This paper focuses on the study of the existence of a mild solution to time and space-fractional stochastic equation perturbed by multiplicative white noise. The required results are obtained by means of Krasnoselskii's fixed point theorem. stochastic equation, mild solution, Expectation, Krasnoselskii' s fixed point theorem. IAM'20: Third conference on informatics and applied mathematics, 21-22 October 2020,Guelma, ALGERIA " djourdem.habib7@gmail.com (H. Djourdem); bouteraa-27@hotmail.fr (N. Boutera) 0000-0002-7992-581X (H. Djourdem); 0000-0002-8772-1315 (N. Boutera)
(
equations subject to the initial condition 0+ [ − ℎ ( )] = Δ ( ) + ⋅ ∇ + ( ) ( ) , ∈ , > 0, ( , 0) = 0 ( ) , ∈ , = 0, and the Dirichlet boundary conditions ( , ) = 0, ∈ , where ⊂
ℝ , ( , ) represents the velocity field of the fluid, the state separable real Hilbert space with inner product ⟨⋅, ⋅⟩, the term ( ) a state dependent random noise, where in completed probability space (Ω, , ) with expectaction ( ) ∈[0, ] is a −
(⋅) takes values in a ( ) =
( ) describes adapted
{ (, ) ( , ) = Γ(1− ) ∫0 ( − ) , 0 < < 1, = 1, where Γ (⋅) stands for the Gamma function.
The existence and non-existence of solutions for the Navier-Stokes equations (NSEs) have been discussed in [13]. Chemin et al. [5] studied the global regularity for the large solutions to the NSEs. Miura [19] focused on the uniqueness of mild solutions to the NSEs. Germain [9] presented the uniqueness criteria for the solutions of the Cauchy problem associated to the NSEs. However, The existence and uniqueness of solutions for the stochastic Navier-Stokes equations (SNSEs) with multiplicative Gaussian noise were proved in [8], [20]. The large deviation principle for SNSEs with multiplicative noise had been established in [23], [27], [31], [32]. Just mention a few, the study of time-fractional Navier-Stokes equations has become a hot topic of research due to its significant role in simulating the anomalous difusion in fractal media [12], [16], [33].
There has been a widespread interest during the last decade in constructing a stochastic integration theory with respect to fractional Brownian motion (FBM) and solving stochastic diferential equations driven by FBM. On the other hand, time-fractional diferential equations are found to be quite efective in modelling anomalous difusion processes as its can characterize the long memory processes [22], [25], [28], [29], [30]. Hence, Burgers equation with time-fractional can be adapted to describe the memory efect of the wall friction through the boundary layer [35]. Furthermore, the analytical solutions of the time- and space-fractional Burgers equations have been investigated by variational iteration method [11] and Adomian decomposition method [21].
The existence of solution for partial neutral integro-diferential equation with infinite delay in infinite dimensional spaces has been extensively studied by many authors. Ezzinbi and al. [7] investigated the existence and regularity of solutions for some partial functional integrodiferential equations in Banach spaces. Cui and Yan [4] investigated the existence of mild solutions for a class of fractional neutral stochastic integro-diferential equations with infinite delay in Hilbert spaces by means of Sadovskii’s fixed point theorem and in another paper [1], Balasubramaniam et al. discussed the existence of mild and strong solutions of semilinear neutral functional diferential evolution equations with nonlocal conditions by using fractional power of operators and Krasnoselskii fixed point theorem, Djourdem and Bouteraa [6] studied the existence of a mild solution to time and space-fractional stochastic equation perturbed by multiplicative with noise via Sadovskii’s fixed point theorem. In particular, the stability theory of stochastic diferential equations has been popularly applied in variety fields of science and technology. Several authors have established the stability results of mild solutions for these equations by using various techniques, we refer the reader to [2], [10], [17], [26].
