##### Department of Mathematics,

University of California San Diego

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### Math 248 - Analysis Seminar

## Benoit Perthame

#### Sorbonne University

## Porous media based models of living tissues and free boundary problems

##### Abstract:

Tissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics.

We will give an overview of the modeling aspects and focus on the links between those mathematical models. Then, we will focus on the `compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form.

The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.

### May 24, 2022

### 11:00 AM

https://ucsd.zoom.us/j/

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