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Invent. math. (2014) 198:833880
DOI 10.1007/s00222-014-0512-5
Received: 31 October 2012 / Accepted: 24 February 2014 / Published online: 20 March 2014 Springer-Verlag Berlin Heidelberg 2014
Abstract In this paper, we prove a local null controllability result for the three-dimensional NavierStokes equations on a (smooth) bounded domain of R3 with null Dirichlet boundary conditions. The control is distributed in an arbitrarily small nonempty open subset and has two vanishing components. Lions and Zuazua proved that the linearized system is not necessarily null controllable even if the control is distributed on the entire domain, hence the standard linearization method fails. We use the return method together with a new algebraic method inspired by the works of Gromov and previous results by Gueye.
Mathematics Subject Classication (2000) 35Q30 93B05 93C10
1 Introduction
1.1 Notations and statement of the theorem
Let T > 0, let be a nonempty bounded domain of R3 of class C and let be a nonempty open subset of . We dene Q
R
Work supported by the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7).
J.-M. Coron
Institut Universitaire de France and Sorbonne Universits, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, Francee-mail: [email protected]
P. Lissy (B)
Sorbonne Universits, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, Francee-mail: [email protected]
Local null controllability of the three-dimensional NavierStokes system with a distributed control having two vanishing components
Jean-Michel Coron Pierre Lissy
R3 by
123
834 J.-M. Coron, P. Lissy
Q := (0, T ) = {(t, x)| t (0, T ) and x }
and we call
:= [0, T ] .
The current point x
R3 is x = (x1, x2, x3). The i-th component of a vector
(or a vector eld) f is denoted f i. The control is u = (u1, u2, u3) L2(Q)3.
We require that the support of u is included in , which is our control domain. We impose that two components of u vanish, for example the rst two:
u1 = 0 and u2 = 0 in Q, (1.1)
so that u will be written under the form (0, 0, 1v) with v L2(Q) from now
on, where 1 :
R is the characteristic function of :
1...