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Stabilization of nonautonomous Navier-Stokes flows under dynamic slip boundary conditions

Buddhika Priyasad, Sérgio S. Rodrigues

TL;DR

This work proves exponential stabilization for the incompressible Navier–Stokes equations under dynamic slip boundary conditions, driving the flow toward a prescribed time-dependent trajectory without relying on spectral properties. A constructive feedback law is developed using oblique projections and a finite, spatially localized set of interior actuators, with the feedback operator expressed through the Helmholtz projection and a Gram-matrix of actuator projections. The analysis introduces an extended shifted Stokes operator and decomposes the dynamics into reaction–convection and nonlinear terms, establishing weak solution existence and energy-based decay via a monotonicity property of the sum of the Stokes operator and the feedback. The main result shows that for sufficiently many actuators and large enough feedback gain, the closed-loop system achieves exponential decay with rate $\mu = \beta^{-1} \alpha$, and crucially, the decay rate can be made arbitrarily large by tightening boundary dynamics (as $\beta \to 0$). The framework accommodates several slip boundary types (Navier, vorticity-type, Neumann) and multi-connected domains, making the approach practically robust for a broad class of dynamic boundary control problems in fluid dynamics.

Abstract

Exponential stabilizability of the incompressible Navier-Stokes equations under dynamic slip boundary conditions toward arbitrary time-dependent trajectories is proven. The feedback control law is constructed explicitly using oblique projections and realized through a finite number of spatially localized interior actuators, without requiring spectral assumptions. The approach extends to various slip boundary condition types (Navier, vorticity-type, and Neumann) and applies to multi-connected domains. Weak solution existence and exponential decay estimates are established, with the stabilization rate depending on the boundary dynamics parameters.

Stabilization of nonautonomous Navier-Stokes flows under dynamic slip boundary conditions

TL;DR

This work proves exponential stabilization for the incompressible Navier–Stokes equations under dynamic slip boundary conditions, driving the flow toward a prescribed time-dependent trajectory without relying on spectral properties. A constructive feedback law is developed using oblique projections and a finite, spatially localized set of interior actuators, with the feedback operator expressed through the Helmholtz projection and a Gram-matrix of actuator projections. The analysis introduces an extended shifted Stokes operator and decomposes the dynamics into reaction–convection and nonlinear terms, establishing weak solution existence and energy-based decay via a monotonicity property of the sum of the Stokes operator and the feedback. The main result shows that for sufficiently many actuators and large enough feedback gain, the closed-loop system achieves exponential decay with rate , and crucially, the decay rate can be made arbitrarily large by tightening boundary dynamics (as ). The framework accommodates several slip boundary types (Navier, vorticity-type, Neumann) and multi-connected domains, making the approach practically robust for a broad class of dynamic boundary control problems in fluid dynamics.

Abstract

Exponential stabilizability of the incompressible Navier-Stokes equations under dynamic slip boundary conditions toward arbitrary time-dependent trajectories is proven. The feedback control law is constructed explicitly using oblique projections and realized through a finite number of spatially localized interior actuators, without requiring spectral assumptions. The approach extends to various slip boundary condition types (Navier, vorticity-type, and Neumann) and applies to multi-connected domains. Weak solution existence and exponential decay estimates are established, with the stabilization rate depending on the boundary dynamics parameters.
Paper Structure (33 sections, 9 theorems, 98 equations, 2 figures)

This paper contains 33 sections, 9 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stabilizability to a reference trajectory
  3. On the literature
  4. Contents
  5. Stabilization of Navier--Stokes equations with dynamic boundary conditions
  6. Dynamic boundary conditions in fluid dynamics
  7. Background and motivation
  8. Stabilization of Navier--Stokes flows
  9. Novel contributions and technical approach
  10. Main result
  11. The actuators
  12. The explicit feedback-input operator
  13. Stabilizability
  14. Mathematical setting
  15. The extended shifted Stokes operator
  16. ...and 18 more sections

Key Result

Theorem 2.2

Given a target trajectory $\widehat{{\mathbf y}} \in L^\infty(\mathbb{R}_+,V)$, there exists $M_*\in{\mathbb N}$ such that for every $M \ge M_*$, there exists $\lambda_* = \lambda_*(M) \ge 0$ such that, for every $\lambda\ge\lambda_*$ the solution to nse-yK-intro satisfies the exponential stabilizat

Figures (2)

  • Figure 1: Placement of square-shaped supports of the actuators $\omega_i$ in a disk.
  • Figure 2: Placement of the supports of the actuators in rectangles (a) and triangles (b).

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Theorem 5.2
  • Remark 5.3
  • Lemma 5.4
  • ...and 10 more