scholarly journals A Rigorous Sharp Interface Limit of a Diffuse Interface Model Related to Tumor Growth

2016 ◽  
Vol 27 (3) ◽  
pp. 847-872 ◽  
Author(s):  
Elisabetta Rocca ◽  
Riccardo Scala
Keyword(s):  
Tumor Growth ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Sharp Interface Limit

scholarly journals Formal asymptotic limit of a diffuse-interface tumor-growth model

2015 ◽  
Vol 25 (06) ◽  
pp. 1011-1043 ◽  
Author(s):  
Danielle Hilhorst ◽  
Johannes Kampmann ◽  
Thanh Nam Nguyen ◽  
Kristoffer George Van Der Zee
Keyword(s):  
Tumor Growth ◽  
Growth Model ◽  
Singular Limit ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Sharp Interface Limit ◽  
Field System ◽  
Tumor Growth Model ◽  
Phase Field System

We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the diffuse-interface model to the identified sharp-interface limit.

scholarly journals Optimal distributed control of a diffuse interface model of tumor growth

Nonlinearity ◽  
2017 ◽  
Vol 30 (6) ◽  
pp. 2518-2546 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Elisabetta Rocca ◽  
Jürgen Sprekels
Keyword(s):  
Tumor Growth ◽  
Distributed Control ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Optimal Distributed Control

scholarly journals Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth

2020 ◽  
Vol 26 ◽  
pp. 71 ◽  
Author(s):  
Matthias Ebenbeck ◽  
Patrik Knopf
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Chemical Potential ◽  
Type Equation ◽  
Field Variable ◽  
Reaction Diffusion Equation ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.

scholarly journals A diffuse interface model for quasi–incompressible flows : Sharp interface limits and numerics, ,

2012 ◽  
Vol 38 ◽  
pp. 54-77 ◽  
Author(s):  
Gonca Aki ◽  
Johannes Daube ◽  
Wolfgang Dreyer ◽  
Jan Giesselmann ◽  
Mirko Kränkel ◽  
...  
Keyword(s):  
Incompressible Flows ◽  
Sharp Interface ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

scholarly journals Analysis of a diffuse interface model of multispecies tumor growth

Nonlinearity ◽  
2017 ◽  
Vol 30 (4) ◽  
pp. 1639-1658 ◽  
Author(s):  
Mimi Dai ◽  
Eduard Feireisl ◽  
Elisabetta Rocca ◽  
Giulio Schimperna ◽  
Maria E Schonbek
Keyword(s):  
Tumor Growth ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

scholarly journals Optimal Control of Treatment Time in a Diffuse Interface Model of Tumor Growth

2017 ◽  
Vol 78 (3) ◽  
pp. 495-544 ◽  
Author(s):  
Harald Garcke ◽  
Kei Fong Lam ◽  
Elisabetta Rocca
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Treatment Time ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

scholarly journals A Comparative Study of the Diffuse-Interface Model and Sharp-Interface Model in the Soldering Related Wetting Spreading Systems

Metals ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 944
Author(s):  
Liu ◽  
Zhang ◽  
Lei ◽  
Li ◽  
Li
Keyword(s):  
Sharp Interface ◽  
Diffuse Interface ◽  
Theoretical Formula ◽  
Interface Model ◽  
Sharp Interface Model ◽  
Diffuse Interface Model ◽  
Formula Derivation ◽  
Dominant Condition ◽  
Special Case

A typical dissolution wetting system, Bi-Sn eutectic filler metal over a Bi substrate in a high-purity argon atmosphere was investigated first using real-time in situ hot stage microscopy for the extensive use of the sharp-interface model and the diffuse-interface model in the modeling of brazing/soldering related wetting systems. Subsequently, the similarities and differences between the aforementioned models in describing the issues of the wetting and spreading interfaces were discussed in terms of soldering definition and theoretical formula derivation. It is noted that (i) the mutual dissolution diffusion between the liquid Bi-Sn solder and Bi substrate were obvious. As a result, the composition and volume of the liquid solder is constantly changing during the wetting and spreading process; (ii) the sharp-interface model is a special case of the diffuse-interface model of the Cahn-Hilliard nonlinear diffuse-equation under the convective dominant condition; (iii) although there are differences between the sharp-interface model and the diffuse-interface model, both of them could be used in brazing/soldering related processes; and, (iv) the agreement between the experimental and simulation results of the sharp-interface model is not as good as that of the diffuse-interface model, which can be attributed to the effects of the elements’ diffusion and the phase transformation.

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