arXiv subjects

Élder J. Villamizar-Roa

Publications and source records attributed to Élder J. Villamizar-Roa.

12 recordsLinked to original sources

arXiv2026

Modeling of an ODE-constrained optimization problem describing tumor dynamics, and numerical approximation via sequential physics-informed neural networks

In this paper, we study an optimal control problem related to an ODE model of glioblastoma growth influenced by the oxygen. The model considers a couple of controls describing the chemotherapy and antiangiogenic therapies. The cost functional aims to reduce the tumor growth, bring the oxygen concentration close to a desired value, and penalize the use of therapies. We solve the optimal control problem, proving the existence of a global optimal control and deriving first-order necessary optimality conditions through the Pontryagin Minimum Principle. For the numerical approximation, we use Physics-Informed Neural Networks (PINNs) to solve the state and adjoint systems, together with a gradient descent method with Armijo line search for the controls. To address the strategy of PINNs we consider the methodology proposed in [15], making a decomposition of the time domain into several subintervals, using different neural networks in each subinterval and enforcing continuity conditions between successive time subintervals. This strategy, called sequential PINN formulations in time, including soft and hard-constrained versions, is compared with the corresponding approximation results of classical solvers and traditional PINNs counterparts.

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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.

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arXiv2024

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller-Segel system

This paper is concerned with the fractional Keller-Segel system in the temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag-Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller-Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition of the initial data.

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arXiv2021

Existence theory of the nonlinear plate equations

This paper is devoted to the theoretical analysis of the nonlinear plate equations in $\mathbb{R}^{n}\times (0,\infty),$ $n\geq1,$ with nonlinearity involving a type polynomial behavior. We prove the existence and uniqueness of global mild solutions for small initial data in $L^{1}(\mathbb{R}^{n})\cap H^s(\mathbb{R}^{n})$-spaces. We also prove the existence and uniqueness of local and global solutions in the framework of Bessel-potential spaces $H^s_p(\mathbb{R}^n)=(I-Δ)^{s/2}L^p(\mathbb{R}^n).$ In order to derive the existence results we develop new time decay estimates of the solution of the corresponding linear problem.

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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 $1<p<2$.

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arXiv2019

Numerical analysis for a chemotaxis-Navier-Stokes system

In this paper we develop a numerical scheme for approximating a $d$-dimensional chemotaxis-Navier-Stokes system, $d=2,3$, modeling cellular swimming in incompressible fluids. This model describes the chemotaxis-fluid interaction in cases where the chemical signal is consumed with a rate proportional to the amount of organisms. We construct numerical approximations based on the Finite Element method and analyze some error estimates and convergence towards weak solutions. In order to construct the numerical scheme, we use a splitting technique to deal with the chemo-attraction term in the cell-density equation, leading to introduce a new variable given by the gradient of the chemical concentration. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed and it is mass-conservative. We obtain uniform estimates and analyze the convergence of the scheme. Finally, we present some numerical simulations to verify the good behavior of our scheme, as well as to check numerically the error estimates proved in our theoretical analysis.

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arXiv2019

On the management fourth-order Schrödinger-Hartree equation

We consider the Cauchy problem associated to the fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in $H^s$-spaces. We also analyze the scaling limit of fast dispersion management and the convergence to a model with averaged dispersions.

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arXiv2019

An optimal control problem for the Navier-Stokes-$α$ system

In this paper we study a distributed optimal control problem for a three-dimensional Navier-Stokes-$α$ model. We prove the solvability of the optimal control problem, and derive first-order optimality conditions by using a Lagrange multipliers Theorem. Finally, considering a velocity tracking control problem for the three-dimensional Navier-Stokes-$α$ model, we analyze the relation of its optimality system to the corresponding one associated to the Navier-Stokes model by proving a convergence theorem, which establishes that, as the length scale $α$ goes to zero, the optimality system of the three-dimensional Navier-Stokes-$α$ model converges to the optimality system associated with the velocity tracking control problem of the Navier-Stokes equations.

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arXiv2018

Time-decay and Strichartz estimates for the Benjamin-Bona-Mahony equation and existence of solutions on modulation spaces

In this paper we derive time-decay and Strichartz estimates for the generalized Benjamin-Bona-Mahony equation on the framework of modulation spaces $M^s_{p,q}.$ We use this results to analyze the existence of local and global solutions of the corresponding Cauchy problem with rough data in modulation spaces. The results improve known results in Sobolev spaces in some sense.

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arXiv2018

Existence theory for the Boussinesq equation in Modulation spaces

In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces $M^{s}_{p^\prime,q}(\mathbb{R}^n),$ $n\geq 1.$ After a decomposition of the Boussinesq equation in a $2\times 2$-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in $ M^s_{p,q}\times D^{-1}JM^s_{p,q}$ spaces for suitable $p,q$ and $s,$ including the special case $p=2,q=1$ and $s=0.$ Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.

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arXiv2017

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of RN , N = 2, 3, for small initial data in Lp-spaces.

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arXiv2017

An Optimal Control Problem for the Steady Nonhomogeneous Asymmetric Fluids

We study an optimal boundary control problem for the two-dimensional stationary micropolar fluids system with variable density. We control the system by considering boundary controls, for the velocity vector and angular velocity of rotation of particles, on parts of the boundary of the flow domain. On the remaining part of the boundary, we consider mixed boundary conditions for the vector velocity (Dirichlet and Navier conditions) and Dirichlet boundary conditions for the angular velocity. We analyze the existence of a weak solution obtaining the fluid density as a scalar function of the stream function. We prove the existence of an optimal solution and, by using the Lagrange multipliers theorem, we state first-order optimality conditions. We also derive, through a penalty method, some optimality conditions satisfied by the optimal controls.

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