Sharp interface limit of a diffuse interface model for tumor-growth

TL;DR

This paper analyzes the sharp interface limit of a diffuse interface tumor-growth model as the interface thickness parameter approaches zero, using matched expansion and energy methods to estimate the solution behavior.

Contribution

It introduces an approximate solution via matched expansion and establishes a smallness estimate for the difference from the true solution using spectral energy methods.

Findings

Explicit behavior of solutions as interface thickness tends to zero

Validation of the approximate solution through energy estimates

Quantitative bounds on the difference between true and approximate solutions

Abstract

We consider the asymptotic limit of a diffuse interface model for tumor-growth when a parameter $\varepsilon$ proportional to the thickness of the diffuse interface goes to zero. An approximate solution which shows explicitly the behavior of the true solution for small $\varepsilon$ will be constructed by using matched expansion method. Based on the energy method, a spectral condition in particular, we establish a smallness estimate of the difference between the approximate solution and the true solution.