+C
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| p\prime
V \prime
. (20)
Since p \geq 2, we have that p\prime \leq 2. Thus,
\=\BbbE
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| p\prime
V \prime
=
= \=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
\varphi \in V \prime ,\| \varphi \| V \prime =1
\Biggl\langle ti+1\int
ti
\~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
dW,\varphi
\Biggr\rangle p\prime
=
= \=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
\varphi \in V \prime ,\| \varphi \| V \prime =1
\left( ti+1\int
ti
\Bigl\langle
\~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
, \varphi
\Bigr\rangle
dW
\right) p\prime
\leq
\leq
\left[ \=\BbbE
\left( ti+1\int
ti
\mathrm{s}\mathrm{u}\mathrm{p}
\varphi \in V \prime ,\| \varphi \| V \prime =1
\Bigl\langle
\~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
, \varphi
\Bigr\rangle
dW
\right) 2
\right]
p\prime
2
.
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 879
Now, thanks to Itô’s isometry, Cauchy – Schwarz’s inequality, estimates (4), (15) and condition (iv)
on G, we deduce that
\=\BbbE
\bigm\| \bigm\| \bigm\| \~GM
ti
\bigl(
ti, u
M (ti)
\bigr)
\Delta Wi
\bigm\| \bigm\| \bigm\| p\prime
V \prime
\leq C\tau
p\prime
2
\Bigl[
E
\Bigl(
1 +
\bigm\| \bigm\| uM (ti)
\bigm\| \bigm\| 2
H
\Bigr) \Bigr] p\prime
2 \leq C\tau
p\prime
2 . (21)
Combining (20), (21) with (17) and (15), we infer that
\=\BbbE
\bigm\| \bigm\| uM (ti+1) - uM (ti)
\bigm\| \bigm\| p\prime
V \prime ds \leq C
\biggl(
\tau p
\prime - 1 + \tau
p\prime
2
\biggr)
, i = 0, . . . ,M - 1,
and, therefore,
\=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
\| \theta \| \leq \tau
T\int
0
\bigm\| \bigm\| uM (t+ \theta ) - uM (t)
\bigm\| \bigm\| p\prime
V \prime ds \leq CT
\biggl(
\tau p
\prime - 1 + \tau
p\prime
2
\biggr)
. (22)
We summarize our findings in the following lemma.
Lemma 1. Under the assumptions (i) – (vi), the sequence
\bigl\{
uM (t)
\bigr\}
M\in \BbbN defined by the rela-
tion (3) satisfies the estimates (15), (16), (18), (19) and (22).
Armed with this lemma, we are able to establish crucial compactness results in the next subsec-
tion.
3.2. Probabilistic compactness results. We start this subsection by introducing some auxiliary
spaces which will be needed for the compactness of probability measures generated by the pair\bigl(
\=W,uM
\bigr)
.
Following [2], for any sequences (\mu n), ( \nu n) such that \mu n, \nu n \geq 0 and \mu n, \nu n \rightarrow 0 as n \rightarrow \infty ,
we define the set U\mu n, \nu n of functions
\varphi \in Lp(0, T ;V ) \cap L\infty (0, T ;H)
such that
\mathrm{s}\mathrm{u}\mathrm{p}
n
1
\nu n
\mathrm{s}\mathrm{u}\mathrm{p}
| \theta | \leq \mu n
\left( T\int
0
\bigm\| \bigm\| \varphi (t+ \theta ) - \varphi (t)
\bigm\| \bigm\| p\prime
V \prime dt
\right)
1
p\prime
<\infty .
We endow U\mu n,\mu n with the norm
\| \varphi \| U\mu n, \nu n
= \mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq T
\bigm\| \bigm\| \varphi (t)\bigm\| \bigm\|
L2(D)
+
\left( T\int
0
\bigm\| \bigm\| \varphi (t)\bigm\| \bigm\| p
V
dt
\right)
1
p
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
n
1
\nu n
\left( \mathrm{s}\mathrm{u}\mathrm{p}
| \theta | \leq \mu n
T\int
0
\bigm\| \bigm\| \varphi (t+ \theta ) - \varphi (t)
\bigm\| \bigm\| p\prime
V \prime dt
\right)
1
p\prime
.
U\mu n, \nu n is a Banach space.
Due to the compact embedding of V into H, we have the following compactness result from [1]
which is interesting in its own right.
Lemma 2. The set U\mu n, \nu n defined above is a compact subset of L2(0, T ;H).
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880 Z. I. ALI, M. SANGO
Let 2 \leq p < \infty and let \scrU \mu n, \nu n be the space consisting of random variables \varphi (t) on
\bigl(
\=\Omega , \=\scrF , \=\BbbP
\bigr)
such that
\=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq T
\bigm\| \bigm\| \varphi (t)\bigm\| \bigm\| 2
H
<\infty ,
\=\BbbE
T\int
0
\bigm\| \bigm\| \varphi (t)\bigm\| \bigm\| p
V
dt <\infty ,
\=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
n
1
\nu n
\left( \mathrm{s}\mathrm{u}\mathrm{p}
| \theta | \leq \mu n
T\int
0
\bigm\| \bigm\| \varphi (t+ \theta ) - \varphi (t)
\bigm\| \bigm\| p\prime
V \prime dt
\right)
1
p\prime
<\infty .
