Transport equation with boundary conditions for free surface localization

During the filling stage of an injection moulding process, which consists i n casting a melt polymer in order to manufacture plastic pieces, the free i nterface between polymer and air has to be precisely described. We set this interface as a zero level set of an unknown function. This function satisf ies a transport equation with boundary conditions, where the velocity field has few regularity properties. In a first part, we obtain existence and uniqueness result for these equati ons, under weaker regularity assumptions than C. Bardos [Bar70], and C. Bar dos, Y. Leroux and J.C. Nedelec [BLN79] in previous articles, but stronger assumptions than R.J. DiPerna and P.L. Lions [DL89b] who studied the case w ithout boundary condition. We also study some regularity properties of the interface. A second part is devoted to an application to injection molding of melt pol ymer. We give a numerical experiment which shows that our method leads to a n accurate localization of interface, which is robust, since it easily hand les changes of topology of the free interface, as bubble formation or fusio n of two fronts of melt polymer.