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Local exact controllability of the Navier-Stokes equations. (English) Zbl 0873.76020

Summary: Let \(\Omega\Subset\mathbb{R}^n\) and \(\widehat u(t,x)\) be a given solution of the equation \(\partial_tu(t,x)+ A(u)= f(t,x)\), \(t\in(0,T)\), \(x\in\Omega\). The equation is called local exact controllable from boundary if for any initial condition \(u_0(x)\) belonging to the \(\varepsilon\)-neighbourhood of the point \(\widehat u(0,\cdot)\) (\(\varepsilon=\varepsilon(\widehat u))\) there exists such boundary control \(\alpha\) that a solution \(u\) of the equation supplied with \(u|_{(0,T)\times\partial\Omega}=\alpha\), \(u|_{t=0}= u_0\) satisfies the condition \(u(T,x)\equiv\widehat u(T,x)\). The local exact boundary controllability of the two-dimensional and three-dimensional Navier-Stokes as well as Boussinesq equations is established in this paper. For two-dimensional Navier-Stokes equations the same property is established also in the case when the control is defined on a boundary’s arbitrary subset. For two-dimensional Euler equations (respectively for Navier-Stokes equations), global exact (respectively approximate) controllability has been shown (with slip boundary conditions for Navier-Stokes equations) by J.-M. Coron [C. R. Acad. Sci., Paris, Ser. I 317, No. 3, 271-276 (1993; Zbl 0781.76013)].

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
93C20 Control/observation systems governed by partial differential equations
35Q30 Navier-Stokes equations

Citations:

Zbl 0781.76013
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