JUNCA , Stéphane

Les exemples classiques de convergence quadratique des moyennes arithmetico-géométrique, harmonico-quadratique, etc ... ne correspondent pas à des points fixes super atractifs comme dans la méthode de Newton mais à une méthode de point fixe dans un cas non contractant! Une explication élémentaire de ce phénomène est donnée.

Le Elo est une méthode de classement inventée pour les tournois d'échecs puis popularisée pour d'autres sports, les joueurs sur internet, le classement de données … Peu d'études mathématiqueson été faites sur le Elo. On commencera à parler du modèle initial au niveau microscopique des joueurs comme une chaîne de Markov puis au niveau...

For hyperbolic systems of one-dimensional conservation laws, the theory of existence of global weak enropy solutions for the initial value problem is generally performed for small initial data in $BV$ or, rarely, in $L^\infty$. In this presentation, the intermediate spaces $BV^s$, 0 < s < 1, are used, $BV=BV^1$ and $L^\infty=BV^0$. The...

The manuscript presents my research on hyperbolic Partial Differential Equations (PDE), especially on conservation laws. My works began with this thought in my mind: ''Existence and uniqueness of solutions is not the end but merely the beginning of a theory of differential equations. The really interesting questions concern the behavior of...

Soient F une famille de suites réelles qui divergent vers +∞, et f une fonction de R dans R. Si f converge en +∞, alors, pour toute suite (x n) n ∈ F, (f (x n)) n converge. La réciproque est généralement fausse sauf si f et F vérifient certaines conditions. Cet article propose des réciproques dans l'esprit du lemme de Croft ainsi que des...

We prove a general result of two‐scale convergence for BV‐functions. We then use this result to show the validity of nonlinear geometric optics for weak solutions to scalar conservation laws when the small perturbation of the initial datum is only BV and periodic with respect to the slow and the fast variable.

The article first studies the propagation of well prepared high frequency waves with small amplitude $\varepsilon$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor (1994), about the maximal...

A popular rating system to measure player strength by pairwise comparison is the Elo's system. This system is widely used to rank players in chess, go and online games. Assuming the independence of the outcomes, it is a Markov chain. The convergence of Elo ratings is studied when the update after each round-robin tournament is a contraction. An...

This talk deals with the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no additional regularity is assumed on the flux. Thus, the well-known $BV$ smoothing effect for $C^2$ uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik is...

Etude de la méthode des moindres carrés pour certaines séries statistiques représentant des perturbations déterministes ou aléatoires de modèle linéaire. La précision de la méthode s'avère étonnante même pour de grandes perturbations.

Quelques exemples et réflexions sur la representation et l'utilisation des nombres et la distance logarithmique

We consider the system of isothermal Euler Equations. For BV solutions, we show that the density converges to the solution to the heat equation when the friction coefficient ε −1 tends to infinity. The same type of results has been obtained in [4]-with totally different methods-for smooth solution to the p-system.

We study the propagation of high frequency oscillations for one dimensional semi-linear hyperbolic systems with small parabolic perturbations. We obtain a new degenerate parabolic system for the profile, and valid an asymptotic development in the spirit of Joly, Métivier and Rauch.

We prove the propagation of oscillations with an asymptotic development for an oscillating initial boundary value problem of semilinear hyperbolic systems in the spirit of J.L.Joly, G.Métivier and J.Rauch. In particular we simplify the Joly-métivier-Rauch's proof for the Cauchy problem. Then, we show the new phenomenon of localised oscillations.

Les notions de pourcentages et d'erreurs relatives induisent naturellement sur $]0, +∞[$ les métriques relatives. Nous présentons quelques propriétés de ces métriques et le rôle central de la distance logarithmique.

An introduction to fractional BV space is provided to obtain weak-entropy solutions of scalar conservation laws. Then, this theory is applied to an example of a $2\times 2$ system in crystallography.

Propagation of elastic waves is studied in a 1D medium containing N cracks modeled by nonlinear jump conditions. The case N = 1 is fully understood. When N > 1, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case N = 2, some mathematical results...

We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We characterize...

The interactions between linear elastic waves and a nonlinear crack with finite compressibility are studied in the one-dimensional context. Numerical studies on a hyperbolic model of contact with sinusoidal forcing have shown that the mean values of the scattered elastic displacements are discontinuous across the crack. The mean dilatation of...

The class of $2 \times 2$ nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional $BV$ spaces $BV^s$ is proved. The exponent $s$ is related to the usual fractional Sobolev derivative. Riemann invariants $w$ and...

International audience

In 1994, Lions, Perthame and Tadmor conjectured an optimal smoothing effect for entropy solutions of multidimensional scalar conservation laws. This effect estimated in fractional Sobolev spaces is linked to the flux nonlinearity. In order to show that the conjectured smoothing effect cannot be exceeded, we use a new definition of a nonlinear...

International audience