\scrU \mu n, \nu n is a Banach space under the norm
\| \varphi \| \scrU \mu n, \nu n
=
\Biggl(
\=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq T
\bigm\| \bigm\| \varphi (t)\bigm\| \bigm\| 2
H
\Biggr) 1
2
+
\left( \=\BbbE
T\int
0
\| \varphi (t)\| pV dt
\right)
1
p
+
+\=\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
n
1
\nu n
\left( \mathrm{s}\mathrm{u}\mathrm{p}
| \theta | \leq \mu n
T\int
0
\bigm\| \bigm\| \varphi (t+ \theta ) - \varphi (t)
\bigm\| \bigm\| p\prime
V \prime dt
\right)
1
p\prime
.
The a priori estimates established in the previous lemmas allow us to assert that for any p \geq 2, and
for \mu n, \nu n such that the series
\sum \infty
n=1
(\mu n)
1/[p\prime (p - 1)]
\nu n
converges, the sequence
\bigl\{
uM (t),M \in \BbbN
\bigr\}
remains in a bounded subset of \scrU \mu n,\nu n .
Next, let S = C(0, T ;\BbbR d) \times L2(0, T ;H) and \scrB (S) the \sigma -algebra of the Borel sets of S . For
each M , we construct the probability measure \Pi M on
\bigl(
S,\scrB (S)
\bigr)
as follows.
Consider the mapping
\varphi : \omega \mapsto \rightarrow
\bigl(
\=W (., \omega ), uM (., \omega )
\bigr)
defined on
\bigl(
\=\Omega , \=\scrF , \=\BbbP
\bigr)
and taking values in
\bigl(
S,\scrB (S)
\bigr)
. Then
\Pi M (A) = \=\BbbP (\varphi - 1(A)) for all A \in \scrB (S).
We now formulate the following key tightness result.
Lemma 3. The family of probability measures \{ \Pi M\} \infty M=1 is tight on
\bigl(
S,\scrB (S)
\bigr)
. That is, for
any \varepsilon > 0, there exist some compact subsets \Sigma \varepsilon \subset C
\bigl(
0, T ;\BbbR d
\bigr)
and Z\varepsilon \subset L2(0, T ;H) such that
\Pi M (\Sigma \varepsilon \times Z\varepsilon ) \geq 1 - \varepsilon \forall M \in \BbbN .
Proof. For the proof, we refer for instance to [2, 32, 33].
The above tightness of the family of probability measures
\bigl(
\Pi M
\bigr)
and Prokhorov’s theorem imply
that \{ \Pi M\} \infty M=1 is relatively compact. Therefore, we can extract a subsequence
\bigl\{
\Pi Mj
\bigr\} \infty
j=1
which
weakly converges to a probability measure \Pi . Hence by Skorokhod’s theorem, there exist a proba-
bility space (\Omega ,\scrF ,\BbbP ) (the expectation of which we denote by \BbbE ) and pairs of random variables
(WMj , u
Mj ) and (W,u) on (\Omega ,\scrF ,\BbbP ) with values in S such that
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 881
the probability law of (WMj , u
Mj ) is \Pi Mj , the probability law of (W,u) is \Pi , (23)
WMj (., \omega ) - \rightarrow W (., \omega ) in C(0, T ;\BbbR d) as j - \rightarrow \infty , \BbbP -a.s., (24)
uMj (., \omega ) - \rightarrow u(., \omega ) in L2(0, T ;H) as j - \rightarrow \infty \BbbP -a.s. (25)
Next, we choose the filtration (\scrF t) by setting
\scrF t = \sigma
\bigl\{
(W (s), u(s)) : 0 \leq s \leq t
\bigr\}
.
It turns out, according to similar reasoning used in [2, 32, 33], that W is a d-dimensional \scrF t-standard
Wiener process.
It follows also that
uMj (ti+1) = uMj (ti) - \tau A
Mj
ti
\bigl(
uMj (ti+1)
\bigr)
+ \~G
Mj
ti
\bigl(
ti, u
Mj (ti)
\bigr)
\Delta WMj ,i, \BbbP -a.s., (26)
where
\Delta WMj ,i =WMj (ti+1) - WMj (ti).
Then
uMj (t) = uMj (ti) if t \in [ti, ti+1), \BbbP -a.s.,
and, therefore, gluing the relations (26) by means of a summation over i = 1, . . . ,Mj - 1, we have
uMj (t) +
t\int
0
As
\bigl(
uMj (s)
\bigr)
ds =
t\int
0
G
\bigl(
s, uMj (s)
\bigr)
dWMj (s), \BbbP -a.s. (27)
as an equality between random variables with values in V \prime for any t \in [0, T ].
We are now in the position to prove Theorem 1. This will be carried through in the next section.
