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1. Stability of optimal traffic plans in the irrigation problem

Author

Antoine Prouff, Antonio De Rosa, Andrea Marchese, Maria Colombo, Paul Pegon, Ecole Polytechnique Fédérale de Lausanne (EPFL), Courant Institute of Mathematical Sciences [New York] (CIMS), New York University [New York] (NYU), NYU System (NYU)-NYU System (NYU), Department of mathematics/Dipartimento di Matematica [Univ. Trento], Università degli Studi di Trento (UNITN), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), Maria Colombo was partially supported by the Swiss National Science Foundation grant 200021_182565. Antonio De Rosa has been supported by the NSF DMS Grant No. 1906451. Andrea Marchese acknowledges partial support from GNAMPA-INdAM., Department of Mathematics [College Park], University of Maryland [College Park], University of Maryland System-University of Maryland System, Bocconi Institute for Data Science and Analytics (BIDSA), Bocconi University [Milan, Italy], Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, M. C. was partially supported by the Swiss National Science Foundation grant 200021_182565. A. D. R. has been supported by the NSF DMS Grant No. 1906451 and the NSF DMS Grant No. 2112311. A. M. acknowledges partial support from GNAMPA-INdAM. A.P. was supported by the Fondation Mathématiques Jacques Hadamard. P.P and A.P. both acknowledge EPFL for hosting them during the semester this paper was prepared., Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)

Subjects
BRANCHED TRANSPORT, Mathematical optimization, Irrigation, TRAFFIC PLANS, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], 01 natural sciences, Stability (probability), symbols.namesake, Mathematics - Analysis of PDEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Computer Science::Networking and Internet Architecture, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, Limit (mathematics), 0101 mathematics, Mathematics - Optimization and Control, Mathematics, TRANSPORTATION NETWORK, BRANCHED TRANSPORT, IRRIGATION PROBLEM, TRAFFIC PLANS, STABILITY, STABILITY, IRRIGATION PROBLEM, Applied Mathematics, TRANSPORTATION NETWORK, 010102 general mathematics, Eulerian path, Flow network, 010101 applied mathematics, Optimization and Control (math.OC), Mathematics - Classical Analysis and ODEs, symbols, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis, Lagrangian, Analysis of PDEs (math.AP)
Abstract

We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [7], extending it to the Lagrangian framework.

Published
2022

2. On existence, stability and many-particle approximation of solutions of 1D Hughes model with linear costs

Author

Andreianov, Boris, Rosini, Massimiliano, Stivaletta, Graziano, Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Peoples Friendship University of Russia [RUDN University] (RUDN), Department of Mathematics and Computer Science, Università degli Studi di Ferrara (UniFE), Uniwersytet Marii Curie-Sklodowskiej = University Marii Curie-Sklodowskiej [Lublin] (UMCS), Department of Information Engineering, Computer Science and Mathematics, Università degli Studi dell'Aquila (UNIVAQ), MDR acknowledges the support of the National Science Centre, Poland, Project 'Mathematics of multi-scale approaches in life and social sciences' No. 2017/25/B/ST1/00051, by the INdAM-GNAMPA Project 2020 'Dalla Buona Posizione alla Teoria dei Giochi nelle Leggi di Conservazione' and by University of Ferrara, FIR Project 2019 'Leggi di conservazione di tipo iperbolico: teoria ed applicazioni', This paper has been supported by the RUDN University Strategic Academic Leadership Program., Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), Dipartimento di Matematica e Informatica = Department of Mathematics and Computer Science [Ferrara] (DMCS), and Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila] (DISIM)

Subjects
MSC 2010: 58J45, 35L65, 35R05, 90B20, Mathematics - Analysis of PDEs, "Hydrodynamic" limit, Hughes model, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Existence, Pedestrian flow, Conservation law, Stability, Many-particle approximation, Moving interface, Analysis of PDEs (math.AP)
Abstract

