A source reconstruction algorithm for the Stokes system from incomplete velocity measurements

Abstract

We consider the inverse problem of determining the spatial dependence of a source of the form f (x) sigma (t) in the Stokes system defined in Omega x (0, T), assuming that sigma (t) is known and f (x) is divergence-free. The only available observation is a single internal measurement of the velocity and the acceleration, for which one of its components is missing. Under adequate hypothesis on sigma we prove uniqueness of this inverse problem and we establish an explicit reconstruction formula. This formula provides the spectral coefficients f(k) of the source f in terms of a family of null controls h((tau)) for the corresponding adjoint system indexed by tau is an element of (0, T].