Recent results in interface motions in the framework of optimal transport: Lecture 1

In the lectures we will discuss recent results obtained on interface motions in the framework of optimal transport. We intend to (time allowing) discuss the following problems:

The Hele-Shaw type flows in the context of tumor growth. Here the flow describe the growth of tumor cells with contact inhibition. The tumor cells then form a congested zone, which evolves by the pressure generated by the constraint on maximal density. We start with a simple mechanical model, and discuss the effects of nutrients and surface tension in the context of minimizing movements. While the well-posedness would be established by minimizing movements, we will also explore qualitative properties of solutions such as regularity of the interface.

The Stefan problem, in the framework of optimal stopping time. Our focus will be on the well-posedness of the supercooled Stefan problem, which describes freezing of supercooled fluid. The interface between the fluid and ice, as it freezes, exhibits a high degree of irregularity. Our goal is to introduce a notion of solutions that are physically meaningful and stable. We will start with a quick introduction of the necessary background on the optimal stopping time between probability measures. We will establish the well-posedness, and discuss qualitative behavior of solutions.

• Lecture 1
• Lecture 2
• Lecture 3