Stabilization of Navier-Stokes Flows

Stabilization of Navier-Stokes Flows presents recent notable progress in the mathematical theory of stabilization of Newtonian fluid flows. Finite-dimensional feedback controllers are used to stabilize exponentially the equilibrium solutions of Navier-Stokes equations, reducing or eliminating turbulence. Stochastic stabilization and robustness of stabilizable feedback are also discussed. The text treats the questions:

. What is the structure of the stabilizing feedback controller?

. How can it be designed using a minimal set of eigenfunctions of the Stokes-Oseen operator?

The analysis developed here provides a rigorous pattern for the design of efficient stabilizable feedback controllers to meet the needs of practical problems and the conceptual controllers actually detailed will render the reader's task of application easier still. Stabilization of Navier-Stokes Flows avoids the tedious and technical details often present in mathematical treatments of control and Navier-Stokes equations and will appeal to a sizeable audience of researchers and graduate students interested in the mathematics of flow and turbulence control and in Navier-Stokes equations in particular. The chief points of linear functional analysis, linear algebra, probability theory and general variational theory of elliptic, parabolic and Navier-Stokes equations are reviewed in an introductory chapter and at the end of chapters 3 and 4.

Auteur
Professor Barbu is a professor with the University Al.I.Cuza (Romania) and member of Romanian Academy. He had visiting professorship positions with several universities in the USA and Europe including the following: Purdue University, Cincinnati University, Virginia University, Ohio University, Bonn University, University of Bologna. He has published a dozen monographs and 170 research papers in the following fields: nonlinear PDEs, control theory of parameter distributed systems and of NavierStokes equations, Stochatic PDEs, integral equations.

Texte du rabat

Stabilization of NavierStokes Flows presents recent notable progress in the mathematical theory of stabilization of Newtonian fluid flows. Finite-dimensional feedback controllers are used to stabilize exponentially the equilibrium solutions of NavierStokes equations, reducing or eliminating turbulence. Stochastic stabilization and robustness of stabilizable feedback are also discussed. The text treats the questions:

• What is the structure of the stabilizing feedback controller?

• How can it be designed using a minimal set of eigenfunctions of the StokesOseen operator?

The analysis developed here provides a rigorous pattern for the design of efficient stabilizable feedback controllers to meet the needs of practical problems and the conceptual controllers actually detailed will render the reader's task of application easier still. Stabilization of NavierStokes Flows avoids the tedious and technical details often present in mathematical treatments of control and NavierStokes equations and will appeal to a sizeable audience of researchers and graduate students interested in the mathematics of flow and turbulence control and in Navier-Stokes equations in particular. The chief points of linear functional analysis, linear algebra, probability theory and general variational theory of elliptic, parabolic and NavierStokes equations are reviewed in an introductory chapter and at the end of chapters 3 and 4.

Contenu
Preliminaries.- Stabilization of Abstract Parabolic Systems.- Stabilization of NavierStokes Flows.- Stabilization by Noise of NavierStokes Equations.- Robust Stabilization of the NavierStokes Equation via the H-infinity Control Theory.