4. Proof of Theorem 1. The proof of Theorem 1 proceeds in several steps.
Step 1. This step is devoted to some weak convergence results. Owing to relation (27), we see
that Lemma 1 holds for the sequence uMj (t); that is, for any p \in [2,\infty ),
\mathrm{s}\mathrm{u}\mathrm{p}
M
\BbbE
T\int
0
\bigm\| \bigm\| uMj (s)
\bigm\| \bigm\| p
V
ds \leq C, (28)
\mathrm{s}\mathrm{u}\mathrm{p}
M
\mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,T ]
\BbbE
\bigm\| \bigm\| uMj (t)
\bigm\| \bigm\| 4
H
\leq C, (29)
\BbbE \mathrm{s}\mathrm{u}\mathrm{p}
\| \theta \| \leq \tau
T\int
0
\bigm\| \bigm\| uMj (t+ \theta ) - uMj (t)
\bigm\| \bigm\| p\prime
V \prime ds \leq CT
\Bigl(
\tau p
\prime - 1 + \tau
p\prime
2
\Bigr)
, (30)
\BbbE
T\int
0
\bigm\| \bigm\| As
\bigl(
uMj (s)
\bigr) \bigm\| \bigm\| p\prime
V \prime ds <\infty , (31)
and similarly
\BbbE
T\int
0
\bigm\| \bigm\| Gs
\bigl(
uMj (s)
\bigr) \bigm\| \bigm\| 4
H
ds <\infty . (32)
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882 Z. I. ALI, M. SANGO
Thus, there exists a new subsequence of uMj (t), which we still denote by the same symbol, such
that
uMj \rightharpoonup u weakly in Lp
\bigl(
\Omega , Lp(0, T ;V )
\bigr)
, (33)
uMj \rightharpoonup u weakly in L4
\bigl(
\Omega , Lr(0, T ;H)
\bigr)
\forall r \in [2,\infty ), (34)
uMj (\omega )\rightharpoonup u(\omega ) weakly star in L\infty (0, T ;H) for almost all \omega \in \Omega .
Furthermore, u satisfies
\BbbE
T\int
0
\| u(t)\| pV dt \leq C,
\BbbE
T\int
0
\| u(t)\| rHdt \leq C \forall r \in [1,\infty ),
\bigm\| \bigm\| u(\omega )\bigm\| \bigm\|
L\infty (0,T ;H)
<\infty , \BbbP -a.s.
It follows from (31) that there exists a random function \chi \in Lp\prime
\bigl(
\Omega , Lp\prime (0, T ;V \prime )
\bigr)
such that up
to extraction of a subsequence
At
\bigl(
uMj (\cdot )
\bigr)
\rightharpoonup \chi (\cdot ) weakly in Lp\prime
\Bigl(
\Omega , Lp\prime (0, T ;V \prime )
\Bigr)
. (35)
Thanks to (29) and Vitali’s theorem, we obtain
uMj - \rightarrow u strongly in L2(\Omega , L2(0, T ;H)) and almost everywhere. (36)
Thus, there exists a new subsequence still denoted as uMj , such that for almost every (t, \omega ), we have
uMj - \rightarrow u strongly inH (with respect to the measure d\BbbP \times dt). (37)
Owing to the condition (v) on G, the estimates (32), the a. e. convergence of uMj to u on \Omega \times [0, T ],
we see that G(s, uMj ) is uniformly integrable in L2
\bigl(
\Omega , L2(0, T ;H)
\bigr)
and G(s, uMj (s)) converges
to G(s, u) a. e. on \Omega \times [0, T ]. Therefore, Vitali’s theorem implies that
G(s, uMj ) \rightarrow G(s, u) strongly in L2
\bigl(
\Omega , L2(0, T ;H)
\bigr)
. (38)
Step 2. We prove in this step the convergence of the stochastic integral
T\int
0
G(t, uMj (t))dWMj (t).
We intend to use integration by parts. But since the integrand is not smooth with respect to t, we
introduce a suitable regularization in order to overcome that obstacle. For that purpose, letting \varrho be
a standard mollifier, we define, for v \in L2(D), the function
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 883
G\varepsilon (t, v) =
1
\varepsilon
T\int
0
\varrho
\biggl(
s - t
\varepsilon
\biggr)
G(s, v)ds;
G\varepsilon is smooth in t and continuous in v, and we have the uniform estimate
\BbbE
T\int
0
\| G\varepsilon (t, v)\| 2Hddt \leq \BbbE
T\int
0
\| G(t, v)\| 2Hddt (39)
and
G\varepsilon (., u) - \rightarrow G(., u) in L2
\Bigl(
\Omega , L2
\Bigl(
0, T ;Hd
\Bigr) \Bigr)
(40)
as \varepsilon \rightarrow 0.
