## Elliptic-Parabolic equations and injection molding

**Abstract:**

*Doubly nonlinear equations arose as a field of interest in the study of an injection molding process. Under assumption on the mold thickness, Navier-Stokes equations were rewritten as a pressure equa- tion which was an elliptic-parabolic equation. This equation could degenerate to a elliptic equation or remain parabolic. We developed a method to prove existence of solution to a wide class of doubly nonlinear equations containing the one of interest [A8, A6]. I studied also some numerical scheme to compute solution of such an equation. In the injection molding pro- cess, a melt plastic was filling a mold initially full of air. We chose a Level Set representation of the interface (without capillarity), where the level-set function was solution of a transport equation with inflow/outflow boundary conditions.*

**Publications:**

[1] Maitre E. et Witomski P. A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations. Nonlinear Anal.-Theory Methods Appl., 50 (2), 223–250, 2002.

[2] Maitre Emmanuel. Numerical analysis of nonlinear elliptic-parabolic equations. ESAIM-Math. Model. Numer. Anal., 36 (1), 143–153, 2002.

[3] Maitre E. et Witomski P. Transport equation with boundary conditions for free surface localization. Numer. Math., 84 (2), 275–303, 1999.

**Gallery:**

Isochrones of front positions during injection molding (1996)

Filling of a complex mold with melt polymer (1996)

Solution of an elliptic-parabolic equation where the solution is discontinuous in time, our algorithm (left) and the method of Jäger and Kacur (right). Our method better captures the discontinuity.