Regularity and nondegeneracy for tumor growth with diffusive nutrient

par Carson Collins (UCLA)

mercredi 15 oct. 2025, 10:30 → 12:00 Europe/Paris

We discuss a model for incompressible tumor growth. The expansion of the tumor region is given by Darcy's law, with a source term coming from a coupled nutrient variable which diffuses and is consumed by the tumor. This model has drawn interest in numerical and biological literature due to a fingering instability in the tumor region which occurs when the nutrient is sparse and diffuses slowly. This is related to the issue that the coupled system does not have a comparison principle, so standard methods to show regularity of Hele-Shaw type flows are unavailable. 

To work around this, we introduce a Hamilton-Jacobi-Bellman interpretation of the pressure, derived from the incompressible limit of the porous medium equation. This gives a Hopf-Lax type inequality for the pressure, which allows us to form barrier arguments to obtain mild nondegeneracy of the tumor motion. Subsequently, we can use connections with obstacle problem theory to show generic-in-spacetime regularity of the tumor region.

This talk is based on joint work with Matt Jacobs and Inwon Kim.