  In this paper we study the Matovich-Pearson equations describing the process
of glass fiber drawing. These equations may be viewed as a 1D-reduction of the
incompressible Navier-Stokes equations including free boundary, valid for the
drawing of a long and thin glass fiber. We concentrate on the isothermal case
without surface tension. Then the Matovich-Pearson equations represent a
nonlinearly coupled system of an elliptic equation for the axial velocity and a
hyperbolic transport equation for the fluid cross-sectional area. We first
prove existence of a local solution, and, after constructing appropriate
barrier functions, we deduce that the fluid radius is always strictly positive
and that the local solution remains in the same regularity class. To the best
of our knowledge, this is the first global existence and uniqueness result for
this important system of equations.
