The degenerate Cahn-Hilliard equation, initially introduced in material science, is nowadays used in several different fields, including biology (tumor growth, cell-cell adhesion) and fluid dynamics (high-friction limit in the Euler-Korteweg equation). In this talk, we discuss several derivations of this equation: via hydrodynamic limit from the Vlasov equation (arXiv:2208.01026, arXiv: 2306.06486), from the nonlocal equation and related interacting particle system in the spirit of Giacomin-Lebowitz work (arXiv:2208.08955, arXiv:2303.11929), and most recently, from the Euler-Korteweg equation (arXiv:2305.01348). In most of these cases, fully rigorous solutions are available only in the case of a torus. So, we will stress fundamental difficulties related to the presence of physical boundaries. This is a joint work with C. Elbar, B. Perthame (Paris), J. A. Carrillo (Oxford), M. Mason (Milan), A. Świerczewska-Gwiazda, and P. Gwiazda (Warsaw).
Jakub Skrzeczkowski obtained his PhD in June 2023 at the Institute of Mathematics of the Polish Academy of Sciences under the supervision of Piotr Gwiazda. During his PhD, he spent more than two years in Paris, working in the group of Benoit Perthame. In September 2023, he joined the Mathematical Institute, University of Oxford, where he works as a postdoctoral research associate in the group of Jose A. Carrillo.