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 1.1 Assume A ∈ P 1 ε ∩ P 2 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) 3 , any g 0 ∈ L 2 (Q), any ϕ T ∈ H and any ψ T ∈ L 2 (Ω), the solution of (3) satisfies…”

Weak-renormalized solutions for a simplified $k-\varepsilon$ model of turbulence

“…Note that, with other or with no boundary conditions, global null controllability results have been established for Navier-Stokes and Boussinesq fluids by Coron [6] and Coron and Fursikov [7] in the two-dimensional case, Fursikov and Imanuvilov [13] and Coron, Marbach and Sueur [8] in the three-dimensional case. Accordingly, it is reasonable to expect results of the same kind when the PDEs in (2) are completed, for instance, with Navier-slip or periodic boundary conditions.…”

“…Lemma 3 Let the assumptions in Lemma 2 be satisfied. There exist positive constants λ 0 , s 0 and C depending on Ω, ω, T , ν, c ν and M such that, for any s ≥ s 0 and λ ≥ λ 0 , any F ∈ L 2 (Q) and any ϕ T ∈ H, the associated solution to (8)…”

“…However, to our knowledge, it is unknown whether it holds for (2)-( 3). If one tries to apply arguments as those above, one finds at once a major difficulty: one is led to a system similar to (8), where linear nonlocal terms appear, for which observability estimates are not clear at all.…”