SeMA DOI 10.1007/s40324-017-0111-7 Remarks concerning the approximate controllability of the 3D Navier–Stokes and Boussinesq systems Enrique Fernández-Cara 1 · Ivaldo T. De Sousa 2 · Franciane B. Viera 2 Received: 28 June 2016 / Accepted: 26 January 2017 © Sociedad Española de Matemática Aplicada 2017 Abstract In this paper, we deal with some 3D systems of the Navier–Stokes kind in a cube or a similar set. We prove some extensions and variants of a result by Guerrero, Imanuvilov and Puel that concerns the (global) approximate boundary controllability. Keywords Null controllability · Approximate controllability · Navier–Stokes system · Boussinesq system Mathematics Subject Classification 35B37 · 93C20 · 35Q30 · 76D05 1 Introduction Let  be the open set  := { x ∈ R 3 : x i ∈ (0, 1), 1 ≤ i ≤ 3 }, whose boundary is denoted by ∂ and let us assume that T > 0. We will set Q :=  × (0, T ),  := ∂ × (0, T ). Partially supported by Grant MTM2013-41286-P (DGI-MINECO, Spain) and CAPES (Brazil). B Enrique Fernández-Cara cara@us.es Ivaldo T. De Sousa tributino_1983@hotmail.com Franciane B. Viera cianev@hotmail.com 1 Dpto. E.D.A.N., Univ. Sevilla, Aptdo. 1160, 41080 Seville, Spain 2 Dpto. Matemática, Univ. Federal do Ceará, Fortaleza, CE, Brazil 123