arXiv subjects

Francisco Guillén-González

Publications and source records attributed to Francisco Guillén-González.

At least 19 recordsLinked to original sources

arXiv2026

Structure-preserving upwind DG scheme for a Cahn-Hilliard-Darcy model of tumor growth

In this work, we develop a structure-preserving numerical scheme for a Cahn-Hilliard-Darcy model that describes tumor growth in a fluid-saturated porous medium. First, we derive a physically consistent model from the general framework proposed in [29] that guarantees mass conservation and pointwise bounds on the phase-field and nutrient variables, with a decreasing energy law. The resulting model couples the evolution of tumor cells via a Cahn-Hilliard equation with a diffusion equation for the nutrients thro chemotactic interactions and extends the model in [1] by introducing the effect of a surrounding fluid described by Darcy's law. Subsequently, we propose a fully discrete scheme that combines an upwind discontinuous Galerkin method in space and a convex splitting strategy in time, which inherits the fundamental properties of the continuous model: mass conservation, pointwise bounds and discrete energy law. Our theoretical analysis is accompanied by numerical experiments that demonstrate the robustness of the proposed scheme and show the influence of the surrounding fluid on the tumor evolution.

math.NA↗

arXiv2026

Optimal control of therapies related to an oxytaxis glioblastoma model

We propose and analyze an optimal control problem associated with a Keller-Segel type parabolic system with chemoattraction, modeling the glioblastoma growth in a bi-dimensional bounded domain, influenced by the presence of oxygen where the controls are two different (chemotherapy and antiangiogenic) therapies. The model considers the random diffusion of tumor cells and oxygen, the movement of cells towards the oxygen gradient (oxytaxis), and reaction terms describing the interaction between cells and oxygen. We establish a mathematical framework to analyze the existence and uniqueness of weak-strong solution of the model and subsequently we analyze an optimal control problem considering a cost functional that minimizes both the tumor growth and the oxygen concentration. We prove the existence of a global optimal solution and derive necessary first-order optimality conditions. Finally, we propose a methodology for approximating the optimal therapies. We use the gradient of the reduced cost functional through the adjoint scheme, and minimize the cost functional implementing the Adam gradient optimization method. Some numerical experiments are provided to demonstrate the effectiveness of the proposed scheme.

math.OC↗

arXiv2025

A convergent time scheme for a chemotaxis-fluids model with potential consumption

The present work deals with a Keller-Segel-Navier-Stokes system with potential consumption, under homogeneous Neumann boundary conditions for cell density and chemical signal, and of Dirichlet type for the velocity field, over a bounded three-dimensional domain. The paper aims to develop a time discretization scheme converging to weak solutions of the system, which are uniformly bounded at infinite time. While global existence results are already known for simplified cases, either in absence of fluid flow or for linear consumption, the existence of global weak solutions for the fully coupled system with potential consumption has remained as an open problem.

math.AP↗

arXiv2024

Property-preserving numerical approximation of a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility

In this paper, we present a new computational framework to approximate a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility that preserves the mass of the mixture, the pointwise bounds of the density and the decreasing energy. This numerical scheme is based on a finite element approximation for the Navier-Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn-Hilliard part. Finally, several numerical experiments such as a convergence test and some well-known benchmark problems are conducted.

math.NA↗

arXiv2024

Mathematical Modelling of Neuroblast Chemotaxis Migration towards the Olfactory Bulb

This article is devoted to the mathematical modeling of migration of neuroblasts, precursor cells of neurons, along the pathway they usually follow before maturing. This pathway is determined mainly by chemotaxis and the heterogeneous mobility of neuroblasts in different regions of the brain. In numerical simulations, the application of novel discontinuous Galerkin methods allows to maintain the properties of the continuous model such as the maximum principle. We present some successful computer tests including parameter adjust to fit real data from rodent brains.

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arXiv2024

Structure preserving finite element schemes for the Navier-Stokes-Cahn-Hilliard system with degenerate mobility

In this work we present two new numerical schemes to approximate the Navier-Stokes-Cahn-Hilliard system with degenerate mobility using finite differences in time and finite elements in space. The proposed schemes are conservative, energy-stable and preserve the maximum principle approximately (the amount of the phase variable being outside of the interval [0,1] goes to zero in terms of a truncation parameter). Additionally, we present several numerical results to illustrate the accuracy and the well behavior of the proposed schemes, as well as a comparison with the behavior of the Navier-Stokes-Cahn-Hilliard model with constant mobility.

