Section: New Results
We have continued our study on composite materials. We have finalized the phd of Xi Lin. We have finished to write the fluid models describing the propagation of an oxyde in the cracks appearing in self-healing composite materials. More precisely, we obtained two families of models, governed by the boundaries conditions. The first one looks like Shallow Water Equations and the second one comes from Lubrication theory. In order to evaluate the relevance of our models, we investigate the well-posedness of our model. In this direction, in  we propose in the full generality a link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF) dissipative entropy introduced to study the lubrications equations. Different dissipative entropies are obtained playing with the drag terms on the viscous shallow water equations. It helps for instance to prove global existence of nonnegative weak solutions for the lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations. Two articles are in preparation.