This paper deals with the one-dimensional formulation of Hughes model for pedestrian flows in the setting of entropy solutions, which authorizes non-classical shocks at the location of the so-called turning curve. We consider linear cost functions, whose slopes $\alpha$ 0 correspond to different crowd behaviours. We prove existence and partial well-posedness results in the framework of entropy solutions. The proofs of existence are based on a a sharply formulated many-particle approximation scheme with careful treatment of interactions of particles with the turning curve, and on local reductions to the well-known Lighthill-Whitham-Richards model. For the special case of BV-regular entropy solutions without non-classical shocks, locally Lipschitz continuous dependence of such solutions on the initial datum $\rho$ and on the cost parameter $\alpha$ is proved. Differently from the stability argument and from existence results available in the literature, our existence result allows for the possible presence of non-classical shocks. First, we explore convergence of the many-particle approximations under the assumption of uniform space variation control. Next, by a local compactness argument that permits to circumvent the possible absence of global BV bounds, we obtain existence of solutions for general measurable data. Finally, we illustrate numerically that the model is able to reproduce typical behaviours in case of evacuation. Special attention is devoted to the impact of the parameter $\alpha$ on the evacuation time.

Published
2021

3. Some analytical solutions for validation of free surface flow computational codes

Author

Bernard Di Martino, Fabien Souillé, Jacques Sainte-Marie, Marie-Odile Bristeau, Anne Mangeney, Numerical Analysis, Geophysics and Ecology (ANGE), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut de Physique du Globe de Paris (IPGP (UMR_7154)), Institut national des sciences de l'Univers (INSU - CNRS)-Université de La Réunion (UR)-Institut de Physique du Globe de Paris (IPG Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), The work presented in this paper was supported in part by the ANR Mimosa (ANR-14-CE01-0012) and ERC contract ERC-CG-2013-PE10-617472 SLIDEQUAKES. The authors also acknowledge the Inria Project Lab 'Algae in Silico' for its financial support., ANR-14-CE01-0012,MIMOSA,Analyse, modélisation et utilisation du bruit sismique large bande: des microséismes aux oscillations libres de la terre(2014), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Institut de Physique du Globe de Paris (IPGP), Institut national des sciences de l'Univers (INSU - CNRS)-IPG PARIS-Université de La Réunion (UR)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de La Réunion (UR)-Université Paris Diderot - Paris 7 (UPD7)-IPG PARIS-Institut national des sciences de l'Univers (INSU - CNRS)

Subjects
010103 numerical & computational mathematics, Analytical solutions, 01 natural sciences, law.invention, Physics::Fluid Dynamics, Free surface Euler equations, symbols.namesake, law, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [MATH]Mathematics [math], Mathematics, Computer simulation, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Mechanical Engineering, Mechanics, Euler system, Condensed Matter Physics, River dynamics, Euler equations, 010101 applied mathematics, Flow (mathematics), Mechanics of Materials, Free surface, symbols, Compressibility, Hydrostatic equilibrium
Abstract

We present several time dependent analytical solutions for the incompressible Euler system with free surface. These analytical solutions give quantitative descriptions of some physical phenomena and can be used as reference solutions when validating numerical simulation codes. They concern fluid flows governed by Euler equations with or withouthydrostatic hypothesis including wet/dry interface, variable density, wide variety of boundary conditions and possibly non-uniform bottom topographies. Some of these analytical solutions concern shallow water type models or free surface Navier-Stokes model i.e. with a viscous fluid.

