Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol 48

2002 ◽  
Vol 55 (4) ◽  
pp. B74-B75 ◽  
Author(s):  
SN Antontsev, ◽  
JI Diaz, ◽  
S Shmarev, ◽  
AJ Kassab,
Keyword(s):  
Free Boundary ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Free Boundary Problems ◽  
Nonlinear Differential Equations ◽  
Nonlinear Pdes ◽  
Energy Methods ◽  
Boundary Problems

MATHEMATICAL ANALYSIS AND CHALLENGES ARISING FROM MODELS OF TUMOR GROWTH

2007 ◽  
Vol 17 (supp01) ◽  
pp. 1751-1772 ◽  
Author(s):  
AVNER FRIEDMAN
Keyword(s):  
Free Boundary ◽  
Differential Equations ◽  
Solid Tumors ◽  
Free Boundary Problems ◽  
The Body ◽  
Multiphase Model ◽  
Spatial Effects ◽  
Boundary Problems ◽  
Fluid Medium ◽  
Tumor Region

In the last four decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. In this paper, we deal with tumor models in which the tumor occupies a well-defined region in space; the boundary of this region is held together by the forces of cell-to-cell adhesion. We shall refer to such tumors as "solid" tumors, although they may sometimes consist of fluid-like tissue, such as in the case of brain tumors (e.g. gliomas) and breast tumors. The most common class of solid tumors is carcinoma: a cancer originating from epithelial cells, that is, from the closely packed cells which align the internal cavities of the body. Models of solid tumors must take spatial effects into account, and are therefore described in terms of partial differential equations (PDEs). They also need to take into account the fact that the tumor region is changing in time; in fact, the tumor region, say Ω(t), and its boundary Γ(t), are unknown in advance. Thus one needs to determine both the unknown "free boundary" Γ(t) together with the solution of the PDEs in Ω(t). These types of problems are called free boundary problems. The models described in this paper are free boundary problems, and our primary interest is the spatial/geometric features of the free boundary. Some of the basic questions we shall address are: What is the shape of the free boundary? How does the free boundary behave as t → ∞? Does the tumor volume increase or shrink as t → ∞? Under what conditions does the tumor eventually become dormant? Finally, we shall explore the dependence of the free boundary on some biological parameters, and this will give rise to interesting bifurcation phenomena. The structure of the paper is as follows. In Secs. 1 and 2 we consider models in which all the cells are of one type, they are all proliferating cells. The tissue is modeled either as a porous medium (in Sec. 1) or as a fluid medium (in Sec. 2). The models are extended in Secs. 3 and 4 to include three types of cells: proliferating, quiescent, and dead. Finally, in Sec. 5 we outline a general multiphase model that includes gene mutations.

scholarly journals On integrability and exact solvability in deterministic and stochastic Laplacian growth

2020 ◽  
Vol 15 ◽  
pp. 3
Author(s):  
Igor Loutsenko ◽  
Oksana Yermolayeva
Keyword(s):  
Free Boundary ◽  
Fluid Mechanics ◽  
Integrable Systems ◽  
Stochastic Models ◽  
Free Boundary Problems ◽  
Exact Results ◽  
Laplacian Growth ◽  
Quantum Integrable Systems ◽  
Boundary Problems ◽  
Exact Solvability

We review applications of theory of classical and quantum integrable systems to the free-boundary problems of fluid mechanics as well as to corresponding problems of statistical mechanics. We also review important exact results obtained in the theory of multi-fractal spectra of the stochastic models related to the Laplacian growth: Schramm-Loewner and Levy-Loewner evolutions.

scholarly journals Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique

1984 ◽  
Vol 148 ◽  
pp. 1-17 ◽  
Author(s):  
G. Ryskin ◽  
L. G. Leal
Keyword(s):  
Finite Difference ◽  
Free Boundary ◽  
Fluid Mechanics ◽  
Free Boundary Problems ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
Boundary Problems ◽  
Boundary Shape ◽  
Curvilinear Coordinate ◽  
Ultimate Equilibrium

We present here a brief description of a numerical technique suitable for solving axisymmetric (or two-dimensional) free-boundary problems of fluid mechanics. The technique is based on a finite-difference solution of the equations of motion on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The overall solution is achieved via a global iterative process, with the condition of balance between total normal stress and the capillary pressure at the free boundary being used to drive the boundary shape to its ultimate equilibrium position.

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