[Paper Review] Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force

Jean‐Michel Coron, Frédéric Marbach+2 more|arXiv (Cornell University)|Jan 5, 2018

Stability and Controllability of Differential Equations36 references3 citations

TL;DR

This paper establishes small-time global exact boundary controllability of the 2D incompressible Navier-Stokes equation in a rectangle with no-slip boundary conditions on top and bottom, by introducing a distributed phantom force that is arbitrarily small in any Sobolev norm. The method combines analytic regularization, the well-prepared dissipation technique, and a novel long-time Cauchy-Kovalevskaya estimate relying only on horizontal analyticity to control large boundary layers.

ABSTRACT

We consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to Jacques-Louis Lions' question about the small-time global exact boundary controllability of the Navier-Stokes equation with the no-slip boundary condition, for which no distributed force is allowed. Our analysis relies on the well-prepared dissipation method already used for Burgers and for Navier-Stokes in the case of the Navier slip-with-friction boundary condition. In order to handle the larger boundary layers associated with the no-slip boundary condition, we perform a preliminary regularization into analytic functions with arbitrarily large analytic radius and prove a long-time nonlinear Cauchy-Kovalevskaya estimate relying only on horizontal analyticity.

Motivation & Objective

• To address Jacques-Louis Lions' open problem on small-time global exact boundary controllability of the Navier-Stokes equation under no-slip boundary conditions.
• To extend prior results that required only smallness in negative Sobolev norms to a stronger smallness condition in arbitrary Sobolev norms.
• To develop a framework that handles the larger boundary layers induced by the no-slip condition through analytic regularization and refined energy estimates.
• To establish global approximate controllability with a distributed force that is uniformly small in Sobolev norms, enabling exact final state control.

Proposed method

• Introduce a distributed phantom force that is arbitrarily small in any Sobolev norm $ H^k $, enabling control without violating smallness constraints.
• Apply a preliminary regularization of initial data into analytic functions with arbitrarily large analytic radius to handle boundary layer instabilities.
• Use the well-prepared dissipation method to control the nonlinear interactions and boundary layer growth in the Navier-Stokes system.
• Perform a long-time nonlinear Cauchy-Kovalevskaya estimate relying solely on horizontal analyticity, inspired by [6, 41], to bound nonlinear terms.
• Decompose the solution into approximate trajectories and remainders, estimating each via Littlewood-Paley theory and frequency localization.
• Employ fast variable scaling and Lebesgue norm estimates to analyze the decay of boundary layer profiles and ensure uniform control.

Experimental results

Research questions

• RQ1 Can the 2D Navier-Stokes equation in a rectangle with no-slip boundary conditions be exactly controlled to zero in small time using only boundary controls and a small distributed force?
• RQ2 Can the size of the distributed force be made arbitrarily small in any Sobolev norm $ H^k $, not just in negative Sobolev spaces?
• RQ3 How can the large boundary layers induced by the no-slip condition be controlled in the context of small-time controllability?
• RQ4 What role does horizontal analyticity play in stabilizing the nonlinear Navier-Stokes system over long timescales?
• RQ5 Can a well-prepared dissipation method be adapted to handle the stronger boundary layer effects of the no-slip condition compared to the Navier slip-with-friction case?

Key findings

• For any $ T > 0 $, initial data $ u_* eq 0 $, and any $ k o exists $, there exists a distributed force $ f_g eq 0 $ such that the solution $ u(T) = 0 $, with $ \|f_g\|_{L^1((0,T);H^k)} \leq \eta $ for any $ \eta > 0 $.
• The distributed force $ f_g $ can be made arbitrarily small in any Sobolev norm $ H^k $, improving prior results that required only smallness in negative Sobolev spaces.
• The method relies on a novel long-time nonlinear Cauchy-Kovalevskaya estimate that depends only on horizontal analyticity, enabling control of nonlinear terms despite large boundary layers.
• The solution is constructed as a weak Leray solution in $ C^0([0,T];L^2_{\text{div}}) \cap L^2((0,T);H^1) $, satisfying the weak formulation with appropriate test functions.
• The analysis establishes uniform bounds on the analytic regularity of the approximate trajectories using frequency localization and exponential weights $ e^{\rho_0 |\partial_x|} $, ensuring stability.
• The boundary layer decay is proven via fast variable scaling and Lebesgue norm estimates, with key decay results relying on $ n \geq 3 $ in the boundary layer profile.