Free boundary problems for Tumor Growth: a Viscosity solutions approach

TL;DR

This paper employs viscosity solutions to analyze free boundary problems in tumor growth models, proving convergence and regularity of interfaces, and providing new insights into the dynamics of propagating tumor boundaries.

Contribution

It introduces a viscosity solutions framework to study limits of porous medium equations with active motion, improving convergence results and understanding interface regularity.

Findings

Proved uniform convergence of density under general initial conditions.

Obtained regularity results for propagating interfaces.

Constructed local radial solutions as barriers for analysis.

Abstract

The mathematical modeling of tumor growth leads to singular stiff pressure law limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uniform convergence of the density under fairly general assumptions on the initial data, thus improving existing results. We also obtain some additional information/regularity about the propagating interfaces, which, in view of the discontinuities, can nucleate and, thus, change topological type. The main tool is the construction of local, smooth, radial solutions which serve as barriers for the existence and uniqueness results as well as to quantify the speed of propagation of the free boundary propagation.