math.NA↗

arXiv2024

Optimal control related to weak solutions of a chemotaxis-consumption model

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain $Ω\subset \mathbb{R}^3$: $$\partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v + f \,v\, 1_{Ω_c},$$ with $s \geq 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$, being $u$ the cell density, $v$ the chemical concentration and $f$ the control acting in the $v$-equation through the bilinear term $f \,v\, 1_{Ω_c}$, in a subdomain $Ω_c \subset Ω$. We address the existence of optimal control restricted to a weak solution setting, where, in particular, uniqueness of state $(u,v)$ given a control $f$ is not clear. Then by considering weak solutions satisfying an adequate energy inequality, we prove the existence of optimal control subject to uniformly bounded controls. Finally, we discuss the relation between the considered control problem and two other related ones, where the existence of optimal solution can not be proved.

math.OC↗

arXiv2024

A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models

In the present review we focus on the chemotaxis-consumption model $\partial_t u - Δu = - \nabla \cdot (u \nabla v)$ and $\partial_t v - Δv = - u^s v$ in $(0,T) \times Ω$, for any fixed $s \geq 1$, endowed with isolated boundary conditions and nonnegative initial conditions, where $(u,v)$ model cell density and chemical signal concentration. Our objective is to present an overview of the related literature and latest results on the aforementioned model concerning the following three distinct research lines we have obtained in [12,24-26]: the mathematical analysis, the numerical analysis and the related optimal control theory with a bilinear control acting on the chemical equation.

math.AP↗

arXiv2023

Convergence of a time discrete scheme for a chemotaxis-consumption model

In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any $s\ge 1$), $$ \partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v \quad \hbox{in $(0,T)\times Ω$,}$$ endowed with isolated boundary conditions and initial conditions, where $(u,v)$ model cell density and chemical signal concentration. The proposed scheme is defined via a reformulation of the model, using the auxiliary variable $z = \sqrt{v + α^2}$ combined with a Backward Euler scheme for the $(u,z)$-problem and a upper truncation of $u$ in the nonlinear chemotaxis and consumption terms. Then, two different ways of retrieving an approximation for the function $v$ are provided. We prove the existence of solution to the time discrete scheme and establish uniform in time \emph{a priori} estimates, yielding the convergence of the scheme towards a weak solution $(u,v)$ of the chemotaxis-consumption model.

math.NA↗

arXiv2023

An optimal control problem subject to strong solutions of chemotaxis-consumption models

We consider a bilinear optimal control problem associated to the following chemotaxis-consumption model in a bounded domain $Ω\subset \mathbb{R}^3$ during a time interval $(0,T)$: $$\partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v + f v 1_{Ω_c},$$ with $s \geq 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$, $u$ being the cell density, $v$ the chemical concentration and $f$ the bilinear control acting in a subdomain $Ω_c \subset Ω$. The existence of weak solutions $(u,v)$ to this model given $f \in L^q((0,T) \times Ω)$, for some $q > 5/2$, has been proved in [F. Guillén-González and A. L. Corrêa Vianna Filho, Optimal Control Related to Weak Solutions of a Chemotaxis-Consumption Model, arXiv:2211.14612, 2022]. In this paper, we study a related optimal control problem in the strong solution setting. First, imposing the regularity criterion $u ^s \in L^q((0,T) \times Ω)$ ($q > 5/2$) for a given weak solution, we prove existence and uniqueness of global-in-time strong solutions. Then, the existence of a global optimal solution can be deduced. Finally, using a Lagrange multipliers theorem, we establish first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers.

math.OC↗

arXiv2023

A structure-preserving upwind DG scheme for a degenerate phase-field tumor model

In this work, we present a modification of the phase-field tumor growth model given in [26] that leads to bounded, more physically meaningful, volume fraction variables. In addition, we develop an upwind discontinuous Galerkin (DG) scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model. Finally, some computational tests in accordance with the theoretical results are introduced. In the first test, we compare our DG scheme with the finite element (FE) scheme related to the same time approximation. The DG scheme shows a well-behavior even for strong cross-diffusion effects in contrast with FE where numerical spurious oscillations appear. Moreover, the second test exhibits the behavior of the tumor-growth model under different choices of parameters and also of mobility and proliferation functions.

math.NA↗

arXiv2023

An unconditionally energy stable and positive upwind DG scheme for the Keller-Segel model

The well-suited discretization of the Keller-Segel equations for chemotaxis has become a very challenging problem due to the convective nature inherent to them. This paper aims to introduce a new upwind, mass-conservative, positive and energy-dissipative discontinuous Galerkin scheme for the Keller-Segel model. This approach is based on the gradient-flow structure of the equations. In addition, we show some numerical experiments in accordance with the aforementioned properties of the discretization. The numerical results obtained emphasize the really good behaviour of the approximation in the case of chemotactic collapse, where very steep gradients appear.