Published
2021

4. Some quasi-analytical solutions for propagative waves in free surface Euler equations

Author

Bristeau, Marie-Odile, Di Martino, Bernard, Mangeney, Anne, Sainte-Marie, Jacques, Souillé, Fabien, Sorbonne Université (SU), Numerical Analysis, Geophysics and Ecology (ANGE), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sciences pour l'environnement (SPE), Université Pascal Paoli (UPP)-Centre National de la Recherche Scientifique (CNRS), Institut de Physique du Globe de Paris, EDF (EDF), The work presented in this paper was supported in part by the ANR Mimosa (ANR-14-CE01-0012) andERC contract ERC-CG-2013-PE10-617472 SLIDEQUAKES., ANR-14-CE01-0012,MIMOSA,Analyse, modélisation et utilisation du bruit sismique large bande: des microséismes aux oscillations libres de la terre(2014), and Centre National de la Recherche Scientifique (CNRS)-Université Pascal Paoli (UPP)

Subjects
analytical solution, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Free Surface Euler Equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Airy wave
Abstract

International audience; This note describes some quasi-analytical solutions for wave propagation in free surface Euler equations and linearized Euler equations. The obtained solutions vary from a sinusoidal form to a form with singularities. They allow a numerical validation of the free-surface Euler codes.; Cette note décrit des solutions quasi analytiques correspondant à la propagation d'ondes dans les équations d'Euler et d'Euler linéarisées à surface libre. Les solutions obtenues varient d'une forme sinusoïdale à une forme présentant des singularités. Elles permettent de valider numériquement les codes de simulation des équations d'Euler à surface libre.

Published
2020

5. Rigorous study of the equilibria of collision kernels appearing in the theory of weak turbulence

Author

Maxime Breden, Laurent Desvillettes, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université Paris Diderot, Sorbonne Paris Cité, Paris, France, Université Paris Diderot - Paris 7 (UPD7), and The research leading to this paper was partly funded by Université SorbonneParis Cité, in the framework of the 'Investissements d'Avenir', convention ANR-11-IDEX-0005. MB also acknowledges partial support from a Lichtenberg Professorship grant ofthe VolkswagenStiftung awarded to C. Kuehn.

Subjects
Keys words: weak turbulence, Complex system, 82C40, FOS: Physical sciences, Type (model theory), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, symbols.namesake, equilibria Mathematical Subject Classification: 76F99, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Mathematical Physics, Physics, 76P05, Turbulence, Mechanical Engineering, 010102 general mathematics, Mathematical Subject Classification: 76F99, 76P05, 82C40, Turbulence theory, Mathematical Physics (math-ph), Collision, equilibria, Functional Analysis (math.FA), Mathematics - Functional Analysis, 010101 applied mathematics, collision kernels, Boltzmann constant, Boltzmann's H-theorem, symbols, Particle, weak turbulence, Analysis, Analysis of PDEs (math.AP)
Abstract

In this paper, we rigorously obtain all the equilibria of collision kernels of type ``two particles give two particles'' appearing in weak turbulence theory under very general assumptions, thus completing the ``equality case'' in Boltzmann's H-theorem for those models. We also provide some rigorous results for collision kernels of type ``two particles give one particle'', under assumptions which include some of the most classical kernels of this type. The method of proof is inspired by the quantitative estimates obtained for the Landau equation by Desvillettes in [J. Funct. Anal. 269:1359-1403, 2015].

Published
2018

6. Multidimensional stability of V-shaped traveling fronts in time periodic bistable reaction–diffusion equations

Author

Wei-Jie Sheng, Department of Mathematics (HIT Harbin Institute of Technology), Harbin Institute of Technology (HIT), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and This paper was supported by NSF of China (11401134) and by China Scholarship Council for a one year visit of Aix Marseille Université.

