Approximate controllability of the Stokes system on cylinders by external unidirectional forces
type: Item
instance of: scholarly article
title: Approximate controllability of the Stokes system on cylinders by external unidirectional forces
zbMATH Open document ID: 0889.93032
DOI: 10.1016/S0021-7824(97)89956-4
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1998-06-29T00:00:00Z
review text: The paper studies the approximate controllability of the Stokes system in \((0,T) \times \Omega\) \[ {\mathbf y}_t -\Delta {\mathbf y} +\nabla p=u \chi_\omega {\mathbf e}, \quad \text{div} {\mathbf y} =0, \quad {\mathbf y} (0,\cdot)=\mathbf{0}, \quad {\mathbf y} |_{\partial \Omega}=\text\textbf{0}. \] Here, \({\mathbf y} (t,x)= (y_1, y_2, y_3)\) and \(p\) denote the flow velocity and the pressure of the flow, respectively; \(\Omega\) is a cylinder \(G\times R^1 \subset {\mathbf R}^3\) with \(G\) a bounded domain in \({\mathbf R}^2\); \(\chi_\omega\) the characteristic function of a subdomain \(\omega \subset \Omega\); \({\mathbf e}\) a given vector in \({\mathbf R}^3\); and \(u=u(t,x)\) a control function. An uncontrollability result is first shown when \(\omega\) is the periodic extension (with respect to the variable \(x_3)\) of an open set of \(\Omega \times(0,L_3)\); \({\mathbf e} =(0,0,1)\); and the solution \({\mathbf y}\) is sought within the \(L_3\)-periodic functions with respect to \(x_3\) with the controls \(u\) \(L_3\)-periodic too. Second, approximate controllability is shown in the space \({\mathcal H}^0\) when \(G\) is a rectangle; and \({\mathbf e}= {\mathbf e}_1= (1,0,0)\), where \({\mathcal H}^0\) is the closure of the set of functions \(w\in C^\infty_0 (\Omega)\) with \(\text{div} {\mathbf w} =0\) in the space \((L^2 (\Omega))^3\).
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 1022295
zbMATH Keywords: cylindrical domain
zbMATH Keywords: approximate controllability
zbMATH Keywords: Stokes system
author: J. I. Díaz
author: Andrei V. Fursikov
reviewed by: Takao Nambu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1022295