On exact controllability for the Navier-Stokes equations

Abstract. We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain Ω with control distributed in a subdomain! Ω Rn; n 2 f2; 3g. The result that we obtained in this paper is as follows. Suppose that v̂(t; x) is a given solution of the Navier-Stokes equations. Let v0(x) be a given initial condition and kv̂(0; )−v0k < " where " is small enough. Then there exists a locally distributed control u; suppu (0; T) ! such that the solution v(t; x) of the Navier-Stokes equations

Inverse problem by Cauchy data on arbitrary subboundary for system of elliptic equations

We consider an inverse problem of determining coefficient matrices in an $N$-system of second-order elliptic equations in a bounded two dimensional domain by a set of Cauchy data on arbitrary subboundary. The main result of the article is as follows: If two systems of elliptic operators generate the same set of partial Cauchy data on an arbitrary subboundary, then the coefficient matrices of the first-order and zero-order terms satisfy the prescribed system of first-order partial differential equations. The main result implies the uniqueness of any two coefficient matrices provided that the one remaining matrix among the three coefficient matrices is known