scholarly journals Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

2021 ◽  
pp. 1-42
Author(s):  
SERGIO FRIGERI ◽  
KEI FONG LAM ◽  
ANDREA SIGNORI
Keyword(s):  
Inverse Problem ◽  
Phase Field ◽  
Necessary Optimality Conditions ◽  
Strong Solutions ◽  
Reaction Diffusion Equation ◽  
Diffuse Interface ◽  
Well Posedness ◽  
Tumour Model ◽  
Degenerate Mobility ◽  
Non Local

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

scholarly journals Optimal control of stochastic phase-field models related to tumor growth

2020 ◽  
Vol 26 ◽  
pp. 104
Author(s):  
Carlo Orrieri ◽  
Elisabetta Rocca ◽  
Luca Scarpa
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Phase Field Model ◽  
Growth Dynamics ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Adjoint System ◽  
Reaction Diffusion Equation ◽  
Cahn Hilliard Equation

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

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 Asymptotic analysis of a tumor growth model with fractional operators

2020 ◽  
Vol 120 (1-2) ◽  
pp. 41-72 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Jürgen Sprekels
Keyword(s):  
Tumor Growth ◽  
Asymptotic Analysis ◽  
Phase Field ◽  
Volume Fraction ◽  
Reaction Diffusion Equation ◽  
Water Volume ◽  
Well Posedness ◽  
Field System ◽  
Phase Field System ◽  
Regularity Results

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn–Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Meth. Biomed. Eng. 28 (2012), 3–24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn–Hilliard equation for the tumor cell fraction φ, coupled to a reaction–diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

scholarly journals Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport

2016 ◽  
Vol 28 (2) ◽  
pp. 284-316 ◽  
Author(s):  
HARALD GARCKE ◽  
KEI FONG LAM
Keyword(s):  
Active Transport ◽  
Tumour Growth ◽  
Chemical Potential ◽  
Reaction Diffusion Equation ◽  
Diffuse Interface ◽  
System Modelling ◽  
Well Posedness ◽  
Source Terms ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derivea prioriestimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

scholarly journals Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

2018 ◽  
Vol 148 (5) ◽  
pp. 957-981
Author(s):  
Tomás Caraballo ◽  
Marta Herrera-Cobos ◽  
Pedro Marín-Rubio
Keyword(s):  
Existence And Uniqueness ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Pullback Attractors ◽  
Non Local ◽  
Bounded Sets

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

scholarly journals Optimal control of time-discrete two-phase flow driven by a diffuse-interface model

2019 ◽  
Vol 25 ◽  
pp. 13 ◽  
Author(s):  
Harald Garcke ◽  
Michael Hinze ◽  
Christian Kahle
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Time Discretization ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Fully Discrete

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in Abelset al.[M3AS22(2012) 1150013]. On the basis of the stable time discretization proposed in Garckeet al.[Appl. Numer. Math.99(2016) 151] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.

scholarly journals Local well-posedness of a quasi-incompressible two-phase flow

2020 ◽  
Author(s):  
Helmut Abels ◽  
Josef Weber
Keyword(s):  
Two Phase Flow ◽  
Type Equation ◽  
Strong Solutions ◽  
Variable Density ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Well Posedness ◽  
Phase Flow ◽  
Two Phase ◽  
Fluid Mixture

AbstractWe show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier–Stokes/Cahn–Hilliard system with a solenoidal velocity field for the mixture, but a variable density of the fluid mixture in the Navier–Stokes type equation. We prove existence of strong solutions locally in time with the aid of a suitable linearization and a contraction mapping argument. To this end, we show maximal $$L^2$$ L 2 -regularity for the Stokes part of the linearized system and use maximal $$L^p$$ L p -regularity for the linearized Cahn–Hilliard system.

scholarly journals Distributed optimal control problems for phase field systems with singular potential

2018 ◽  
Vol 26 (2) ◽  
pp. 71-85
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Gabriela Marinoschi ◽  
Elisabetta Rocca
Keyword(s):  
Control Problem ◽  
Phase Field ◽  
Singular Potential ◽  
Necessary Optimality Conditions ◽  
Logarithmic Potential ◽  
Diffuse Interface ◽  
Distributed Optimal Control ◽  
Distributed Optimal Control Problems ◽  
Field Systems ◽  
The Cost

Abstract In this paper we review some results obtained for a distributed con- trol problem regarding a class of phase field systems of Caginalp type with logarithmic potential. The aim of the control problem is forcing the location of the diffuse interface to be as close as possible to a pre- scribed set. However, due to some discontinuity in the cost functional, we have to regularize it and solve the related control problem for the approximation. We discuss the necessary optimality conditions.

scholarly journals Non-local Solvable Birth–Death Processes

2021 ◽  
Author(s):  
Giacomo Ascione ◽  
Nikolai Leonenko ◽  
Enrica Pirozzi
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Limit Distribution ◽  
Spectral Decomposition ◽  
Strong Solutions ◽  
Stochastic Representation ◽  
Discrete Approximations ◽  
Local Difference ◽  
Non Local ◽  
Birth Death

AbstractIn this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

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