Integrating by parts, we get
t\int
0
G\varepsilon
\bigl(
s, uMj (s)
\bigr)
dWMj (s) = G\varepsilon (t, uMj )WMj (t) -
t\int
0
G\varepsilon \prime \bigl( s, uMj (s)
\bigr)
WMj (s)ds. (41)
By Fubini’s theorem, Burkholder – Davis – Gundy’s inequality and (39), we obtain
\BbbE
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
G\varepsilon
\bigl(
s, uMj (s)
\bigr)
dWMj (s)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
H
\leq \BbbE
t\int
0
\bigm\| \bigm\| G\varepsilon
\bigl(
s, uMj (s)
\bigr) \bigm\| \bigm\| 2
Hdds \leq C. (42)
Similarly,
t\int
0
G\varepsilon (s, u)dW (s) = G\varepsilon (t, u)W (t) -
t\int
0
G\varepsilon \prime (s, u)W (s)ds. (43)
Owing to (38), we have that
G\varepsilon (t, uMj ) - \rightarrow G\varepsilon (t, u) a. e. in \Omega \times (0, T ). (44)
It then follows from the definition of G\varepsilon (G\varepsilon \prime (t, \cdot ) is still continuous in \cdot ), (41) and (24) that
\mathrm{l}\mathrm{i}\mathrm{m}
j - \rightarrow \infty
t\int
0
G\varepsilon (s, uMj )dWMj (s) = G\varepsilon (s, u)W (t) -
t\int
0
G\varepsilon \prime (s, u)W (s)ds (45)
for almost all \omega . Hence, by (43) and (45), we get
\mathrm{l}\mathrm{i}\mathrm{m}
j - \rightarrow \infty
t\int
0
G\varepsilon (s, uMj )dWMj (s) \rightarrow
t\int
0
G\varepsilon (s, u)dW (s) (46)
for almost all \omega .
By (42), the sequence of stochastic integrals
\biggl( \int t
0
G\varepsilon (s, uMj )dWMj (s)
\biggr)
i\in \BbbN
is uniformly bounded
in L2(\Omega , H) for any t \in [0, T ], then it is uniformly integrable in the space Lr(\Omega , H) for any
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884 Z. I. ALI, M. SANGO
1 \leq r < 2. Combining this with (46), we are able to use Vitali’s theorem in order to obtain that
t\int
0
t\int
0
G\varepsilon (s, uMj )dWMj (s) \rightarrow
t\int
0
G\varepsilon (s, u)dW (s) strongly in Lr(\Omega , H). (47)
On the other hand, we also have that
t\int
0
G\varepsilon (s, uMj )dWMj (s)\rightharpoonup \psi (t) weakly in L2(\Omega , H),
for some random function \psi . Therefore,
t\int
0
G\varepsilon (s, uMj )dWMj (s)\rightharpoonup \psi (t) weakly in Lr(\Omega , H) for 1 \leq r < 2.
Since the convergence (47) holds also weakly in Lr(\Omega , H), we get
\psi (t) =
t\int
0
G\varepsilon (s, u)dW (s),
by uniqueness of weak limits. Thus,
t\int
0
G\varepsilon (s, uMj )dWMj (s)\rightharpoonup
t\int
0
G\varepsilon (s, u)dW (s) weakly in L2(\Omega , H).
This can be expressed as: for fixed \varepsilon let j tends to \infty to have, for any \kappa \in L2(\Omega , H),
\BbbE
\left( \kappa , t\int
0
G\varepsilon (s, uMj )dWMj (s)
\right) \rightarrow \BbbE
\left( \kappa , t\int
0
G\varepsilon (s, u)dW (s)
\right) . (48)
We obviously have that the sequence
\biggl( \int t
0
G\varepsilon (s, uMj )dWMj (s)
\biggr)
j\in \BbbN
is uniformly bounded in L2(\Omega , H).
Thus, there exists \eta \in L2(\Omega , H) such that for any \kappa \in L2(\Omega , H)
\BbbE
\left( \kappa , t\int
0
G\varepsilon (s, uMj )dWMj (s)
\right) \rightarrow \BbbE (\kappa , \eta ) as j \rightarrow \infty .
Lastly we need to prove that
t\int
0
G(s, u)dW (s) = \eta . For that purpose, we rewrite (48) as follows:
\BbbE
\left( \kappa , t\int
0
G(s, uMj )dWMj (s) -
t\int
0
G(s, u)dW (s)
\right) = I\varepsilon 1 + I\varepsilon 2 + I\varepsilon 3 , (49)
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 885
where \kappa is an arbitrary element of L2(\Omega , H) and
I\varepsilon 1 = \BbbE
\left( \kappa , t\int
0
\bigl[
G(s, uMj ) - G\varepsilon (s, uMj )
\bigr]
dWMj (s)
\right) ,
I\varepsilon 2 = \BbbE
\left( \kappa , t\int
0
\bigl[
G\varepsilon (s, u) - G(s, u)
\bigr]
dW (s)
\right) ,
I\varepsilon 2 = \BbbE
\left( \kappa , t\int
0
G\varepsilon (s, uMj )dWMj (s) -
t\int
0
G\varepsilon (s, u)dW (s)
\right) .
By Burkholder – Davis – Gundy’s inequality
I\varepsilon 1 \leq \BbbE \| \kappa \| L2(D)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
\bigl[
G(s, uMj ) - G\varepsilon (s, uMj )
\bigr]
dWMj (s)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
H
\leq
\leq C\BbbE
\left[ t\int
0
\bigm\| \bigm\| G(s, uMj ) - G\varepsilon (s, uMj )
\bigm\| \bigm\| 2
Hd ds
\right]
1
2
and
I\varepsilon 2 \leq C\BbbE
\left[ t\int
0
\bigm\| \bigm\| G(s, u) - G\varepsilon (s, u)
\bigm\| \bigm\| 2
Hdds
\right]
1
2
.
