arXiv subjects

Konstantina Trivisa

Publications and source records attributed to Konstantina Trivisa.

29 records · Page 2Linked to original sources

arXiv2017

On a free boundary problem for polymeric fluids: Global existence of weak solutions

We investigate the stability and global existence of weak solutions to a free boundary problem governing the evolution of polymeric fluids. We construct weak solutions of the two-phase model by performing the asymptotic limit of a macroscopic model governing the suspensions of rod-like molecules (known as Doi-Model) in compressible fluids as the adiabatic exponent $γ$ goes to $\infty.$ The convergence of these solutions, up to a subsequence, to the free-boundary problem is established using techniques in the spirit of Lions and Masmoudi \cite{LionsMasmoudi-1999}.

math.AP↗

arXiv2017

On the vanishing viscosity approximation of a nonlinear model for tumor growth

We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Darcy's equation. The domain occupied by the tumor in this setting is a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on the vanishing viscosity of the Brinkman's regularization. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data.

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arXiv2016

Analysis and Simulation on a Model for the Evolution of Tumors under the Influence of Nutrient and Drug Application

We investigate the evolution of tumor growth relying on a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the role of nutrient and drug application in the progression of tumors. We construct an explicit convergent numerical scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization and anisotropies in the development of tumor shape irregularities are discussed.

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arXiv2015

A convergent explicit finite difference scheme for a mechanical model for tumor growth

Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on imaging analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy's law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of Lions (Mathematical Topics in Fluid Dynamics, 1998).

math.NA↗

arXiv2015

On a nonlinear model for tumor growth in a cellular medium

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the "tumor" is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the "waste" and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles \cite{Enault-2010}, \cite{LiLowengrub-2013} where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions \cite{Lions-1998} (see also \cite{Feireisl-book, DT-MixedModel-2013}).

math.AP↗

arXiv2015

On a nonlinear model for tumor growth with drug application

We investigate the dynamics of a nonlinear system modeling tumor growth with drug application. The tumor is viewed as a mixture consisting of proliferating, quiescent and dead cells as well as a nutrient in the presence of a drug. The system is given by a multi-phase flow model: the densities of the different cells are governed by a set of transport equations, the density of the nutrient and the density of the drug are governed by rather general diffusion equations, while the velocity of the tumor is given by Brinkman's equation. The domain occupied by the tumor in this setting is a growing continuum $Ω$ with boundary $\partial Ω$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation. Both the solutions and the domain are rather general, no symmetry assumption is required and the result holds for large initial data. This article is part of a research program whose aim is the investigation of the effect of drug application in tumor growth.

math.AP↗

arXiv2014

Relative entropy in hyperbolic relaxation for balance laws

We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [A. Tzavaras, Commun. Math. Sci., 3-2, 2005] for systems of hyperbolic conservation laws.

math.AP↗

arXiv2012

Hydrodynamic limit of the kinetic Cucker-Smale flocking model

The hydrodynamic limit of a kinetic Cucker-Smale model is investigated. In addition to the free-transport of individuals and the Cucker-Smale alignment operator, the model under consideration includes a strong local alignment term. This term was recently derived as the singular limit of an alignment operator due to Motsch and Tadmor. The model is enhanced with the addition of noise and a confinement potential. The objective of this work is the rigorous investigation of the singular limit corresponding to strong noise and strong local alignment. The proof relies on the relative entropy method and entropy inequalities which yield the appropriate convergence results. The resulting limiting system is an Euler-type flocking system.

math.AP↗

arXiv2012

On strong local alignment in the kinetic Cucker-Smale model

In two recent papers the authors study the existence of weak solutions and the hydrodynamic limit of kinetic flocking equations with strong local alignment. The introduction of a strong local alignment term to model flocking behavior was formally motivated in these papers as a limiting case of an alignment term proposed by Motsch and Tadmor. In this paper, we rigorously justify this limit, and show that the considered equation is indeed a limit of the Motsch-Tadmor model when the radius of interaction goes to zero. The analysis involves velocity averaging lemmas and several $L^p$ estimates.

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arXiv2012

Existence of weak solutions to kinetic flocking models

We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under consideration include the kinetic Cucker-Smale equation with possibly non-symmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor. The main tools employed in the analysis are the velocity averaging lemma and the Schauder fixed point theorem along with various integral bounds.

math.AP↗

arXiv2010

On the Dynamics of a Fluid-Particle Interaction Model: The Bubbling Regime

This article deals with the issues of global-in-time existence and asymptotic analysis of a fluid-particle interaction model in the so-called bubbling regime. The mixture occupies the physical space $Ω\subset \mathbb{R}^3$ which may be unbounded. The system under investigation describes the evolution of particles dispersed in a viscous compressible fluid and is expressed by the conservation of fluid mass, the balance of momentum and the balance of particle density often referred as the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually by action-reaction principle. We show that solutions exist globally in time under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, we prove the large-time stabilization of the system towards a unique stationary state fully determined by the masses of the initial density of particles and fluid and the external potential.

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