Navier-Stokes Equation on the Rectangle: Controllability by Means of Low Mode Forcing

Approximate Controllability for Navier–Stokes Equations in 3D Rectangles Under Lions Boundary Conditions

“…Using (IH.C-eq.18), (19), (24), and the induction hypothesis (IH-Cn1n2-eq.25), we conclude that Z j(n),n with n = (n 1 , n 2 , q + 1). Finally, we obtain Z j(n),n ∈ G q for all n ∈ R q+1 3 .…”

“…Next we introduce two fruitful lemmas which play the main role in the proof below. These lemmas help us to avoid explicit and complicated computations of operator B(a, b) as some works before in 2D cases (see [15,18,19,21]). In Lemma 3.3, we denote…”

“…Therefore one of the families {n, z α 1 , z γ 1 } or {n, z α 1 , z δ 1 } is linearly independent and, by Lemma 3.4, it follows that Z {1,2},(1,l,q+1) ∈ G q . By induction, using (19), (22), and the induction hypothesis (IH-C1l-eq.23) it follows that…”

Gevrey regularity for Navier–Stokes equations under Lions boundary conditions

“…where [Rod06] it is considered the usual Laplacian ∆ in (0, a) × (0, b) and here (cf. the discussion following Equation (2)) we consider the Laplace-de Rham operator…”

Controllability of nonlinear PDE’s: Agrachev–Sarychev approach

“…They studied the 2D NavierStokes equations on a torus controlled by a finite-dimensional external force and proved the properties of approximate controllability and exact controllability in observed projections. These results were later extended to the Euler and Navier-Stokes systems on various 2D and 3D manifolds; see [AS06,Rod06,Shi06,Rod07,AS07,Shi07].…”