Passing to the limit as \varepsilon - \rightarrow 0 in the above inequalities and using (40), we get that \mathrm{l}\mathrm{i}\mathrm{m}\varepsilon - \rightarrow 0
\bigl(
I\varepsilon 1 +
+ I\varepsilon 2
\bigr)
= 0. By (48), we have
I\varepsilon 3 = \BbbE
\left( \kappa , t\int
0
G\varepsilon (s, uMj )dWMj (s) -
t\int
0
G\varepsilon (s, u)dW (s)
\right) \rightarrow 0.
Thus, it follows from (49) that
t\int
0
G(s, uMj )dWMj (s)\rightharpoonup
t\int
0
G(s, u)dW (s) weakly in L2(\Omega , H). (50)
Step 3. In this step, we prove that \chi = At(u). This will close the arguments leading to the
complete proof of Theorem 1, since passing to the limit in equation (27) and using the convergences
(35), (37), (50), we have that
u (t) +
t\int
0
\chi (s) ds =
t\int
0
G (s, u (s)) dW (s) , \BbbP -a.s. in V \prime . (51)
Then the needed relation will follow from the definition of pseudomonotone operators as given
in the introduction.
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886 Z. I. ALI, M. SANGO
Let us prove (51). We recall the equation (27)
uMj (t) +
t\int
0
As
\bigl(
uMj (s)
\bigr)
ds =
t\int
0
G
\bigl(
s, uMj (s)
\bigr)
dWMj (s), \BbbP -a.s. in V \prime . (52)
Testing the equation (52) with uMj - u with respect to the inner product of H, we get
t\int
0
\bigl(
uMj (s) - u(s), duMj
\bigr)
+
t\int
0
\bigl\langle
As
\bigl(
uMj (s)
\bigr)
, uMj (s) - u(s)
\bigr\rangle
ds =
=
t\int
0
\bigl(
G
\bigl(
s, uMj (s)
\bigr)
, uMj (s) - u(s)
\bigr)
dWMj (s).
By Burkholder – Davis – Gundy’s inequality and (36), it is clear that
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
E
t\int
0
\bigl(
G(s, uMj (s)), uMj (s) - u(s)
\bigr)
dWMj (s) = 0.
Therefore, (51) will follow, if we can show the following lemma.
Lemma 4. Under our conditions
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f} \BbbE
t\int
0
\bigl(
uMj (s) - u(s), duMj
\bigr)
\geq 0.
This result is crucial for our work. It is the stochastic version of a result by Landes (see [21]).
The proof is partly based on the following integration by parts of stochastic integrals:
\bigl(
uMj (t), u(t)
\bigr)
=
t\int
0
\bigl(
duMj (s), u(s)
\bigr)
+
+
t\int
0
\bigl(
du(s), uMj (s)
\bigr)
+
\bigl\langle \bigl\langle
uMj , u
\bigr\rangle \bigr\rangle H
t
, (53)
where the last term denotes the quadratic covariation of uMj and u, namely, if \{ ei\} i\in \BbbN is an or-
thonormal basis of H, \bigl\langle \bigl\langle
uMj , u
\bigr\rangle \bigr\rangle H
t
=
=
\infty \sum
i=1
\left( t\int
0
G
\bigl(
s, uMj (s)
\bigr)
dWMj (s), ei
\right) \left( t\int
0
G(s, u(s))dW (s), ei
\right) =
=
\left( t\int
0
G(s, uMj (s))dWMj (s),
t\int
0
G(s, u(s))dW (s)
\right) .
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 887
In view of (50),
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\BbbE
\bigl\langle \bigl\langle
uMj , u
\bigr\rangle \bigr\rangle H
t
= \BbbE
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
G(s, u(s))dW (s)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
H
=
= \BbbE
t\int
0
\bigm\| \bigm\| G(s, u(s))\bigm\| \bigm\| 2
H
ds, (54)
thanks to Fubini’s theorem and Itô’s isometry.
Next, we show that
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\BbbE
t\int
0
\bigl(
duMj (s), u(s)
\bigr)
= \mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\BbbE
t\int
0
\bigl(
du(s), uMj (s)
\bigr)
. (55)
Testing (52) with u, and using the convergences (35) and (50), it readily follows that
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\BbbE
t\int
0
\bigl(
duMj (s), u(s)
\bigr)
= - \BbbE
t\int
0
\langle \chi (s), u\rangle + \BbbE
t\int
0
\bigl(
G(s, u(s)), u
\bigr)
dW (s).
Similarly
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\BbbE
t\int
0
\bigl(
du(s), uMj (s)
\bigr)
= - \BbbE
t\int
0
\langle \chi (s), u\rangle + \BbbE
t\int
0
\bigl(
G(s, u(s)), u
\bigr)
dW (s).
Hence, (55) holds.