math.NA↗

arXiv2023

Uniform in time solutions for a chemotaxis with potential consumption model

In this work we investigate the following chemo-attraction with consumption model in bounded domains of \, $\mathbb{R}^N$ ($N=1,2,3$): $$ \partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v $$ where $s\ge 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$. The main novelty in the model is the nonlinear potential consumption term $u^sv$. Through the convergence of solutions of an adequate truncated model, two main results are established; existence of uniform in time weak solutions in $3D$ domains, and uniqueness and regularity in $2D$ (or $1D$) domains. Both results are proved imposing minimal regularity assumptions on the boundary of the domain.

math.AP↗

arXiv2022

An upwind DG scheme preserving the maximum principle for the convective Cahn-Hilliard model

The design of numerical approximations of the Cahn-Hilliard model preserving the maximum principle is a challenging problem, even more if considering additional transport terms. In this work we present a new upwind Discontinuous Galerkin scheme for the convective Cahn-Hilliard model with degenerate mobility which preserves the maximum principle and prevents non-physical spurious oscillations. Furthermore, we show some numerical experiments in agreement with the previous theoretical results. Finally, numerical comparisons with other schemes found in the literature are also carried out.

math.NA↗

arXiv2020

Comparison of two finite element schemes for a chemo-repulsion system with quadratic production

In this paper we propose two fully discrete Finite Elements (FE) schemes for a repulsive chemotaxis model with quadratic production term. The first one (called scheme UV) corresponds to the backward Euler in time with FE in space approximation; while the second one (called scheme US$_\varepsilon$) is obtained as a modification of the scheme US proposed by [Guillén-González et al.], by applying a regularization procedure. We prove that the schemes UV and US$_\varepsilon$ have better properties than the FE scheme US. Specifically, we prove that, unlike the scheme US, the scheme UV is energy-stable in the primitive variables of the model, under a "compatibility" condition on the FE spaces. On the other hand, the scheme US$_\varepsilon$ is energy-stable with respect to the same modified energy of the scheme US, and an "approximated positivity" property holds (which is not possible to prove for the schemes US and UV). Additionally, we study the well-posedness of the schemes and the long time behaviour obtaining exponential convergence to constant states. Finally, we compare the numerical schemes throughout several numerical simulations.

math.DS↗

arXiv2020

On a bi-dimensional chemo-repulsion model with nonlinear production

In this paper, we study the following parabolic chemo-repulsion with nonlinear production model: $$ \left\{ \begin{array}{rcl} \partial_tu-Δu&=&\nabla\cdot(u\nabla v),\\ \partial_tv-Δv+v&=&u^p+fv\, 1_{Ω_c}. \end{array} \right. $$ This problem is related to a bilinear control problem, where the state $(u,v)$ is the cell density and the chemical concentration respectively, and the control $f$ acts in a bilinear form in the chemical equation. For $2D$ domains, we first consider the case of quadratic signal production ($p=2$), proving the existence and uniqueness of global strong state solution for each control, and the existence of global optimum solution. Afterwards, we deduce the optimality system for any local optimum via a Lagrange multiplier Theorem, proving regularity of the Lagrange multipliers. Finally, we consider the case of signal production $u^p$ with $1<p<2$.

math.AP↗

arXiv2018

A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem

In this paper we study a bilinear optimal control problem associated to a 3D chemo-repulsion model with linear production. We prove the existence of weak solutions and we establish a regularity criterion to get global in time strong solutions. As a consequence, we deduce the existence of a global optimal solution with bilinear control and, using a Lagrange multipliers theorem, we derive first-order optimality conditions for local optimal solutions.

math.OC↗

arXiv2018

Unconditionally energy stable fully discrete schemes for a chemo-repulsion model

This work is devoted to study unconditionally energy stable and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: Find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, such that $$ \left\{ \begin{array} [c]{lll} \partial_t u - Δu - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ Ω,\ t>0,\\ \partial_t v - Δv + v = u \ \ \mbox{in}\ Ω,\ t>0, \end{array} \right. $$ in a bounded domain $Ω\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we propose three fully discrete Finite Element (FE) approximations. The first one is a nonlinear approximation in the variables $(u,v)$; the second one is another nonlinear approximation obtained by introducing ${\boldsymbolσ}=\nabla v$ as an auxiliary variable; and the third one is a linear approximation constructed by mixing the regularization procedure with the energy quadratization technique, in which other auxiliary variables are introduced. In addition, we study the well-posedness of the numerical schemes, proving unconditional existence of solution, but conditional uniqueness (for the nonlinear schemes). Finally, we compare the behavior of such schemes throughout several numerical simulations and provide some conclusions.

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