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    • Stabilizability of Two-Dimensional Navier-Stokes Equations with Help of A Boundary Feedback Control

    A. V. Fursikov

    Journal of Mathematical Fluid Mechanics（2001）

    Moscow MV Lomonosov State Univ

    Cited 77|Views1
    Abstract
    For 2D Navier—Stokes equations defined in a bounded domain \( \Omega \) we study stabilization of solution near a given steady-state flow \( \hat v(x) \) by means of feedback control defined on a part \( \Gamma \) of boundary \( \partial\Omega \). New mathematical formalization of feedback notion is proposed. With its help for a prescribed number \( \sigma > 0 \) and for an initial condition v 0(x) placed in a small neighbourhood of \( \hat v(x) \) a control u(t,x'), \( x' \in \Gamma \), is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier—Stokes equations satisfies the inequality: \( \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 \). To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.
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    Keywords. Oseen equations,Navier—Stokes equations,stabilization,extension operator,stable invariant manifold.
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