The main contribution of this paper is to establish the existence of mild solution for the
problem (
In this section, we give some notions and certain important preliminaries, which will be used by in the subsequent discussions. Let (Ω, , ,
{ } ≥0) be a filtered probability space with a normal ifltration, where is a probability measure on (Ω, ) and is the Borel −algebra . Let { } ≥0 satisfying that 0
contains all -null sets. Theoperator is the infinitesimal generator of a strongly continuous semigroup on a separable real Hilbert space . ∞ and ( ) by the usual Lebesgue and Sobolev space, respectively. We assume that is the negative Laplacian −Δ in a bounded domain with zero Dirichlet boundary conditions in Hilbert space = 2 ( ), which are given since the operator is self-adjoint, i.e., there exist the eigenvectors corresponding to eigenwhere { } =1 norm
1
Let 20 = 2 ( 2 ( ) , )
where = ⟨, ⟩ with the inner product ⟨⋅, ⋅⟩ in 2 ( ), the norm ‖
Then we can rewrite the equation (
‖ ‖ = ‖ 2 ‖‖, the bilinear ‖ ( ) = (, ) {
( ) , ≥ 0} is a -Wiener process with linear bounded covarience operator such 1, 2, …, then the Wiener process is given by ∞ =1 ∞ √ 1 be a Hilbert-Schmidt space of operators from 2 ( ) to with the ‖ ‖ 20 = ‖ ‖ ‖ 1 ‖ 2 ‖‖ = ∞ ∑ i.e.,
, there exists a constant dependent on such that
We shall also need the following result with respect to the operator (see [28]). For any > 0, an analytic semigroup ( ) = − , ≥ 0 is generated by the operator on ‖ [0, ] × Ω → 02 which satisfies in which ( ) denotes the Banach space of all bounded operators from to itself.
Next we will introduce the following lemma to estimate the stochastic integrals, which contains the Burkhoder-Davis-Gundy’s inequality.
[15] For any 0 ≤ 1 < 2 ≤ and ≥ 2 and for any predictable stochastic process ∶ then, we have ⎢⎜∫ ‖ ( )‖220 ⎞⎟⎟⎠ 2 ⎤⎥⎥⎥⎦ < ∞, ⎡⎛ ⎢ ⎜ ⎢⎝0 ⎣ ⎡‖‖ 2 ⎢‖ ⎢‖‖ 1 ⎣ ‖ ⎢‖∫ ( )
‖ ⎤ ⎡⎛ 2 ( )‖‖‖‖ ⎥⎥ < ( ) ⎢⎢⎜∫ ‖ ( )‖220 ⎞⎟⎟⎠ 2 ⎥⎤⎥⎥⎦ .
‖‖ ⎥⎦ ⎢⎣⎜⎝ 1
Ispired by the definition of the mild solution to the time-fractional diferential equations (see [24], [34]), we give the following definition of mild solution for our time-fractional stochastic equation. lowing integral equation is satisfied
An -adapted stochastic process ( ( ) , ∈ [0, ]) is called a mild solution to (
+ ∫ ( − )
−1
where the generalized Mittag-Lefler operators ( ) and , ( ) are defined, respectively, by
Mainardi’s Wright-type function with ∈ (
( ) ‖‖ ≤ ∫ ( ) ‖ ( ) ‖
∞ 0 ∞ 0 ∞ ∞ 0 0 = ∞
∞ ≤ ∫ 0
∞
≤ ∫
[3] For any ∈ (
. ( ) ≥ 0 ∫ 0 ∞
( ) = Γ (1 + ) Γ (1 + )
, − 2 −
( ) ‖ ‖ Γ (1 − ) Γ (1 − ) − 2 ‖ ‖ , ∈ 2 ( ) ,
− 2 1− ( ) ‖ ‖ for all ≥ 0.
