Exact Controllability of a Faedo–Galërkin Scheme for the Dynamics of Polymer Fluids

Abstract

We describe the dynamics of fluids with scattered polymer chains through a multi-field model accounting for weakly non-local inertia and second-neighborhood interactions due to chain entanglements; viscous effects appear at both macroscopic and polymer representations. We consider a linearized version of the pertinent balance equations. For it, we prove exact controllability of the pertinent Faedo-Galërkin scheme on the basis of Hilbert's uniqueness method in combination with an appropriate fixed point argument.

On the exact controllability of a Galerkin scheme for 3D viscoelastic fluids with fractional Laplacian viscosity and anisotropic filtering

“…[36,37]) or other similar systems for the description of fluid models (e.g., besides some families viscoelastic fluids, also micropolar fluids and polymer fluids, see Refs. [4,19,38]).…”

“…[36, 37]) or other similar systems for the description of fluid models (e.g., besides some families viscoelastic fluids, also micropolar fluids and polymer fluids, see Refs. [4, 19, 38]). To prove Equation (), we take into account the following adjoint system (here $$\mathbf {p}_t=\partial _t\mathbf {p}$$): $$$$\begin{equation} \begin{aligned} & -(I-\alpha ^2 \Delta _h)\mathbf {p}_t +\nu \Lambda ^{2\beta } A_h\mathbf {p}+ \varpi \circ A_h \mathbf {p}- \mu (\mathbf {h}\cdot \nabla )\mathbf {p}+\nabla q(t,x)=\mathbf {0},\\ & \qquad x\in \mathbb {T}^3,\, t>0, \\ &(\nabla \cdot \mathbf {p})(t,x)=0,\,\, x\in \mathbb {T}^3,\, t>0,\\ & \mathbf {p}(T, x) = (I - \alpha ^2\Delta _h)^{-1}\mathbf {g}, \,\, x\in \mathbb {T}^3, \end{aligned} \end{equation}$$$$where $$\mathbf {g}\in E$$ and …”

“…where 𝐴 1∕2 ℎ = (𝐼 − 𝛼 2 Δ ℎ ) 1∕2 , 𝐸 = span{𝐞 1 , … , 𝐞 𝑑 𝑁 } is a suitable finite-dimensional space, and 𝐰 𝑛 (𝑥, 𝑡) = ∑ 𝑑 𝑁 𝑗=1 𝜚 𝑛,𝑗 (𝑡)𝐞 𝑗 (𝑥), with 𝜚 𝑛,𝑗 (𝑡) coefficient functions (see, e.g., Refs. [4,19,38]). Also, 𝐏(𝐰 0 ) = ∑ 𝑑 𝑁 𝑗=1 (𝐰 0 , 𝐞 𝑗 )𝐞 𝑗 .…”

“…Because of the nature of the considered system, as in the cases treated in Refs. [4, 37, 38], it is not evident how to achieve the exact controllability for the Equations () and () with arbitrary target functions (see also Bisconti and Mariano [19] for further comments). In this paper, we restrict ourselves to analyzing the exact controllability of a suitable Galerkin's approximation scheme $$\lbrace \mathbf {w}^n\rbrace$$ for Equations () and () on $$\mathbb {T}^3$$, and for $$t>0$$, that is $$$$\begin{equation} \begin{aligned} &{\left((I - \alpha ^2\Delta _h)\partial _t\mathbf {w}^n, \mathbf {e}\right)} + {\left((\mathbf {w}^n\cdot \nabla )\mathbf {w}^n,\mathbf {e}\right)} + \nu {\left(\Lambda ^{\beta }A_h^{1/2}\mathbf {w}^n, \La...$$…”

“…[4], the authors consider the problem of the exact controllability of Galerkin's approximations of micropolar fluids. A similar analysis, in the case of polymer fluids, is addressed in Bisconti and Mariano [19]. The recent paper [32] addresses the problem of controllability of an ideal MHD system with a control prescribed on the boundary.…”