Since uMj (t) weakly converges to u(t), by (28) for almost every t, we deduce from (53) – (55)
that
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\BbbE
t\int
0
\bigl(
duMj (s), u(s)
\bigr)
=
1
2
\BbbE
\left( \bigm\| \bigm\| u(t)\bigm\| \bigm\| 2
H
-
t\int
0
\bigm\| \bigm\| G(s, u(s))\bigm\| \bigm\| 2
H
ds
\right) . (56)
By Itô’s formula,
\BbbE
t\int
0
\bigl(
duMj (s), uMj (s)
\bigr)
=
=
1
2
\BbbE
\left( \bigm\| \bigm\| uMj (t)
\bigm\| \bigm\| 2
H
-
t\int
0
\bigm\| \bigm\| \bigm\| G(s, uMj (s))
\bigm\| \bigm\| \bigm\| 2
H
ds
\right) . (57)
We then deduce from (57), (38), Fatou’s lemma and the weak semicontinuity of the norm \| \cdot \| H
that
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f} \BbbE
t\int
0
\bigl(
duMj (s), uMj (s)
\bigr)
=
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888 Z. I. ALI, M. SANGO
=
1
2
\mathrm{l}\mathrm{i}\mathrm{m}
j\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f} \BbbE
\left( \bigm\| \bigm\| \bigm\| uMj (t)
\bigm\| \bigm\| \bigm\| 2
H
-
t\int
0
\bigm\| \bigm\| \bigm\| G(s, uMj (s))
\bigm\| \bigm\| \bigm\| 2
H
ds
\right) \geq
\geq 1
2
\BbbE
\left( \bigm\| \bigm\| u(t)\bigm\| \bigm\| 2
H
-
t\int
0
\bigm\| \bigm\| \bigm\| G(s, u(s))\bigm\| \bigm\| \bigm\| 2
H
ds
\right) . (58)
The Lemma 4 now follows from (56) and (58), using the subadditivity property of \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} . This
subsequently proves (51), and we can deduce from the definition of pseudomonotone operators that
\chi (t) = At(u). We therefore conclude that u is a weak solution of problem (P ).
Theorem 1 is proved.
Remark 1. The arguments used in the proof of our main result readily apply to the non vanishing
initial value (u(0) \not = 0) case and to the case when the process W (t) is a cylindrical Hilbert space-
valued Wiener process. We omitted these generalities in order to focus on the key ideas leading to
the settling of the main problematic. Since pseudomonotone operators arise naturally in variational
inequalities, the approach developed here is a decisive stepping stone for the generalization of our
result to stochastic variational inequalities featuring pseudomontone operators.
In the next section, we provide an example of application of our main result which includes seve-
ral important particular cases of stochastic partial differential equations arising in applied sciences.
5. Example of application of Theorem 1. From now on, we set V = Wm,p
0 (D), H = L2(D)
and V \prime = W - m,p\prime (D) with p\prime , the conjugate of p and p \geq 2. By Rellich – Kondrachov embedding
theorem, V is compactly embedded in H which in its turn is continuously embedded in V
\prime
. As
applications of the theory developed in the paper, we consider the higher-order stochastic quasilinear
parabolic problem
(P )
\left\{
du+
\bigl[
At(u) + g(t, x, u)
\bigr]
dt = f(t, x)dt+G(t, u)dW (t) in QT ,
D\alpha u = 0 = 0 on (0, T )\times \partial D for | \alpha | \leq m - 1,
u(x, 0) = 0 in D,
where T > 0 is fixed real number, D is a bounded domain in \BbbR n, QT is the cylinder (0, T )\times D, the
stochastic process u = u(t, x) and the standard d-dimensional Wiener process W together with the
probability space on which they are defined are the unknowns, the functions f, G and g are given,
At is an elliptic operator of order 2m in the generalized divergence form, that is,
At(u) =
\sum
| \beta | \leq m
( - 1)| \beta | D\beta A\beta (t, x, u,Du, . . . ,D
mu),
with the functions A\beta satisfying the Carathéodory conditions, that is each A\beta (t, x, \xi ) is measurable
in (t, x) and continuous in \xi . Here \xi is an element of the vector space \BbbR N of m-jets on \BbbR n which
assumes the representation \xi = \{ \xi \alpha : | \alpha | \leq m\} . To each \xi , there corresponds a couple (\eta , \zeta ), with
\eta = \{ \eta \alpha : | \alpha | \leq m - 1\} and \zeta = \{ \zeta \alpha : | \alpha | = m\} . Let Q = [0, T ] \times D and 2 \leq p < \infty . We now
formulate the conditions on A\beta following Browder [11].
(i) For each multiindex \beta with | \beta | \leq m, A\beta (t, x, \xi ) is Carathéodory; that is, it is measurable in
(t, x) on Q = [0, T ] \times D for each fixed m-jet \xi =
\bigl\{
\xi \alpha : | \alpha | \leq m
\bigr\}
and continuous in \xi for almost
all (t, x). In addition, there exist a constant c0 > 0 and nonnegative function h0 \in Lp\prime
\bigl(
0, T ;Lp\prime (Q)
\bigr)
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 889
such that \bigm| \bigm| A\beta (t, x, \xi )
\bigm| \bigm| \leq c0
\bigl\{
| \xi | p - 1 + h0(t, x)
\bigr\}
for all (t, x) \in [0, T ]\times D and all m-jets \xi .