The operators and { ( )} ≥0 and { , ( )} ≥0
in (
For any ( ) and , ( ) are linear and bounded operators. Moreover, for 0 < < 1 and 0 ≤ < 2, there exists a constant > 0 such that ( ) and , ( ) are defined, respectively, by ‖ ( ) ‖ ≤ − 2 ‖ ‖ , ‖‖ ,
( ) ‖‖ ≤ − 2 ‖ ‖ .
For > 0 and 0 ≤ < 2, by means of Lemma 2 and Lemma 2, we have
2 Γ (2 − 2 ) 0 Γ (1 + (1 − 2 ))
( 2 − 1) 2 ‖ ‖ , ∈ 2 ( ) .
⎞
⎟
⎠
}
It is obviously to see that the term
and
3. Existence results
and we define the following space
In this section, we present our main results on the existence of mild solutions of problem (
( 1) is the infinitesimal generator of a strongly continuous semigroup { ( ) , ≥ 0} on . We will also suppose that the operator ( ) , > 0 is compact. ( 2) The function ∶ Ω× → 02 satisfies the following global Lipshitz and growth conditions: and and satisfies the following properties 0 be a real number, then the bounded bilinear operator ∶ 2 ( ) → −1 ( )
Our main results is based on the following Krasnoselskii fixed points theorem [14]. [14] Let a closed, bounded, convex and nonemty subset of . Consider the be a Banach space, operators 1
and 2 ( ) 1 + 2
∈ ( ) 1
such that whenever , is a contraction mapping. ( ) 2 is compact and continuous.
.
In the proof of main result, we need the following Lemmas. Assume that conditions ( 1) and
equation (
0
⎛
⎝0
‖
(
By using also the Holder inequality and Lemma 2, we obtain −1 = 2 ∫ [‖ ( )‖ ] , (13) ‖ ‖ ‖ ‖0 ⎡⎛ ⎢ ⎜ ⎢⎝0 ⎣ ⎛ ⎝0 ⎡ ∞ ⎢ 2 .
So, we conclude Φ3 ( ) ⊂ .
Next, we show that Φ3 ( ) ⊂ . By ( 1), ( 5) and from (13), we have
Then Φ3 is continuous and maps into . The continuity of Φ3 follows from ( 4). ⎡ ∞ ⎢ ⎣0 ⎡ ∞ ⎣0 −1 −1 Now, we set = 1 + 2, where ⎥⎥ and for ∈ [0, ].
We set 0 0 ( ) ( 2 ) ( ) = ∫ ( − ) For any ≥ 2, by vertue of Lemma 2, it follows that
[‖ 1‖ ] = [ ‖ ( 2) 0 − ( 2) 0‖ ] bounded. operators 1 is strongly continuous. Assume ( 2) , ( 4), ( 5) hold and 0 < < ≤ 2, ≥ 2, then the operator 2 is uniformly −1 − ∫ ( 1 − ) −1
1
−1
From Lemma 11, using the estimate (
1 0
1 + ∫ [( 2 − ) −1 − ( 1 − )
2 that is the operator 2 is uniformly bounded. and 0 1
2 0 1 0 −1
1
0
2
1
= ∫ ( 1 − )
3 = ∫ ( 2 − )
−1
, ( 2 − ) ( ( ))
− ∫ ( 1 − )
−1
, ( 1 − ) ( )
( )
( )
1
0
2
1
⎡‖ 1
‖
⎣‖0
⎢‖∫ [( 2 − )
‖
( )
−1
( )
For the first term 21 in (15), applying the assumptions ( 5) and Lemma 2 and Holder inequality,
Using the assumptions ( 5) and Lemma 2 and Holder inequality, we have
and
( ∈[0, ]
[
‖ ( )‖
2
1]
)
( 2 − 1)
(1− )−2
2
We shall employ Theorem 3. For better readability, we divide the proof into two steps.
argument to obtain (
and ∈ (0, ). By the assumptions ( 2) and similar
where 0 = ℎ < 1.
Hence 1 is a contraction on . ‖ ‖ ‖ ‖0
‖
‖
‖
‖
‖
2
which implies
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