Small-time global exact controllability of the Navier–Stokes equation with Navier slip-with-friction boundary conditions

Abstract

In this work, we investigate the small-time global exact controllability of the Navier–Stokes equation, both towards the null equilibrium state and towards weak trajectories.We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time global exact controllability holds. Our analysis relies on the controllability of the Euler equation combined with asymptotic boundary layer expansions. Choosing the boundary controls with care enables us to guarantee good dissipation properties for the residual boundary layers, which can then be exactly canceled using local techniques.

Remarks on local controllability for the Boussinesq system with Navier boundary condition

“…Recently, Coron et al have proved a global exact controllability result for the Navier-Stokes and Navier-type conditions (for small time), see [5]. A challenging problem would be to use the Boussinesq system proposed in this Note in order to apply and prove analogous results to [5].…”

“…Theorem 2. Assume A ∈ P 1 ε ∩ P 2 for some ε > 0 and (0, p, θ) satisfying (4)- (5). There exists a constant λ 0 , such that for any λ ≥ λ 0 there exist two constants C (λ) > 0 increasing on A P 1 ε ∩P 2 and s 0 (λ) > 0 such that for any j ∈ {1, 2}, any a > 0, any g ∈ L 2 (Q) N , any g 0 ∈ L 2 (Q), any ϕ T ∈ H and any ψ T ∈ L 2 (Ω), the solution of (3) satisfies…”

On Current Developments in Partial Differential Equations

“…In the late 1980's, Jacques-Louis Lions introduced in [86] (see also [87][88][89]) the question of the controllability of fluid flows in the sense of how the Navier-Stokes system can be driven by a control of the flow on a part of the boundary to a wished plausible state, say a vanishing velocity. Lions' problem has been solved in [30] by Coron, Marbach and Sueur in the particular case of the Navier slip-with-friction boundary condition. In its original statement with the no-slip Dirichlet boundary condition, it is still an important open problem in fluid controllability.…”

Local null controllability of a fluid–rigid body interaction problem with Navier slip boundary conditions

“…Moreover, in [23], the authors established the local controllability with N − 1 scalar controls. With Navier-slip conditions on the fluid equations, global null controllability is obtained for the weak solution in [11] such that the controls are only located on a small part of the domain boundary. Concerning controllability results of fluid-structure systems with Dirichlet boundary conditions, in dimension 2, we mention the paper [7], where the authors proved the null controllability in velocity and the exact controllability for the position of the rigid body assuming some geometric properties for the solid and provided that the initial conditions are small enough, more precisely a condition of smallness on the H 3 norm of the initial fluid velocity is needed.…”

“…There are several possible extensions to this work. First let us recall that in [11], the authors obtain the global exact controllability of the Navier-Stokes system with Navier boundary conditions. One of their ingredients is to use the local exact null controllability of [22].…”