(ii) If we divide up the m-jet \xi into its pure mth order part \zeta and the lower-order jets \eta , then\sum
| \beta | =m
\Bigl[
A\beta (t, x, \eta , \zeta ) - A\beta (t, x, \eta , \~\zeta )
\Bigr]
(\zeta \beta - \~\zeta \beta ) > 0
for \zeta \beta \not = \~\zeta \beta and for all (t, x) \in Q.
(iii) There exist a constant c1 > 0 and a positive function h1 \in L1(Q) such that\sum
| \beta | \leq m
A\beta (t, x, \xi )\xi \beta \geq c1| \xi | p - h1(t, x)
for all (t, x) \in Q and all \xi .
(iv) g(t, x, u) is Carathéodory. It satisfies the sign condition rg(t, x, r) \geq 0, g(t, x, 0) = 0 and
g \in L\infty (Q).
(v) The intensity of the noise G(t, u) : [0, T ] \times L2(D) - \rightarrow (L2(D))d is continuous in (t, u),
and there exists a positive constant C such that\bigm\| \bigm\| G(t, u)\bigm\| \bigm\|
(L2(D))d
\leq C
\bigl(
1 + \| u(t)\| L2(D)
\bigr)
.
(vi) We assume f(t, x) is measurable in Q and there exists a positive constant C such that
T\int
0
\| f(t)\| p
\prime
Lp\prime (D)
dt \leq C.
We consider the operator family \scrA t : Wm,p
0 (D) - \rightarrow W - m,p\prime (D), defined by
\langle \scrA t(u), v\rangle =
\sum
| \beta | \leq m
\int
D
A\beta (t, x, u,Du, . . . ,D
mu)D\beta vdx
for any u, v \in Wm,p
0 (D) and for any t \in [0, T ].
We note that under the conditions (i) – (iii), the operator \scrA t is pseudomonotone, as proved by
Browder in [10] (see also [6, 8]). Alongside \scrA t, we consider the operator S : V \rightarrow V \prime such that
\langle S(u), v\rangle =
\int
D
g(t, x, u(x))v(x)dx.
Based on the above conditions, as in [41], we have the following crucial result.
Lemma 5. The operators \scrA t and S induce the operator
\scrA t + S : Lp(0, T ;V ) \rightarrow Lp\prime (0, T ;V \prime ),
which is pseudomonotone.
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890 Z. I. ALI, M. SANGO
Now we can invoke Theorem 1, to infer that problem (P ) has at least a martingale solution
u \in L2
\Bigl(
\Omega , L\infty \bigl( 0, T ;L2(D)
\bigr) \Bigr)
\cap Lp
\Bigl(
\Omega , Lp
\bigl(
0, T ;Wm,p
0 (D)
\bigr) \bigr)
,
in the sense of Definition 1.
The result of this example is to the best of our knowledge new for higher-order quasilinear
stochastic parabolic equations of pseudomonotone type. The merit of the general approach undertaken
in the work, is that the arguments are independent of the order of the equations. In the second-order
case, arguments relying on truncation functions may lead to particular versions of our result, but
it is well-know that such arguments break in the higher order case, due to lack of appropriate
corresponding notion of truncation.
6. Closing remarks. We are deeply grateful to one of the reviewers for her/his insighful
comments and for suggesting that we compare our work to those of Liu and Röckner on local
monotonicity which she/he brought to our attention. This section is devoted to that task.
The main difference between our work and those of Liu and Röckner [22 – 24] is that while we
are dealing in the paper with genuine pseudomonotone operators, as defined by Brezis and Browder
[5, 9, 10], for the class of stochastic evolution equations studied, they consider local monotonicity,
generalizing the results of Pardoux [27], Krylov and Rozovskii [20].
Namely, they assume the following condition appearing in these papers as
(H2) (local monotonicity)
2\langle A(t, v1) - A(t, v2), v1 - v2\rangle V \ast \times V +
\bigm\| \bigm\| B(t, v1) - B(t, v2)
\bigm\| \bigm\| 2
2
\leq
\leq
\bigl(
K + \rho (v2)
\bigr)
| v1 - v2| 2H ,
where \rho : V \rightarrow [0,+\infty ) is a measurable function and locally bounded in V .
In [24] under (H2) and some additional conditions such as coercivity and growth conditions,
they establish unique strong probabilistic solution for evolution equations of the type
dXt = A(t,Xt)dt+B(t,Xt)dWt.
Condition (H2) is further weakened in [23] to
(H \prime \prime
2 ) (local monotonicity)
2\langle A(t, v1) - A(t, v2), v1 - v2\rangle V \ast \times V +
\bigm\| \bigm\| B(t, v1) - B(t, v2)
\bigm\| \bigm\| 2
2
\leq
\leq (f + \eta (v1) + \rho (v2))| v1 - v2| 2H ,
where f \in L1
\bigl(
[0, T ],\BbbR
\bigr)
, \eta , \rho : V \rightarrow [0,+\infty ) are measurable functions and locally bounded in V.