Subjects
Current (mathematics), Bistability, 010102 general mathematics, Mathematical analysis, Time periodic V-shaped traveling fronts, 01 natural sciences, Stability (probability), 010101 applied mathematics, Bistable, Computational Mathematics, Computational Theory and Mathematics, Two-dimensional space, Reaction diffusion equations, Modeling and Simulation, Stability theory, Bounded function, Reaction–diffusion system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Multidimensional stability, Mathematics
Abstract

International audience; This paper deals with the multidimensional stability of time periodic V-shaped traveling fronts in bistable reaction–diffusion equations. It is well known that time periodic V-shaped traveling fronts are asymptotically stable in two dimensional space. In the current study, we further show that such fronts are asymptotically stable under spatially decaying initial perturbations in Rn with n≥3. In particular, we show that the fronts are algebraically stable if the initial perturbations belong to L1 in a certain sense. Furthermore, we prove that there exists a solution oscillating permanently between two time periodic V-shaped traveling fronts, which implies that time periodic V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Finally we show that time periodic V-shaped traveling fronts are only time global solutions of the Cauchy problem if the initial perturbations lie between two time periodic V-shaped traveling fronts.

Published
2016

7. A Ginzburg-Landau model with topologically induced free discontinuities

Author

Goldman, Michael, Merlet, Benoît, Millot, Vincent, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), We thank R. Badal, M. Cicalese, L. De Luca, and M. Ponsiglione for telling us about their result [BCLP16] and giving us an early access to a preprint version. We also thank M. Dos Santos for pointing out the paper [ILR00]. The authors have been supported by the Agence Nationale de la Recherche through the grants ANR-12-BS01-0014-01 (Geometrya), ANR-14-CE25-0009-01 (MAToS), and by the PGMO research project COCA. BM was partially supported by the INRIA team RAPSODI and the Labex CEMPI (ANR-11-LABX-0007-01)., ANR-12-BS01-0014,GEOMETRYA,Théorie géométrique de la mesure et applications(2012), ANR-14-CE25-0009,MAToS,Analyse des singularités topologiques dans quelques problèmes issus de la physique mathématique(2014), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe

Subjects
Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Condensed Matter::Superconductivity, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

We study a variational model which combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory, a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1/m$ with $m\geq 2$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg-Landau model, the energy is parameterized by a small length scale $\varepsilon>0$. We perform a complete $��$-convergence analysis of the model as $\varepsilon\downarrow0$ in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small $\varepsilon>0$, the minimizers of the original problem have the same structure away from the limiting vortices.

Published
2017
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8. Différentes approches pour la modélisation de l’électroporation des membranes cellulaires

Author

Clair Poignard, Silve, A., Wegner, L., Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Modélisation Mathématique pour l'Oncologie (MONC), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institute for Pulsed Power and Microwave Technology [Karlsruhe], Karlsruher Institut für Technologie (KIT), This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the 'Investments for the future', Programme IdEx Bordeaux–CPU (ANR-10-IDEX-03-02).Numerical simulations presented in this paper were carried out using the PLAFRIM exper- imental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil R ́egional d’Aquitaine, FeDER, Universit ́e de Bordeaux and CNRS(see https://plafrim.bordeaux.inria.fr/).C.P. is partly granted by the Plan Cancer project DYNAMO (9749 Inserm), and the PlanCancer project NUMEP (11099 Inserm). C.P. is also member of the European lab of CNRS, LEA EBAM, on electroporation., Inria Bordeaux Sud-Ouest, Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux]

Subjects
modèles biophysiques pour la formation de pores, Cell electroporation modeling, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], biophysical models for pore formation, mathematical models for membrane electroporation, Modélisation de l’éectroporation cellulaire, modèles mathématiques pour l’électroporation des membranes cellulaires
Abstract

