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

TL;DR

This paper proves the exponential stabilization of nonautonomous Navier-Stokes flows with dynamic slip boundary conditions using explicit feedback control, applicable to various boundary types and domains.

Contribution

It introduces a novel explicit feedback control law for stabilizing Navier-Stokes flows under dynamic slip boundary conditions without spectral assumptions.

Findings

Achieved exponential stabilization toward time-dependent trajectories.

Constructed control law using oblique projections and localized actuators.

Extended results to multiple slip boundary condition types and complex domains.

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.