arXiv subjects

Paulo Amorim

Publications and source records attributed to Paulo Amorim.

At least 19 recordsLinked to original sources

arXiv2026

A structured model of vector-borne disease with within-host viral load and antibody dynamics

We present an epidemiological model for vector-borne diseases that includes within-host viral load and antibody dynamics using structured transport equations. By incorporating the internal dynamics into the infected and recovered host compartments, the formulation introduces nonlinearities and nonlocalities. We establish analytical properties, including well-posedness and mass conservation, and characterize its characteristic curves. Furthermore, we derive a simplified Uniform Host Response (UHR) model featuring delay-type terms. For both the full and UHR frameworks, the basic reproduction number is determined and shown to serve as a threshold for the existence of an endemic equilibrium, and is related to the linear stability of the disease-free state. Finally, numerical experiments, parameterized specifically for Dengue fever, demonstrate how within-host mechanisms influence population-level epidemiological outcomes.

math.AP↗

arXiv2026

The obstacle problem for scalar conservation laws with nonlocal dynamics

In this article, we present a method to find a solution to a one-dimensional nonlocal conservation law that respects a space-dependent mapping, referred to as the obstacle. This is achieved by generalizing existing results for the local conservation law: We consider a relaxation of the velocity, that explicitly depends on the obstacle. We prove existence of solutions to the relaxed problem and show that, as the relaxation mapping converges to a Heaviside-type function, the corresponding solutions converge to a weak solution of a discontinuous nonlocal conservation law. Moreover, we can characterize the limiting flux in several cases.\\ The paper concludes with a numerical study that illustrates the aforementioned convergence.

math.AP↗

arXiv2024

The obstacle problem for linear scalar conservation laws with constant velocity

In this contribution, we present a novel approach for solving the obstacle problem for (linear) conservation laws. Usually, given a conservation law with an initial datum, the solution is uniquely determined. How to incorporate obstacles, i.e., inequality constraints on the solution so that the resulting solution is still "physically reasonable" and obeys the obstacle, is unclear. The proposed approach involves scaling down the velocity of the conservation law when the solution approaches the obstacle. We demonstrate that this leads to a reasonable solution and show that, when scaling down is performed in a discontinuous fashion, we still obtain a suitable velocity - and the solution satisfying a discontinuous conservation law. We illustrate the developed solution concept using numerical approximations.

math.AP↗

arXiv2023

On the existence and partial stability of standing waves for a nematic liquid crystal director field equation

In this paper, following the studies of Amorim and al. in Partial Differ.Equ. Appl. '23, we consider some new aspects of the motion of the director field of a nematic liquid crystal submitted to a magnetic field and to a laser beam. In particular, we study the existence and partial orbital stability of special standing waves, in the spirit of Cazenave and Lions and Hadj Selem and al. in Milan J. Math. '14 and we present some numerical simulations.

math.AP↗

arXiv2015

The obstacle-mass constraint problem for hyperbolic conservation laws. Solvability

In this work we introduce the obstacle-mass constraint problem for a multidimensional scalar hyperbolic conservation law. We prove existence of an entropy solution to this problem by a penalization/viscosity method. The mass constraint introduces a nonlocal Lagrange multiplier in the penalized equation, giving rise to a nonlocal parabolic problem. We determine conditions on the initial data and on the obstacle function which ensure global in time existence of solution. These are not smoothness conditions, but relate to the propagation of the support of the initial data.

math.AP↗

arXiv2015

Modeling ant foraging: a chemotaxis approach with pheromones and trail formation

We consider a continuous mathematical description of a population of ants and simulate numerically their foraging behavior using a system of partial differential equations of chemotaxis type. We show that this system accurately reproduces observed foraging behavior, especially spontaneous trail formation and efficient removal of food sources. We show through numerical experiments that trail formation is correlated with efficient food removal. Our results illustrate the emergence of trail formation from simple modeling principles.

nlin.AO↗

arXiv2015

Analysis of a chemotaxis system modeling ant foraging

In this paper we analyze a system of PDEs recently introduced in [P. Amorim, {\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail formation}], in order to describe the dynamics of ant foraging. The system is made of convection-diffusion-reaction equations, and the coupling is driven by chemotaxis mechanisms. We establish the well-posedness for the model, and investigate the regularity issue for a large class of integrable data. Our main focus is on the (physically relevant) two-dimensional case with boundary conditions, where we prove that the solutions remain bounded for all times. The proof involves a series of fine \emph{a priori} estimates in Lebesgue spaces.

