On the Existence and long time behaviour of \(H^{1}\)-Weak Solutions for \(2, 3d\)-Stochastic \(3^{rd}\)-Grade Fluids Equations

In the present work, we investigate stochastic third grade fluids equations in a \(d\)-dimensional setting, for \(d = 2, 3\). More precisely, on a bounded and simply connected domain \(\mathcal{D}\) of \(\mathbb{R}^d\), \(d = 2,3\), with a sufficiently regular boundary \(\partial \mathcal{D},\) we c...

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Bibliographic Details
Published in:arXiv.org 2023-11
Main Authors: Nouira, Raya, Cipriano, Fernanda, Tahraoui, Yassine
Format: Article
Language:English
Subjects:
Asymptotic properties
Boundary conditions
Conjugation
Dirichlet problem
Fluid flow
Incompressible flow
Mathematical analysis
Sobolev space
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Summary:In the present work, we investigate stochastic third grade fluids equations in a \(d\)-dimensional setting, for \(d = 2, 3\). More precisely, on a bounded and simply connected domain \(\mathcal{D}\) of \(\mathbb{R}^d\), \(d = 2,3\), with a sufficiently regular boundary \(\partial \mathcal{D},\) we consider incompressible third grade fluid equations perturbed by a multiplicative Wiener noise. Supplementing our equations by Dirichlet boundary conditions and taking initial data in the Sobolev space \(H^1(\mathcal{D})\), we establish the existence of global stochastic weak solutions by performing a strategy based on the conjugation of stochastic compactness criteria and monotonicity techniques. Furthermore, we study the asymptotic behaviour of these solutions, as \(t \to \infty\).
ISSN:2331-8422