Exact boundary zero controllability of three-dimensional Navier-Stokes equations

abstractIn a bounded three-dimensional domain Ω a solenoidal initial vector fieldv0(x)∈H3 (Ω) is given. We construct a vector fieldz(t, x) defined on the lateral surface [0,T]×ϖΩ of the cylinder [0,T]×Ω which possesses the following property: the solutionv(t, x) of the boundary value problem for the Navier-Stokes equation with the initial valuev0(x) and the boundary Dirichlet conditionz(t, x) satisfies the relationv(T, x)≡0 at the instantT. Moreover, $$\parallel v(t, \cdot )\parallel _{H^3 (\Omega )} \leqslant c\exp \left( { - k/(T - t)^2 } \right) as t \to T$$ , wherec>0,k>0 are certain constants.