Local exact controllability of the two-dimensional Navier–Stokes equations

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 $\omega \subset \Omega$.