Exact controllability for the three-dimensional Navier-Stokes equations with the Navier slip boundary conditions

Exact Internal Controllability for the Two-Dimensional Magnetohydrodynamic Equations

“…The proof of the above lemma is identical to that of its three-dimensional version (see [3] or [4]). In a completely similar way we can obtain the corresponding estimate for ∇r.…”

“…The proof of this theorem is quite long and can be found in [4]. (See also [3] for a related situation.) We emphasize that there is no significant difference between the two-dimensional and three-dimensional cases.…”

“…Estimate (3.9) was essentially obtained by Imanuvilov in [5]. (We refer the reader to Lemma 2.2 in [6] or Lemma 3.1 in [3], too. )…”

“…Eight of them (I 3 through I 10 ) can be estimated identically as the corresponding terms in the three-dimensional case. We refer the reader to [2] (see also [3]) for the expression of those estimates and for other details. So, let us examine the other two terms (which contain ∆D).…”

“…Now we add inequalities (3.13) and (3.21). Because of the different signs before the corresponding surface integrals in I 2 , I 2 , I 3 , I 3 , I 5 , I 5 , I 9 , and I 9 (see [2] or [3] for the definition of the other quantities), all those surface integrals are cancelled. (The different signs come from the odd exponents of the powers of λ.)…”