In [23], Liu shows that (H \prime \prime
2 ) together with some additional conditions imply pseudomonotonicity
of A(t, \cdot ) in the same sense as ours. So pseudomonotonicity as considered by us is weaker than the
above (H \prime \prime
2 ) version of local monotonicity. But the results of [23] are essentially deterministic since
they are established for evolution equations of the form
u\prime (t) = A(t, u(t)) + b(t). (59)
For the applications of (H \prime \prime
2 ) to SDEs, Liu announced in [23] the investigation stochastic evolu-
tion equations of the form
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PROBABILISTIC WEAK SOLUTIONS FOR NONLINEAR STOCHASTIC EVOLUTION PROBLEMS . . . 891
dXt = A(t,Xt)dt+BdNt,
with Levy type additive noise. This project was fulfilled in [22] jointly with Röckner, for the equation
dXt = A(t,Xt)dt+B(t)dWt (60)
driven by Wiener noise and B(t) is a functional depending on t only, by reducing (60) to the
deterministic like evolution equation (59), thanks to a transformation made possible by the special
form of the random forcing B(t)dWt. Hence the results for equation (60) are derived from those of
this deterministic equation established in [23].
The results of [24] were extended to the case of SPDEs driven by Levy noises in [14].
These facts show that the existence result for the genuine pseudomonotone operator that we
prove was never established, to the best of our knowledge, in a previous paper and neither in Liu and
Röckner’s works.
The stochastic evolution equation that we consider in its current form can’t be reduced to a
deterministic like equation.
Due to lack of monotonicity and local monotonicity, Galerkin’s method is not applicable to our
stochastic evolution problem due to unavailability of a Carathéodory like existence result for the cor-
responding system of stochastic ordinary differential equations arising in the Galerkin approximation
scheme. This compels us to use Gyöngy – Millet’s numerical scheme.
Due to lack of local Lipschitzity on the intensity of the noise in our equation, the natural solution
is martingale like (probabilistic weak in the sense of Skorokhod) as considered by us. The papers
considering local monotonicity which implicitly implies local Lipschitzity of the intensity of the
noise, are able to establish strong solutions.
A key point in our work is a sign condition introduced by Landes [21] for deterministic equations
involving pseudomonotone operators that we successfully extend to the stochastic case.
Our work therefore generalize the results obtained for SPDEs under local monotonicity and
methodologically, our approach is different as well.
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Received 01.01.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 7
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| id | umjimathkievua-article-2286 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:21:50Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/0c/05e5d104ebdff8b006899098400c170c.pdf |
| spelling | umjimathkievua-article-22862022-10-24T09:23:10Z Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators Ali, Z. I. Sango, M. Ali, Z. I. Sango, M. . UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these equations despite numerous works involving various generalizations of the monotonicity condition. Key to our work is a sign result for the Ito differential of an approximate solution that we establish, as well as several compactness results of the analytic and probabilistic nature, and a characterization of pseudomonotone operators due to F. E. Browder. УДК 519.21 Імовiрнiснi слабкi розв’язки нелiнiйних стохастичних еволюцiйних задач, що мiстять псевдомонотоннi оператори Вивчається важливий клас стохастичних нелiнiйних еволюцiйних задач iз псевдомонотонними елiптичними частинами. Встановлено iснування ймовiрнiсних слабких (або мартингальних) розв’язкiв. На сьогоднi не iснує теорiї розв’язностi, розробленої для таких рiвнянь, незважаючи на те, що є багато робiт, в яких вивчаються рiзнi узагальнення умов монотонностi. Ключем до нашої роботи є знаковий результат для диференцiала Iто наближеного розв’язку, який ми встановлюємо, а також кiлька результатiв щодо компактностi аналiтичної та ймовiрнiсної природи i характеризацiя псевдомонотонних операторiв по Ф. E. Браудеру Institute of Mathematics, NAS of Ukraine 2022-08-09 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2286 10.37863/umzh.v74i7.2286 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 7 (2022); 871 - 892 Український математичний журнал; Том 74 № 7 (2022); 871 - 892 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2286/9273 Copyright (c) 2022 M SANGO |
| spellingShingle | Ali, Z. I. Sango, M. Ali, Z. I. Sango, M. Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title | Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title_alt | Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title_full | Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title_fullStr | Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title_full_unstemmed | Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title_short | Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| title_sort | probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators |
| topic_facet | . |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2286 |
| work_keys_str_mv | AT alizi probabilisticweaksolutionsfornonlinearstochasticevolutionproblemsinvolvingpseudomonotoneoperators AT sangom probabilisticweaksolutionsfornonlinearstochasticevolutionproblemsinvolvingpseudomonotoneoperators AT alizi probabilisticweaksolutionsfornonlinearstochasticevolutionproblemsinvolvingpseudomonotoneoperators AT sangom probabilisticweaksolutionsfornonlinearstochasticevolutionproblemsinvolvingpseudomonotoneoperators |