Séminaire Equations aux dérivées partielles
organisé par l'équipe Modélisation et contrôle
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Roxana Sublet
Modelling collective cell dynamics
6 octobre 2026 - 14:00Salle de conférences IRMA
This work focuses on the mathematical modeling of cellular tissue dynamics. We first propose an individual-based model that provides the dynamics of the positions, velocities, and polarities of the cells, idealized as hard spheres. Cells interact with each other through contact forces, smooth attraction, and polarity alignment. The present work is an extension of the model proposed in [3] and validated by experiments on cellular rings. We will study the well-posedness of a regularized version. We are particularly interested in the congested regime and the impact of apoptosis on the jammed state. Indeed, cell apoptosis corresponds to programmed cell death: when cells leave the tissue, they induce local contractions but also enable cellular rearrangements. To this end, we add to the previous model a microscopic description of apoptotic and proliferation events. Numerical simulations are performed to show the impact of apoptosis on collective cell dynamics. Next, we derive a macroscopic description, following the methodology proposed in [1] and [2]. We start from a mean-field dynamics of the kinetic distribution function in phase space (position, polarity, radius), where contact forces have been replaced with repulsion forces. We then introduce a specific time and space rescaling and identify the equilibrium distribution functions, which are parameterized by two macroscopic quantities: the density and the mean polarity. Based on the Generalized Collision Invariant (GCI) method [2], we are then able to identify their dynamics: the resulting description can be seen as a modified Self-Organized Hydrodynamics (SOH) model. We finally discuss the obtained model and highlight the effect of the apoptotic events on the dynamics. This is a joint work with Laurent Navoret (Université de Strasbourg) and Marcela Szopos (Université Paris Cité). It has also been carried out in collaboration with Romain Levayer (Institut Pasteur) and Daniel Riveline (IGBMC, Université de Strasbourg) in the context of the ANR project MAPEFLU. References [1] Degond, P., Dimarco, G., Mac, T. B. N., and Wang, N. Macroscopic models of collective motion with repulsion. Communications in Mathematical Sciences, 13(6), 1615–1638 (2015). [2] Degond, P., and Motsch, S. Continuum limit of self-driven particles with orientation interaction. Mathematical Models and Methods in Applied Sciences, 18(supp01), 1193–1215 (2008). [3] Vecchio, S. L., Pertz, O., Szopos, M., Navoret, L., and Riveline, D. Spontaneous rotations in epithelia as an interplay between cell polarity and boundaries. Nature Physics (2024). -
Guillaume Delay
Un cadre général pour l'analyse numérique des inéquations variationnelles elliptiques de première espèce
13 octobre 2026 - 14:00Salle de conférences IRMA
Nous présentons dans ce travail un cadre unifié pour les inéquations variationnelles elliptiques de première espèce. Ce cadre regroupe plusieurs problèmes issus de la mécanique du contact. Il repose sur une formulation mixte faisant intervenir un multiplicateur de Lagrange. Nous proposons une discrétisation basée sur une méthode de Galerkine. Des discrétisations conformes et non conformes peuvent être considérées. Le résultat principal de ce travail établit la convergence de la méthode de discrétisation mixte sous certaines hypothèses. Ces hypothèses doivent ensuite être vérifiées pour chaque méthode de discrétisation que nous souhaitons étudier. Nous proposons d'appliquer ce cadre général à un problème de contact elliptique, discrétisé à l’aide d’une méthode d’éléments finis et d’une méthode hybride d’ordre élevé (HHO). Dans les deux cas, une discrétisation mixte P2 − P0 est considérée. Des estimations d’erreur a priori sur la solution numérique sont fournies pour les deux discrétisations. Ces méthodes de discrétisation constituent une variante des travaux précédents [1] et [2]. Dans [1], des fonctions « bulle » additionnelles sont introduites afin d’assurer la stabilité inf-sup du problème, tandis que nous démontrons ici que ces inconnues supplémentaires peuvent être supprimées sous certaines hypothèses sur le maillage. Dans [2], l’analyse porte uniquement sur la variable primale. Le multiplicateur de Lagrange est introduit à des fins de simulation. Dans notre travail, l’analyse numérique prend en compte le multiplicateur de Lagrange, qui est choisi dans un espace de dimension plus petite. Ceci est un travail en collaboration avec Jad Dabaghi. [1] T. Gustafsson, R. Stenberg, J. Videman. Mixed and stabilized finite element methods for the obstacle problem, SIAM J. Numer. Anal. 55 (2017), no. 6, 2718–2744. [2] M. Cicuttin, A. Ern, T. Gudi. Hybrid high-order methods for the elliptic obstacle problem, J. Sci. Comput. 83 (2020), paper no. 8, 18 -
Damiano Lombardi
Séminaire
3 novembre 2026 - 14:00Salle de conférences IRMA
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Vincent Boulard
Optimal geometric barriers for observability of heat semigroups on metric measure spaces
10 novembre 2026 - 14:00Salle de conférences IRMA
Abstract : Observability inequalities for the heat equation exhibit an exponential cost at small times. Is this exponential behavior an artefact of the usual Carleman and spectral methods, or is it forced by the underlying geometry? We show that it is genuinely geometric: any weighted observability inequality must satisfy an exponential decay rate controlled by the maximal distance to the observation set. As a consequence, we answer an open question raised by Ervedoza and Zuazua concerning geometric lower bounds on the exponential rate appearing in infinite-time integrated observability inequalities for heat equations. Our results are established in a general geometric metric-measure setting and apply, in particular, to Riemannian manifolds, Schrödinger operators, sub-Riemannian manifolds, and Laplacians on metric graphs. -
Luca Nenna
Title: Diffusion with sticky boundaries: optimal transport and gradient flows
17 novembre 2026 - 14:00Salle de conférences IRMA
Abstract: Diffusion processes with sticky boundaries describe particles that move inside a domain but may also spend a positive amount of time on its boundary, where they undergo a tangential diffusion. At the PDE level, this leads to a coupled system of bulk and boundary diffusion equations, with mass exchange between the two. In this talk, I will discuss the optimal transport geometry associated with these processes. Starting from their short-time large deviations, I will explain how the relative strength of bulk and boundary diffusion determines the effective transport cost. A sharp transition occurs when diffusion along the boundary becomes faster than in the interior: travelling along the boundary can then provide a shortcut, leading to a transport geometry that differs from the usual Euclidean one. I will then discuss the connection with entropy gradient flows and the variational formulation of the corresponding bulk–boundary diffusion equations, highlighting the difficulties caused by the coexistence of interior and boundary mass. The talk is based on joint work with Jean-Baptiste Casteras and Léonard Monsaingeon. -
Frédérique Lecourtier
Séminaire
24 novembre 2026 - 14:00Salle de conférences IRMA
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Daria Hrebenshchykova
TBA
1 décembre 2026 - 14:00A confirmer
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Claire Schnoebelen
TBD
8 décembre 2026 - 14:00Salle de conférences IRMA