We consider a Cahn-Hilliard-Darcy system for an incompressible mixture of two fluids. The relative concentration difference is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential, while the fluid velocity obeys a Darcy’s law depending on the Korteweg force. Systems of this kind are used to describe fluid flow in a porous medium as well as to model solid tumor growth through diffuse interfaces. We will report a series of results obtained in collaboration with S. Frigeri and M. Grasselli. Besides the existence of a global weak solution which satisfies an energy identity, we will discuss the existence of a strong solution, regularity properties, and uniqueness issues. Further works on similar systems with sources will also be mentioned.

学术报告海报20240514_Cavaterra.pdf