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Local exact controllability of the two-dimensional Navier–Stokes equations
A. V. Fursikova,
Yu. S. Èmanuilovb a M. V. Lomonosov Moscow State University
b Moscow State Forest University
Abstract:
Let
$\Omega \subset \mathbb R^2$ be a bounded domain with boundary
$\partial \Omega$ consisting of two disjoint closed curves
$\Gamma _0$ and
$\Gamma _1$ such that
$\Gamma _0$ is connected and
$\Gamma _1\ne \varnothing$. The Navier–Stokes system $\partial _tv(t,x)-\Delta v+(v,\nabla )v+\nabla p=f(t,x)$,
$\operatorname {div}v=0$ is considered in
$\Omega$ with boundary and initial conditions $(v,\nu )\big |_{\Gamma _0}=\operatorname {rot}v\big |_{\Gamma _0}=0$ and
$v\big|_{t=0}=v_0(x)$ (here
$t\in (0,T)$,
$x\in \Omega$, and
$\nu$ is the outward normal to
$\Gamma_0$). Let
$\widehat v(t,x)$ be a solution of this system such that
$\widehat v$ satisfies the indicated boundary conditions on
$\Gamma_0$ and $\|\widehat v(0,\,\cdot \,)-v_0\|_{W^2_2(\Omega )}<\varepsilon$, where
$\varepsilon =\varepsilon (\widehat v)\ll 1$. Then the existence of a control
$u(t,x)$ on
$(0,T)\times \Gamma _1$ with the following properties is proved: the solution
$v(t,x)$ of the Navier–Stokes system such that $(v,\nu )\big |_{\Gamma _0}=\operatorname {rot}v\big |_{\Gamma _0}=0$,
$v\big |_{t=0}=v_0(x)$ and
$v\big |_{\Gamma _1}=u$, coincides with
$\widehat v(T,\,\cdot \,)$ for
$t = T$, that is,
$v(T,x)=\widehat v(T,x)$. In particular, if
$f$ and
$\widehat v$ do not depend on
$t$ and
$\widehat v(x)$ is an unstable steady-state solution, then it follows from the above result that one can suppress the occurrence of turbulence by some control
$\alpha$ on
$\Gamma_1$. An analogous result is established in the case when
$\Gamma _0=\partial \Omega$ and
$\alpha(t,x)$ is a distributed control concentrated in an arbitrary subdomain
$\omega \subset \Omega$.
UDC:
517.977.1
MSC: 76D05,
35B37,
93B05,
93C20 Received: 04.03.1996
DOI:
10.4213/sm160
Fulltext:
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