math.AP↗

arXiv2014

A continuous model of ant foraging with pheromones and trail formation

We propose and numerically analyze a PDE model of ant foraging behavior. Ant foraging is a prime example of individuals following simple behavioral rules based on local information producing complex, organized and ``intelligent'' strategies at the population level. One of its main aspects is the widespread use of pheromones, which are chemical compounds laid by the ants used to attract other ants to a food source. In this work, we consider a continuous description of a population of ants and simulate numerically the foraging behavior using a system of PDEs of chemotaxis type. We show that, numerically, this system accurately reproduces observed foraging behavior, such as trail formation and efficient removal of food sources.

math.AP↗

arXiv2013

A Numerical Approach to Scalar Nonlocal Conservation Laws

We address the study of a class of 1D nonlocal conservation laws from a numerical point of view. First, we present an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are lead to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.

math.NA↗

arXiv2012

The linear stability of shock waves for the nonlinear Schr\"odinger-Inviscid Burgers system

We investigate the coupling between the nonlinear Schr\"odinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schr\"odinger--Burgers system, when the reference solution is an entropy--satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.

math.AP↗

arXiv2012

Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.

math.NA↗

arXiv2012

Convergence of a numerical scheme for a coupled Schr\"odinger--KdV system

We prove the convergence in a strong norm of a finite difference semi-discrete scheme approximating a coupled Schr\"odinger--KdV system on a bounded domain. This system models the interaction of short and long waves. Since the energy estimates available in the continuous case do not carry over to the discrete setting, we rely on a suitably truncated problem which we prove reduces to the original one. We present some numerical examples to illustrate our convergence result.

math.NA↗

arXiv2012

A nonlinear model describing a short wave long wave interaction in a viscoelastic medium

In this paper we introduce a system coupling a nonlinear Schrödinger equation with a system of viscoelasticity, modeling the interaction between short and long waves, acting for instance on media like plasmas or polymers. We prove the existence and uniqueness of local (in time) strong solutions and the existence of global weak solutions for the corresponding Cauchy problem. In particular we extend previous results in [Nohel \emph{et. al.}, Commun. Part. Diff. Eq., 13 (1988)] for the quasilinear system of viscoelasticity. We finish with some numerical computations to illustrate our results.

math.AP↗

arXiv2012

On a nonlocal hyperbolic conservation law arising from a gradient constraint problem

In some models involving nonlinear conservation laws, physical mechanisms exist which prevent the formation of shocks. This gives rise to conservation laws with a constraint on the gradient of the solution. We approach this problem by studying a related conservation law with a spatial nonlocal term. We prove existence, uniqueness and stability of solution of the Cauchy problem for this nonlocal conservation law. In turn, this allows us to provide a notion of solution to the conservation law with a gradient constraint. The proof of existence is based on a time-stepping technique, and an $L^1$-contraction estimate follows from stability results of Karlsen and Risebro.

math.AP↗

arXiv2010

A geometric approach to error estimates for conservation laws posed on a spacetime

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.

math.AP↗

arXiv2009

Convergence of numerical schemes for short wave long wave interaction equations

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schr\"odinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.

math.NA↗

arXiv2008

Computing Gowdy spacetimes via spectral evolution in future and past directions

We consider a system of nonlinear wave equations with constraints that arises from the Einstein equations of general relativity and describes the geometry of the so-called Gowdy symmetric spacetimes on T3. We introduce two numerical methods, which are based on pseudo-spectral approximation. The first approach relies on marching in the future time-like direction and toward the coordinate singularity t=0. The second approach is designed from asymptotic formulas that are available near this singularity; it evolves the solutions in the past timelike direction from "final" data given at t=0. This backward method relies a novel nonlinear transformation, which allows us to reduce the nonlinear source terms to simple quadratic products of the unknown variables. Numerical experiments are presented in various regimes, including cases where "spiky" structures are observed as the coordinate singularity is approached. The proposed backward strategy leads to a robust numerical method which allows us to accurately simulate the long-time behavior of a large class of Gowdy spacetimes.

gr-qc↗

arXiv2007

Finite volume schemes on Lorentzian manifolds

We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry accurately and generalize techniques and estimates that were known in the (flat) Euclidian setting, only. The strong convergence of the scheme then is then a consequence of the well-posed theory recently developed by Ben-Artzi and LeFloch for conservation laws on manifolds.

math.NA↗