A central limit theorem and moderate deviation principle for the stochastic 2D Oldroyd model of order one
type: Item
instance of: scholarly article
title: A central limit theorem and moderate deviation principle for the stochastic 2D Oldroyd model of order one
zbMATH Open document ID: 1471.60100
author: Manil T. Mohan
published in: Stochastic and Partial Differential Equations. Analysis and Computations
publication date: 2021-08-12T00:00:00Z
review text: The author considers the Oldroyd model of order one for non-Newtonian fluid flows written as \(\frac{\partial u^{\varepsilon }}{\partial t}(t,x)-\mu \Delta u^{\varepsilon }(t,x)-(\beta \ast \Delta u^{\varepsilon })(t,x)+(u^{\varepsilon }(t,x)\cdot \nabla )u^{\varepsilon }(t,x)+\nabla p^{\varepsilon }(t,x)=f(t,x)+\sqrt{\varepsilon }\sigma (t,u^{\varepsilon }(t,x))\frac{\partial W}{\partial t}(t,x)\), \(\nabla \cdot u^{\varepsilon }(t,x)=0\), \(\int_{G}p^{\varepsilon }(t,x)dx=0\), and posed in \((0,\infty )\times G\), where \(G\) is a bounded and open domain in \(\mathbb{R}^{2}\) with smooth boundary \(\partial G\).\ Here \(u^{\varepsilon }(t,x)\) denotes the velocity field, \(p^{\varepsilon }(t,x)\) the pressure field, \(f\) a deterministic external force and \(W\) an \(\mathbb{H}\)-valued Wiener process, where \(\mathbb{H}\) is the completion of \(\mathcal{V}=\{u\in C_{0}^{\infty }(G,\mathbb{R}^{2}):\nabla \cdot u=0\}\) with respect to the \(\mathbb{L} ^{2}(G)\)-norm, \(\mathbb{H}=\{u\in L^{2}(G):\nabla \cdot u=0\), \(u\cdot n\mid _{\partial G}=0\}\), if \(G\) is bounded.\ The initial and boundary conditions \( u^{\varepsilon }(0,x)=u^{0}(x)\) and \(u^{\varepsilon }(t,x)=0\) on \(\partial G\) are imposed. The author introduces the Helmholtz-Hodge orthogonal projection \(P_{\mathbb{H}}:\mathbb{L}^{2}(G)\rightarrow \mathbb{H}\), the operator \(A\) defined as \(Au=-P_{\mathbb{H}}\Delta u\), \(u\in D(A)=\mathbb{V}\cap \mathbb{H} ^{2}(G)\), where \(\mathbb{V}\) is the completion of \(\mathcal{V}=\{u\in C_{0}^{\infty }(G,\mathbb{R}^{2}):\nabla \cdot u=0\}\) with respect to the \( \mathbb{H}^{1}(G)\)-norm, and the trilinear form \(b(.,.,.):\mathbb{V}\times \mathbb{V}\times \mathbb{V}\rightarrow \mathbb{R}\) defined as \( b(u,v,w)=\int_{G}(u(x)\cdot \nabla )v(x)\cdot w(x)dx\), from which he deduces an operator \(B\). He gathers properties of these operators. Applying the Helmholtz-Hodge orthogonal projection to the above equation, he derives the abstract formulation \(du^{\…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 76A10
zbMATH DE Number: 7381507
zbMATH Keywords: Oldroyd model
zbMATH Keywords: non-Newtonian fluid flow
zbMATH Keywords: Helmholtz-Hodge orthogonal projection
zbMATH Keywords: strong solution
zbMATH Keywords: existence and uniqueness result
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: central limit result
zbMATH Keywords: moderate deviation principle
MaRDI profile type: Publication
full work available at URL: s40072-020-00176-5
OpenAlex ID: W3042863306
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DOI: 10.1007/S40072-020-00176-5
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