Cell electroporation is a complex phenomenon, which consists in the emergence of defects in cell membranes subjected to electric pulses. Since the end of the 90’s biophysical models have been developed to explain and predict the conditions for cell electroporation. However the recent biological data, in particular those dealing with the influence of the repetition rate of the pulses challenge these biophysical models. In this chapter we introduce different biophysical models of electropore formation and we discuss their mathematical basis and their advantages and disadvantages. We also present the phenomenological modeling, which consists in designing the model on an empirical basis thanks to the experience. The aim of the chapter is to introduce the reader to different ways of modeling cell membrane electroporation, and to provide some possible directions to obtain a more reliable theory of electroporation in accordance with the experiments and with a justified theoretical basis.; Le but de ce rapport est de présenter les différentes approches biophysiques et phénoménologiques développées pour d´écrire l’électroperméabilisation de la membrane cellulaire. Nous repartons des modèles simples de circuits électriques équivalents, et celui de Schwan décrivant la membrane comme un condensateur mis en parallèle avec une résistance. Ensuite nous présentons les principes de la théorie de l’électroporation développée dans les années 90’ à partir des énergies libres des membranes, et qui a fait l’objet depuis de quelques modifications. Enfin nous présentons les approches phénoménologiques récentes qui permettent de comparer les données expérimentales et les simulations numériques. L’objectif est présenter les avantages et les désavantages de chaque approche afin d’ouvrir de possibles directions de recherches pour obtenir une théorie de l’électroporation plus en accord avec les observations, tout en se fondant sur une base théorique solide.

Published
2016

9. Patient specific simulation of tumor growth, response to the treatment and relapse of a lung metastasis: a clinical case

Author

Julien Jouganous, Jean Palussière, Thierry Colin, François Cornelis, Olivier Saut, Modélisation Mathématique pour l'Oncologie (MONC), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Hôpital Pellegrin, CHU Bordeaux [Bordeaux]-Groupe hospitalier Pellegrin, Institut Bergonié [Bordeaux], UNICANCER, This study has been carried out within the frame of the LABEX TRAIL, ANR-10-LABX-0057 with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the 'Investments for the future' Programme IdEx (ANR-10-IDEX-03-02).Experiments presented in this paper were carried out using the PLAFRIM experimental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil R ́egional d’Aquitaine, FeDER, Universit ́e de Bordeaux and CNRS (see https://plafrim.bordeaux.inria.fr/)., Inria Bordeaux, ANR-10-IDEX-0003,IDEX BORDEAUX,Initiative d'excellence de l'Université de Bordeaux(2010), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Oncology, medicine.medical_specialty, Pathology, Radiofrequency ablation, Angiogenesis, medicine.medical_treatment, Lung metastasis, [SDV.CAN]Life Sciences [q-bio]/Cancer, 010103 numerical & computational mathematics, 01 natural sciences, law.invention, Lesion, 03 medical and health sciences, Tumor growth modeling, 0302 clinical medicine, law, Internal medicine, medicine, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Tumor growth, 0101 mathematics, Clinical data assimilation, Chemotherapy, business.industry, Partial Differential Equations, Patient specific, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 030220 oncology & carcinogenesis, Clinical case, Medical imaging, medicine.symptom, business
Abstract

In this paper, a parametrization strategy based on reduced order methods is presented for tumor growth PDE models. This is applied to a new simple spatial model for lung metastasis including angiogenesis. The goal is to help clinicians monitoring tumors and eventually predicting their evolution or response to a particular kind of treatment. To illustrate the whole approach, a clinical case including the natural history of the lesion, the response to a chemotherapy, and the relapse before a radiofrequency ablation is presented.

Published
2015

10. A Theory of L 1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

Author

Kenneth H. Karlsen, Boris Andreianov, Nils Henrik Risebro, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Center of Mathematics for Applications [Oslo] (CMA), Department of Mathematics [Oslo], Faculty of Mathematics and Natural Sciences [Oslo], University of Oslo (UiO)-University of Oslo (UiO)-Faculty of Mathematics and Natural Sciences [Oslo], University of Oslo (UiO)-University of Oslo (UiO), and This paper was written as part of the research program on Nonlinear Partial Dierential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo, which took place during the academic year 2008-09.

