Journal of Mathematical Fluid Mechanics

, Volume 7, Issue 1, pp 108–152

Navier–Stokes Equations: Controllability by Means of Low Modes Forcing

Original Paper

DOI: 10.1007/s00021-004-0110-1

Cite this article as:
Agrachev, A.A. & Sarychev, A.V. J. math. fluid mech. (2005) 7: 108. doi:10.1007/s00021-004-0110-1


We study controllability issues for 2D and 3D Navier–Stokes (NS) systems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and continuous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric.

Mathematics Subject Classification (2000).

35Q30 93C20 93B05 93B29 


Navier–Stokes equation controllability geometric control 

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.SISSA-ISASTriesteItaly
  2. 2.V. A. Steklov Mathematical InstituteMoscowRussia
  3. 3.DiMaDUniversity of FlorenceFirenzeItaly

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