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On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. (English) Zbl 0872.93040
The paper studies the approximate controllability for a Navier-Stokes equation in a bounded domain \(\Omega\subset\mathbb{R}^2\). Let \(\Omega^\#\) and \(\Gamma^\#\) be subsets of \(\Omega\) and \(\Gamma=\partial\Omega\), respectively, on which all control actions are exerted. It is assumed that \(\Omega^\#\cup\Gamma^\#\neq\emptyset\). In other words, there is no control in \(\Omega\backslash\Omega^\#\) and on \(\Gamma\backslash\Gamma^\#\). Thus the equation for \(y= (y^1,y^2)\) is described by \[ y_t-\Delta y+(y\cdot\nabla) y+\nabla p=0\quad\text{in }(\overline\Omega\backslash\Omega^\#)\times [0,T], \]
\[ \text{div }y= 0\quad\text{on }\overline\Omega\times [0,T], \]
\[ y\cdot n=\sigma y\cdot\tau+\text{curl }y=0\quad \text{on }(\Gamma\backslash\Gamma^\#)\times [0,T], \]
\[ y(\cdot,0)= y_0\quad\text{in }\overline\Omega. \] Here, \(n\) denotes the outward unit normal vector; \(\tau\) the unit tangent vector on \(\Gamma\); and \(\sigma\in C^\infty(\Gamma,\mathbb{R})\). For any \(T>0\) and any \(y_0,y_1\in C^\infty(\overline\Omega,\mathbb{R}^2)\) satisfying \[ \text{div }y_i= 0\quad\text{in }\overline\Omega,\quad y_i\cdot n=\sigma y\cdot\tau+ \text{curl }y_i=0\quad\text{on }\Gamma\backslash\Gamma^\#,\quad i=1,2, \] the following approximate controllability result is shown: There is a sequence of controls such that the corresponding solutions \(y^k(\cdot, T)\) are arbitrarily close to \(y_1\) in some topology as \(k\to\infty\).
Reviewer: T.Nambu (Kobe)

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
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