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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)

##### MSC:
 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability
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##### References:
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