arXiv subjects

João Paulo Dias

Publications and source records attributed to João Paulo Dias.

6 recordsLinked to original sources

arXiv2026

Existence of Complementary and Variational Weak Solutions to Obstacle Problems for a Quasilinear Wave Equation

We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two obstacles problem. In contrast with the linear setting, for both strictly quasilinear cases we obtain continuous solutions in a weak complementary sense, which moreover satisfy a weak entropy condition in the free region where the string is not in contact with the obstacles. We further show that, in both the one and two obstacle cases, these solutions are variational solutions in a hyperbolic sense without the viscosity term.

math.AP↗

arXiv2024

Parametric probabilistic approach for cumulative fatigue damage using double linear damage rule considering limited data

This work proposes a parametric probabilistic approach to model damage accumulation using the double linear damage rule (DLDR) considering the existence of limited experimental fatigue data. A probabilistic version of DLDR is developed in which the joint distribution of the knee-point coordinates is obtained as a function of the joint distribution of the DLDR model input parameters. Considering information extracted from experiments containing a limited number of data points, an uncertainty quantification framework based on the Maximum Entropy Principle and Monte Carlo simulations is proposed to determine the distribution of fatigue life. The proposed approach is validated using fatigue life experiments available in the literature.

cs.CE↗

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↗

arXiv2023

On short wave-long wave interactions\\ in the relativistic context: Application to the Relativistic Euler Equations

In this paper we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless $1+3$-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the $1+3$-dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. This is done by means of the introduction of natural auxiliary dependent variables, rendering the discussion totally similar to the non-relativistic case. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T.~Kato (1975) on quasi-linear symmetric hyperbolic systems.

math.AP↗

arXiv2022

On short wave-long wave interactions in the relativistic context

In this paper we introduce models of short wave-long wave interactions in the relativistic setting. In this context the nonlinear Schrödinger equation is no longer adequate for describing short waves and is replaced by a nonlinear Dirac equation. Two specific examples are considered: the case where the long waves are governed by a scalar conservation law; and the case where the long waves are governed by the augmented Born-Infeld equations in electromagnetism.

math.AP↗

arXiv2013

Finite-time blowup for a complex Ginzburg-Landau equation with linear driving

In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $α>0$, $γ\in \R$ and $-π/2<θ<π/2$. By convexity arguments we prove that, under certain conditions on $α,θ,γ$, a class of solutions with negative initial energy blows up in finite time.

math.AP↗