On a Hele-Shaw Flow Problem with Free and Solid Boundary Components

The purpose of this work is to study a one phase Hele-Shaw fluid flow occupying a time variable domain (Omega (t)), due to the injection of the fluid with a constant rate at a single point of the initial domain (Omega (0)), and in the presence of a fixed solid body (Omega _0). We show the short time existence and uniqueness of the solution for the corresponding boundary value problem in the three dimensional case and in the absence of surface tension.