A central limit theorem and moderate deviation principle for the stochastic 2D Oldroyd model of order one
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 \(du^{\varepsilon }(t)=-\mu \lbrack Au^{\varepsilon }(t)+(\beta \ast Au^{\varepsilon })(t)+B(u^{\varepsilon }(t))-f(t)]dt+\sqrt{ \varepsilon }\sigma (t,u^{\varepsilon }(t))dW(t)\), with the initial condition \(u^{\varepsilon }(0)=u_{0}\). For given \(u^{0}\in \mathbb{H}\) and \( f\in L^{2}(0,T;\mathbb{V}^{\prime })\), a \(\mathbb{H}\)-valued \((\mathcal{F} _{t})_{t\geq 0}\) adapted continuous process \(u^{\varepsilon }(\cdot )\) is called a strong solution to the above abstract formulation if the process \( u^{\varepsilon }\in L^{4}(\Omega ;L^{\infty }(0,T;\mathbb{H}))\cap L^{2}(\Omega ;L^{2}(0,T;\mathbb{V}))\) is such that \(u^{\varepsilon }(\cdot )\) has a \(\mathbb{V}\)-valued modification, which is progressively measurable with continuous paths in \(\mathbb{H}\) for all \(t\in \lbrack 0,T]\), \(\mathbb{P }\)-a.s., and for every \(t\in \lbrack 0,T]\), as an element of \(\mathbb{V} ^{\prime }\), \(\mathbb{P}\)-a.s.: \(u^{\varepsilon }(t)=u^{0}-\int_{0}^{t}\mu \lbrack Au^{\varepsilon }(t)+(\beta \ast Au^{\varepsilon })(t)+B(u^{\varepsilon }(t))-f(t)]dt+\sqrt{\varepsilon }\int_{0}^{t}\sigma (t,u^{\varepsilon }(t))dW(t)\). Here \((\Omega ,\mathcal{F},\mathbb{P})\) is a complete probability space equipped with an increasing family of sub-sigma fields \(\{\mathcal{F}_{t}\}_{0\leq t\leq T}\) of \(\mathcal{F}\) satisfying \( \mathcal{F}_{0}\) contains all elements \(F\in \mathcal{F}\) with \(\mathbb{P} (F)=0\), \(\mathcal{F}_{t}=\mathcal{F}_{t+}=\cap _{s>t}\mathcal{F}_{s}\) for \( 0\leq t\leq T\). Such a strong solution is called unique if \(\widetilde{u} ^{\varepsilon }(t)\) is another strong solution, then \(\mathbb{P}\{\omega \in \Omega :u^{\varepsilon }(t)=\widetilde{u}^{\varepsilon }(t)\), for all \(t\in \lbrack 0,T]\}=1\). The author quotes from a previous paper he wrote [Stochastic Anal. Appl. 38, No. 1, 1--61 (2020; Zbl 1431.35105)] the existence of a unique strong solution. Under some hypotheses on \(\sigma \) and \(f\), the author proves the existence of a constant \(\varepsilon _{0}>0\) such that for all \(0<\varepsilon \leq \varepsilon _{0}\), the estimate \( \mathbb{E}[\sup_{t\in \lbrack 0,T]}\left\Vert u^{\varepsilon }(t)-u^{0}(t)\right\Vert _{\mathbb{H}}^{2}+2\mu \int_{0}^{T}\left\Vert u^{\varepsilon }(t)-u^{0}(t)\right\Vert _{\mathbb{V}}^{2}dt]\leq \varepsilon C\), for some constant \(C\) which depends on \(u^{0}\) and \(f\). The author uses an Itô formula and estimates on \(u^{\varepsilon }\) and \(u^{0}\). His main result proves a central limit result which involves the unique strong solution \( \mathfrak{U}^{0}\) to a stochastic linearized 2D Oldroyd model. In the last part of his paper, the author proves a moderate deviation principle for this problem, proving that \(\frac{1}{\sqrt{\varepsilon }\lambda (\varepsilon )} (u^{\varepsilon }-u^{0})\) obeys a large deviation principle on \(C([0,T]; \mathbb{H})\cap L^{2}(0,T;\mathbb{V})\) with speed \(\lambda ^{2}(\varepsilon ) \).\ He here uses a variational method based on a weak convergence approach developed by \textit{A. Budhiraja} and \textit{P. Dupuis} [Probab. Math. Stat. 20, No. 1, 39--61 (2000; Zbl 0994.60028)] and Skorokhod's representation theorem.
scientific article; zbMATH DE number 7381507
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