A tumor growth model of Hele-Shaw type as a gradient flow

¹ Istituto Nazionale di Alta Matematica, Sede SNS Pisa, Italy.
² Lab. des Math., Université Paris-Sud, Orsay, France.
³ Université Paris-Dauphine, PSL Research University, CNRS, CEREMADE, 75016 Paris, France.

^* Corresponding author: simone.dimarino@altamatematica.it

Received: 11 February 2019
Accepted: 9 April 2020

Abstract

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.