[Paper Review] On the controllability of the Navier-Stokes equation in spite of boundary layers

Jean‐Michel Coron, Frédéric Marbach+1 more|arXiv (Cornell University)|Mar 21, 2017

Stability and Controllability of Differential Equations5 citations

TL;DR

This paper establishes small-time global exact null controllability of the 2D and 3D incompressible Navier-Stokes equations with Navier slip-with-friction boundary conditions on part of the boundary, despite the presence of boundary layers. By combining asymptotic expansion techniques and enhanced dissipation, the authors construct a weak Leray solution that vanishes at any positive time T, proving that arbitrary initial data can be driven to zero via boundary control on an open subset of the boundary.

ABSTRACT

In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier slip-with-friction boundary condition except on a part of the boundary. This under-determination encodes that one has control over the remaining part of the boundary. We prove that for any initial data, for any positive time, there exists a weak Leray solution which vanishes at this given time.

Motivation & Objective

• To resolve Jacques-Louis Lions' open problem on small-time global exact null controllability of the Navier-Stokes equations.
• To address the challenge of boundary layers in the Navier-Stokes system under partial boundary control.
• To establish controllability when the uncontrolled boundary part satisfies Navier slip-with-friction conditions rather than no-slip conditions.
• To extend the understanding of control mechanisms in viscous incompressible flows with non-ideal boundary conditions.
• To provide a constructive framework for driving weak Leray solutions to zero at any positive time T.

Proposed method

• Utilizes an asymptotic expansion in a small parameter ε to decompose the velocity field into leading-order and higher-order components.
• Introduces a two-scale expansion: u = u⁰ + εu¹ + v + r^ε, where u⁰ and u¹ are regularized components and v is a convection field.
• Employs a boundary layer correction via a projection method that adjusts finite-dimensional moments to compensate for nonhomogeneous boundary data.
• Applies enhanced dissipation effects to control the remainder term r^ε, leveraging the ε-scaling in the viscous term to tame nonlinearities.
• Uses energy estimates on the remainder equation to show ‖r^ε(T/ε)‖_{L²} = o(1), ensuring vanishing at time T.
• Relies on the existence of weak Leray solutions and the under-determined nature of the system due to partial boundary control.

Experimental results

Research questions

• RQ1 Can the 2D and 3D incompressible Navier-Stokes equations be driven to zero at any positive time T using boundary control on a non-empty open subset of the boundary?
• RQ2 Does the presence of Navier slip-with-friction boundary conditions on the uncontrolled part allow for global exact null controllability despite boundary layer formation?
• RQ3 Can the enhanced dissipation effect be harnessed to control the remainder term in a multi-scale asymptotic expansion of the Navier-Stokes solution?
• RQ4 Is it possible to construct a weak Leray solution that vanishes at time T for any initial data in L²_σ(Ω) under partial boundary control?
• RQ5 Can the strategy used for Navier conditions be extended to the more challenging no-slip case, particularly in favorable geometric settings?

Key findings

• The paper proves that for any initial data u₀ ∈ L²_σ(Ω) and any T > 0, there exists a weak Leray solution to the Navier-Stokes equations that vanishes at time T.
• The result holds for both 2D and 3D smooth, bounded, simply connected domains with Navier slip-with-friction conditions on the uncontrolled boundary part.
• The control is achieved via boundary data on an open subset Σ of the boundary, while the rest of the boundary is subject to Navier conditions.
• The remainder term r^ε in the asymptotic expansion satisfies ‖r^ε(T/ε)‖_{L²(Ω)} = o(1), ensuring vanishing at time T.
• The method relies on a two-scale expansion and enhanced dissipation to stabilize the remainder and compensate for nonhomogeneous boundary data.
• The construction is robust to the size of initial data and time horizon T, confirming small-time global exact null controllability under the given boundary conditions.