Inverse Source Problem for the Stokes System

Abstract

We consider a Stokes system with external force term:

$$\frac{{\partial y}}{{\partial t}} = \Delta y - \nabla p + r\left( t \right)f\left( x \right),\nabla \cdot y = 0in\left( {0,T} \right) \times \Omega $$

and

$${y_{\left| {\left( {0,T} \right) \times \partial \Omega } \right.}} = 0,$$

where Ω ⊂ ℝⁿ, n = 2,3 is a bounded domain. We discuss an inverse source problem of determining f = (f ₁,..., f _n ) from ${y_{\left| {\omega \times \left( {0,T} \right)} \right.}},{p_{\left| {\omega \times \left( {0,T} \right)} \right.}},y\left( {\theta ,\cdot } \right)$ and p(θ, ·), provided that ω ⊂Ω is an arbitrary domain, θ >0 and real-valued r are given. Our main result is the Lipschitz stability in the inverse problem and the proof is based on a Carleman estimate for the Stokes system.