Global-in-time solutions for the isothermal Matovich-Pearson equations

Eduard Feireisl (1) , Philippe Laurencot (2) , Andro Mikelic (3)

1 Mathematical Institute [Praha] - Academy of Sciences
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 ICJ - Institut Camille Jordan

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.

Mots clés: Fiber drawing; Matovich-Pearson equations; incompressible free boundary Navier-Stokes equations; non-local transport equation; iterated comparison; barrier function

Domaines: Equations aux dérivées partielles [math.AP]

Eduard Feireisl, Philippe Laurencot, Andro Mikelic. Global-in-time solutions for the isothermal Matovich-Pearson equations. Nonlinearity, 2011, 24, pp.277-292. ⟨10.1088/0951-7715/24/1/014⟩. ⟨hal-00488835⟩