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

Abstract

A saturating set consisting eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The explicit saturating set leads to the approximate controllability for Navier-Stokes equations in 3D cylinders under Lions boundary conditions. MSC2010: 93B05, 35Q30, 93C20.

Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension

“…He also considered the Burgers equation on the real line in [20] and on a bounded interval with Dirichlet boundary conditions in [21]. Rodrigues [16] proved approximate controllability of the 2D NS system on a rectangle with Lions boundary conditions, and with Phan [14] they generalised that result to the 3D case. In the papers [11,12], Nersisyan considered 3D Euler system for incompressible and compressible fluids, and Sarychev [17] considered the 2D defocusing cubic Schrödinger equation.…”

Approximate controllability for Navier–Stokes equations in \rm3D cylinders under Lions boundary conditions by an explicit saturating set

“…Lions boundary condition is a particular case of Navier boundary conditions. For works and motivations concerning Lions and Navier boundary conditions (in both 2D and 3D cases) we refer to [6,10,11,16,17,30,31] and references therein.…”

“…In [17], the approximate controllability also follows from the existence of a (L, D(A))-saturating set. For any given length triplet L = (L 1 , L 2 , L 3 ) of a 3D rectangle, we presented an explicit (L, D(A))-saturating set C for the 3D rectangle Ω = (0, L 1 ) × (0, L 2 ) × (0, L 3 ) (which will be recalled below).…”

“…Here we follow the definition of saturating set as in the previous work (see [17]) because it leads to some advantages in computations.…”

“…To construct the eigenfunctions Z j(k),k , Lemma 3.8 (a similar version of Lemma 3.1 in [17]) is not enough to prove the linear independence. Therefore another Lemma 3.9 will be introduced and used mostly in the proof.…”