Uniform Controllability of a Stokes Problem with a Transport Term in the Zero-Diffusion Limit

Abstract

In this paper we consider a Stokes system with Navier-slip boundary conditions. The main results concern the behaviour of the cost of null controllability with respect to the diffusion coefficient when the control acts in the interior. In particular, we prove in (0, π) 2 that for a sufficiently large time the cost decays exponentially as the diffusion coefficient vanishes, whereas in (0, π) 3 we prove that for most of the control domains and for any time T > 0 the cost explodes exponentially as the diffusion coefficient vanishes.

Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

“…More recently, better approximations have been given for the optimal time in which the cost of the control decays: the lower bound was improved first in [34] through complex analysis and properties of the entire functions, and more recently in [28] through semi-classical and spectral analysis; and the upper bound was improved in [20,32] (in the first one through complex analysis and, in the second one, by transforming the transportdiffusion equation into a heat equation with just a diffusive term). As for similar results, several results have been obtained for the KdV equation (see [22,23,6,7,9]), the Burgers equation (see [21]), the Stokes system (see [2]), an artificial advection-diffusion problem (see [11,12]), the heat equation on networks (see [3]), and a fourth-order parabolic equation (see [8,35,26]).…”

“…In addition, a spectral decomposition has been used in [2] to obtain the dissipation estimate in a transport-diffusion Stokes system. Indeed, in this paper we follow the philosophy of [2] of using as much information as possible about the spectral decomposition, with the contribution that now while proving the Carleman estimate we work directly in the symmetrized system, that is, in (2.4).…”

“…When T is small the cost of the null controllability of (1.1) explodes exponentially if ε → 0, which can be proved, for instance, as in [24] and [2]. Thus, in this paper we focus on large values of T .…”

Local null controllability of the penalized Boussinesq system with a reduced number of controls

“…Let Ω be such that Hypothesis 2 holds, ω ⊂ Ω be a nonempty open set and let m ≥ 2. Then, there are ε 0 > 0, C > 0 and λ 0 ≥ 1 such that if T > 0, ε ∈ (0, ε 0 ), (ϕ T , ψ T ) ∈ L 2 (Ω) 4 , λ ≥ λ 0 , and s ≥ e Cλ (T m + T 2m ), we have…”

“…• Considering other systems, the study of controllability problems in which the control has a reduced number of components has been an active topic of research recently. In particular, for the Stokes and Navier-Stokes systems we can consult for instance the following papers: [29,18,22,12,8,13,9,23,4]. For more results on the controllability of linear parabolic systems with a reduced number of controls, see the survey [1] and the references therein.…”