Existence of a nonstationary Poiseuille solution

The nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution of the inverse problem for the heat equation. The existence and uniqueness of such nonstationary Poiseuille solution with the prescribed flux $F(t)$ of the velocity field is studied. It is proved that under some compatibility conditions on the initial data and flux $F(t)$ the corresponding inverse problem has a unique solution in Holder spaces.