arXiv subjects

Robert Roussarie

Publications and source records attributed to Robert Roussarie.

4 recordsLinked to original sources

arXiv2026

Existence and uniqueness of traveling fronts for a free interface model of autoignition in reactive jets

In this paper we consider a one-dimensional reaction-diffusion model with piecewise continuous reaction term that describes propagation of autoignition fronts in reactive co-flow jets in a certain parametric regime. The model is reduced to a free boundary problem with two interfaces. It is shown that this problem admits permanent traveling front solution which is unique up to translations. The result is obtained using dynamical system approach employing Stable Manifold Theorem and the Melnikov integral as the main tools.

math.AP↗

arXiv2026

Slow-fast system in Rosales-Majda combustion model with fractional order kinetics

We consider traveling wave solutions of a one-dimensional model for detonation waves derived by Rosales and Majda, when the reaction order $α$ belongs to $[0,1)$. The chemical kinetics is a simplified Arrhenius law or a Heaviside function. The model in the reaction zone is a slow-fast dynamical system for a vector representing temperature and mass fraction, which depends on the velocity $c$ and small viscosity $β$. Our goal in this paper is to study the bifurcation diagram in the $(β,c)$ parameter space and identify the nature of the trajectories corresponding to viscous shock waves. The demonstrations are based on a variety of techniques including the Poincaré-Bendixson theorem and the Fenichel theory. Theoretical results are confirmed by numerical computations.

math.AP↗

arXiv2020

Existence of a traveling wave solution in a free interface problem with fractional order kinetics

In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 < $α$ < 1. We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-C 1 vector field in dimension 2. The latter is treated by qualitative topo-logical methods based on the Poincar{é}-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the origin is proved. A significant result is that the settling time to reach the origin is finite, meaning that the trailing interface is finite in contrast to the case $α$ = 1, but in accordance with $α$ = 0. Finally, the integro-differential system is solved via a fixed-point method.

math.AP↗

arXiv2015

Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems

In this paper we introduce new methods to prove the finite cyclicity of some graphics through a triple nilpotent point of saddle or elliptic type surrounding a center. After applying a blow-up of the family, yielding a singular 3-dimensional foliation, this amounts to proving the finite cyclicity of a family of limit periodic sets of the foliation. The boundary limit periodic sets of these families were the most challenging, but the new methods are quite general for treating such graphics. We apply these techniques to prove the finite cyclicity of the graphic $(I_{14}^1)$, which is part of the program started in 1994 by Dumortier, Roussarie and Rousseau (and called DRR program) to show that there exists a uniform upper bound for the number of limit cycles of a planar quadratic vector field. We also prove the finite cyclicity of the boundary limit periodic sets in all graphics but one through a triple nilpotent point at infinity of saddle, elliptic or degenerate type (with a line of zeros) and surrounding a center, namely the graphics $(I_{6b}^1)$, $(H_{13}^3)$, and $(DI_{2b})$.

math.DS↗