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

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