Subjects
Pure mathematics, Admissibility, Adapted entropy, 01 natural sciences, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Boundary trace, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discontinuous flux, Germ, Uniqueness, Mathematics - Numerical Analysis, 0101 mathematics, Scalar conservation law, Mathematics, Conservation law, Finite volume method, Mechanical Engineering, 010102 general mathematics, Primary 35L65, Secondary 35R05, Numerical Analysis (math.NA), 010101 applied mathematics, Convergence of numerical approximations, Uniqueness criteria, Finite volume scheme, Piecewise, Dissipative system, Adapted viscosity, Entropy solution, Vanishing viscosity, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: (CL) u_t + f(x; u)_x = 0; f(x; u) = f^l(u), x < 0 and f(x;u)= f^r(u); x > 0; where the fluxes f^l and f^r are mainly assumed to be continuous. Developing the ideas of a number of preceding works (Baiti and Jenssen [14], Audusse and Perthame [12], Garavello et al. [35], Adimurthi et al. [3], Buerger et al. [21]), we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are piecewise constant weak solutions of the form c(x) = c^l 1l_{x0}. We refer to such a family as a "germ". It is well known that (CL) admits many different L1 contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specic attention to the "vanishing viscosity" germ, which is a way to express the "Gamma-condition" of Diehl [32]. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} (in the spirit of Vol'pert [80]) and in the form of a family of global entropy inequalities (following Kruzhkov [50] and Carrillo [22]). We characterize those germs that lead to the L1-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specic viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

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11. Kinetic theory of plasmas

Author

Thierry Magin, Marc Massot, Benjamin Graille, Center for Turbulence Research [Stanford] (CTR), Stanford University, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion (EM2C), Université Paris Saclay (COmUE)-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec, Papers presented during the AVT-162 RTO AVT/VKI Lecture Series held at the von Karman Institute, Rhode St. Genèse, Belgium., CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)-Université Paris Saclay (COmUE), and The authors acknowledge support from two scientific departments of CNRS (French Center for Scientific Research): MPPU (Mathématiques, Physique, Planète et Univers) and ST2I (Sciences et Technologies de l'Information et de l'Ingénierie), through a 2007-2008 PEPS project entitled: ``Analyse et simulation de problèmes multi-échelles : applications aux plasmas froids, à la combustion et aux écoulements diphasiques'.

Subjects
Electromagnetic field, Physics, plasmas in thermal nonequilibrium, Internal energy, 82C40, 76X05, 41A60, [SPI.FLUID]Engineering Sciences [physics]/Reactive fluid environment, [SPI.PLASMA]Engineering Sciences [physics]/Plasmas, Non-equilibrium thermodynamics, Electron, 01 natural sciences, Boltzmann equation, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], 010305 fluids & plasmas, Magnetic field, Classical mechanics, [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], conservation equations, multicomponent transport properties, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Kinetic theory, 010306 general physics, Boltzmann's entropy formula, Scaling
Abstract

International audience; In the present study, we derive from kinetic theory a unified fluid model for multi- component plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and reactive collisions. Given the strong disparity of mass between the electrons and heavy particles, such as molecules, atoms, and ions, we conduct a dimen- sional analysis of the Boltzmann equation and introduce a scaling based on a multiscale perturbation parameter equal to the square root of the ratio of the electron mass to a characteristic heavy-particle mass. We then generalize the Chapman-Enskog method, em- phasizing the role of the perturbation parameter on the collisional operator, the streaming operator, and the collisional invariants of the Boltzmann equation. The system is exam- ined at successive orders of approximation, each corresponding to a physical time scale. At the highest approximation order investigated, the multicomponent Navier-Stokes regime is reached for the heavy particles and is coupled to first-order drift-diffusion equations for the electrons. The transport coefficients are then calculated in terms of bracket opera- tors whose mathematical structure allows for positivity properties to be determined and Onsager's reciprocal relations to hold. The transport coefficients exhibit an anisotropic behavior when the magnetic field is strong enough. We also give a complete description of the Kolesnikov effect, i.e., the crossed contributions to the mass and energy transport fluxes coupling the electrons and heavy particles. Finally, the first and second laws of ther- modynamics are proved to be satisfied by deriving a total energy equation and an entropy equation.