Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of Free Boundaries in the Incompressible Limit of Tumor Growth Models

Jiajun Tong, Yuming Paul Zhang

TL;DR

This paper provides an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit, and derives upper bounds for the Hausdorff dimensions of the free boundaries and shows that the limiting free boundary has finite $(d-1)-dimensional Hausdorff measure.

Abstract

We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-$m$ strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite $(d-1)$-dimensional Hausdorff measure.

Convergence of Free Boundaries in the Incompressible Limit of Tumor Growth Models

TL;DR

This paper provides an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit, and derives upper bounds for the Hausdorff dimensions of the free boundaries and shows that the limiting free boundary has finite $(d-1)-dimensional Hausdorff measure.

Abstract

We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in- strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite -dimensional Hausdorff measure.
Paper Structure (23 sections, 27 theorems, 295 equations)

This paper contains 23 sections, 27 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Uniform strict expansion of the support relative to streamlines
  3. Convergence of the free boundaries
  4. Hausdorff dimensions of the free boundaries
  5. Other related works
  6. Organization of the paper
  7. Preliminaries
  8. Notations
  9. Assumptions
  10. Preliminary results
  11. The Aronson-Bénilan Estimate
  12. Expansion of Positive Sets along Streamlines
  13. Uniform Estimates for Strict Expansion
  14. Strict expansion at the initial time
  15. Strict expansion after the initial time
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\underline{\varrho}$ and $\bar{\varrho}$ be, respectively, a sub-solution and a super-solution to main with bounded, non-negative and compactly supported initial data $\underline{\varrho}^0$ and $\bar{\varrho}^0$. If $\underline{\varrho}^0\leq \bar{\varrho}^0$, then $\underline{\varrho}\leq \ba

Theorems & Definitions (56)

  • Definition 2.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Remark 3.1
  • ...and 46 more