Navier-Stokes Equation on a Plane Bounded Domain

Continuity Properties for Controllability

Sérgio S. Rodrigues

SISSA-ISAS, via Beirut 2-4, Trieste 34014,

ABSTRACT. We study some continuity properties of the Navier-Stokes equations on C3 domains. These properties are important for the study of controllability issues as we may see in the works done in [AGR 05] for periodic boundary conditions and, in [ROD 06a] for Lions boundary conditions. Here we show that the same continuity properties hold in the C3 bounded domains having a boundary with a finite number of connected components. Navier and no-slip boundary conditions are considered. As a corollary of the continuity properties it follows that the existence of a finite saturating set of vector fields implies controllability on observed component and L2- approximate controllability of the equation. These controllabilities are both obtained by means of finite dimensional forcing taking values in the space spanned by the vector fields in the saturating set.

KEYWORDS: incompressible fluid, 2D Navier-Stokes system, controllability

1. Introduction

Following part of the work done in [AGR 05] and [ROD 06a] we study continuity properties of the 2D Navier-Stokes (N-S) equation. We deal with the system

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