### Plenary Lecture 14: Free boundary problems for mechanical models of tumor growth

**Vázquez, JL ***(Universidad Autonoma de Madrid)*

Friday 27 June 2014, 15:00-15:45

Seminar Room 1, Newton Institute

#### Abstract

Mathematical models of tumor growth, now commonly used, present several
levels of complexity, both in terms of the biomedical ingredients and the
mathematical description. The simplest ones contain competition for space
using purely fluid mechanical concepts. Another possible ingredient is the
supply of nutrients. The models can describe the tissue either at the level
of cell densities, or at the scale of the solid tumor, in this latter case
by means of a free boundary problem.
We first formulate a free boundary model of Hele-Shaw type, a variant
including growth terms, starting from the description at the cell level and
passing to a certain singular limit which leads to a Hele-Shaw type problem.
A detailed mathematical analysis of this purely mechanical model is
performed. Indeed, we are able to prove strong convergence in passing to the
limit, with various uniform gradient estimates; we also prove uniqueness for
the limit problem. At variance with the classical Hele-Shaw problem, here
the geometric motion governed by the pressure is not sufficient to
completely describe the dynamics.
Using this theory as a basis, we go on to consider a more complex model
including nutrients. Here, technical difficulties appear, that reduce the
generality and detail of the description.
We prove uniqueness for the system, a main mathematical difficulty.
Joint work with Benoit Perthame, Paris, and Fernando Quiros, Madrid

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