Perturbations et singularités

On considère un type de problèmes dépendant d'un petit paramètre eps qui sont bien posés pour eps>0 mais dont la limite eps=0 est mal posé (les conditions aux limites sur une partie du bord ne sont pas adaptées aux équations). Les solutions deviennent de plus en plus singulières lorsque eps décroît (phénomène de complexification, qui fait intervenir le nouveau paramètre émergeant log eps). Application aux coques minces et autres perspectives

Rigorous and heuristic treatment of sensitive singular perturbations arising in elliptic shells

We consider singular perturbations of elliptic systems depending on a parameter ε such that, for ε = 0 the boundary conditions are not adapted to the equation (they do not satisfy the Shapiro - Lopatinskii condition). The limit holds only in very abstract spaces out of distribu- tion theory involving complexification and non-local phenomena. This system appears in the thin shell theory when the middle surface is el- liptic and the shell is fixed on a part of the boundary and free on the rest. We use a heuristic reasoning applying some simplifications which allow to reduce the original problem in a domain to another problem on its boundary. The novelty of this work is that we consider systems of partial differential equations while in our previous work we were dealing with single equations

Etude théorique et numérique des singularités pour les coques elliptiques

Cette communication est consacrée à l'étude des singularités dans le cas des coques elliptiques. Dans le cas de coques bien inhibées, nous mettrons ensuite en évidence deux types de singularités dont l'une n'est pas classique : il s'agit d'une singularité logarithmique qui apparaît lorsque le domaine de chargement présente des coins. Dans le cas des coques elliptiques mal-inhibées (présentant une partie du bord libre), le problème limite ne satisfait plus la condition de Shapiro-Lopatinskii. Dans ce cas, le problème limite est mal posé et ses solutions ne sont plus contenues dans l'espace des distributions. On observe alors au cours du processus asymptotique un phénomène de complexification : les déplacements au voisinage du bord libre deviennent de plus en plus amples et oscillants. L'étude théorique sera complétée par des simulations numériques utilisant un logiciel éléments finis MODULEF couplé avec un logiciel de maillage adaptatif anisotrope BAMG. Cette technique permet d'approcher précisément les singularités prédites par la théorie avec un nombre réduit d'éléments

Autour de l’évolution biologique. Réflexions d’un physicien

Ce texte est la version écrite d’un exposé devant la Société de Biologie le 17 février 2016. Il contient des réflexions d’un scientifique non biologiste sur la problématique de l’évolution biologique, le type de causalité qu’elle met en œuvre et les idées qu’elle suscite, notamment sur le caractère constructif et structurant de phénomènes tels que la prédation, le rôle de la stabilité et des attracteurs. Cela conduit à une réflexion plus vaste sur la dialectique, cadre général des phénomènes évolutifs, qui dépasse la logique formelle de l’instantanéité

Remarks and examples on transient processes and attractors in biological evolution

We present a model for the competition of two biological entities into the same species (polyphasie, clonal/sex, cancerous cells), the first one with a birth ratio higher than the second when the resources are abundant, whereas the situation is reversed for scarce resources. The first one rapidly exhausts the resources, improving growth of the second, leading to a auto-sustained cyclic process (ESS = Evolutionary Stable Strategy). We use known models of population dynamics for three agents: two phases asexual and sexual (for instance) of the same species and one of resources. The main feature of the model (for certain values of the parameters) is the very long and entangled transient process, which involves a long period where one of the forms is practically absent, before emerging again to join a stable cycle which implies preservation of both forms. This model should throw some light on the biological problem of the maintenance of sexuality in competition with asexual clones, as well as on the alternated fast growth versus latency in cancer tumors

Various kinds of sensitive singular perturbations

We consider variational problems of P. D. E. depending on a small parameter ϵ when the limit process ϵ↓0 implies vanishing of the higher order terms. The perturbation problem is said to be sensitive when the energy space of the limit problem is out of the distribution space, so that the limit problem is out of classical theory of P. D. E. We present here a review of the subject, including abstract convergence theorems and two very different model problems (the second one is presented for the first time). For each one we prove the sensitive character and we give a formal asymptotics for the behavior ϵ↓0

New schemes of dynamic preservation of diversity. Remarks on stability and topology

International audienceWe address the biological dynamics problem of the persistence of several species in conditions of non-existence of an equilibrium, including an example of stabilization by predation and the very controversial "competitive exclusion", (which depends on the precise definition of persistence). We give normal forms for various examples of such (essentially dynamical) persistence and comments on the involved topology, which implies the presence of exceptional heteroclinic connections binding equilibria on the boundary

On internal and boundary layers with unbounded energy in thin shell theory. Parabolic characteristic and noncharacteristic cases

International audienceWe consider the system of equations of Koiter shell theory - in a slightly simplified form - in the case when the limit problem for small thickness is parabolic, i.e., when the directions of the principal curvatures of the middle surface coincide everywhere. Under loadings that do not belong to the dual of the limit energy space, the solution energy grows without limit as the thickness tends to zero and concentrates on internal or boundary layers. We consider both the cases when the singular loadings are applied along a non-characteristic curve or along a characteristic curve. We prove convergence in the layers