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14,196 results on '"[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]"'

1. Asymptotic stability of scalar multi-D inviscid shock waves

Author: Serre, Denis, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics - Analysis of PDEs, Algebra and Number Theory, contraction semigroup, 35B40, asymptotic stability. MSC2010: 35L65, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], shock waves, Geometry and Topology, Scalar conservation laws, Burgers equation, Analysis of PDEs (math.AP)
Abstract: In several space dimensions, scalar shock waves between two constant states u $\pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.
Published: 2023

2. An Arbitrary Order and Pointwise Divergence-Free Finite Element Scheme for the Incompressible 3D Navier–Stokes Equations

Author: Marien-Lorenzo Hanot, Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Numerical Analysis, Applied Mathematics, Numerical Analysis (math.NA), Finite Element, exterior calculus, incompressible Navier-Stokes, Computational Mathematics, Mathematics - Analysis of PDEs, de Rham complex, 35Q30 (Primary) 65N30, 76D07, 76M10 (Secondary), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Hodge decomposition, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: In this paper we discretize the incompressible Navier-Stokes equations in the framework of finite element exterior calculus. We make use of the Lamb identity to rewrite the equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity. We propose a discretization on a large class of finite elements, including arbitrary order polynomial spaces readily available in many libraries. The main advantage of this discretization is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness ensures pressure robustness. We focus the analysis on a class of linearized equations for which we prove well-posedness and provide a priori error estimates. The results are validated with numerical simulations., 26 pages, 6 figures, added detailed proofs in appendices
Published: 2023

3. Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory

Author: Ervedoza, Sylvain, Le Balc’h, Kévin, 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), Control And GEometry (CaGE ), 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), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)
Subjects: Quantitative unique continuation, Applied Mathematics, Control, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Elliptic equations, Analysis
Abstract: The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in 2-d and observed from a non-empty open subset ω ⊂ Ω. More precisely, for every real-valued bounded potential V, our main result shows that, when Ω has a finite number of holes, the observability constant of the elliptic operator −∆ + V, with domain H^2 ∩ H^1_0(Ω), is of the form C exp (C |V|^{1/2} \log^{1/2}(|V|)) where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov, in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane. The main difference and additional difficulty is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.
Published: 2023

4. Quadratic behaviors of the 1D linear Schrödinger equation with bilinear control

Author: Bournissou, Mégane, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)
Subjects: Mathematics - Analysis of PDEs, Applied Mathematics, Exact controllability, Power series expansion, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Schrödinger equation, Bilinear control, Analysis
Abstract: We study its controllability around the ground state when the linearized system is not controllable and wonder whether the quadratic term can help to recover the directions lost at the first order. More precisely, in this paper, we formulate assumptions under which the quadratic term induces a drift which prevents the small-time local controllability (STLC) of the system in appropriate spaces. For finite-dimensional systems, quadratic terms induce coercive drifts in the dynamic, quantified by integer negative Sobolev norms, along explicit Lie brackets which prevent STLC.In the context of the bilinear Schr{\"o}dinger equation, the first drift, quantified by the $H^{-1}$-norm of the control, was already observed and used to deny STLC with controls small in $L^\infty$. In this article, we improve this result by denying STLC with controls small in $W^{-1, \infty}$. Furthermore, for any positive integer $n$, we formulate assumptions under which one may observe a quadratic drift quantified by the $H^{-n}$-norm of the control and we use it to deny STLC in suitable spaces.
Published: 2023

5. The condensation of a trapped dilute Bose gas with three-body interactions

Author: Phan Thành Nam, Julien Ricaud, Arnaud Triay, Mathematisches Institut [München] (LMU), and Ludwig-Maximilians-Universität München (LMU)
Subjects: Condensed Matter::Quantum Gases, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 35J10, 82B10, 82D05, FOS: Physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General Earth and Planetary Sciences, Mathematical Physics (math-ph), Mathematical Physics, General Environmental Science
Abstract: We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting via a three-body interaction potential of the form $N\, V(N^{1/2}(x-y,x-z))$. In the limit $N\to \infty$, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schr\"odinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the $N$-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schr\"odinger minimizer is unique up to a complex phase., Comment: 57 pages
Published: 2023

6. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: Linearized system

Author: Jean Cauvin-Vila, Virginie Ehrlacher, Amaury Hayat, Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), The authors acknowledge support from the ANR project COMODO (ANR-19-CE46-0002) which funds the Ph.D of Jean Cauvin-Vila., and ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019)
Subjects: Feedback stabilization, Mathematics - Analysis of PDEs, Boundary control, Backstepping, Applied Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Parabolic PDEs, Cross-diffusion systems, Exponential stability, Analysis, Analysis of PDEs (math.AP)
Abstract: We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time., Comment: 37 pages, no figure
Published: 2023

7. Fast and stable schemes for non-linear osmosis filtering

Author: Calatroni, L., Morigi, S., Parisotto, S., Recupero, G. A., Morphologie et Images (MORPHEME), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Biologie Valrose (IBV), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Signal, Images et Systèmes (Laboratoire I3S - SIS), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica [Bologna], Alma Mater Studiorum Università di Bologna [Bologna] (UNIBO), Department of Applied Mathematics and Theoretical Physics [Cambridge] (DAMTP), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)
Subjects: Computational Mathematics, Computational Theory and Mathematics, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 68U10, 94A08, 49K20, 65M05, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We consider a non-linear variant of the transport-diffusion osmosis model for solving a variety of imaging problems such as shadow/soft-light removal and compact data representation. The non-linear behaviour is encoded in terms of a general scalar function g with suitable properties, which allows to balance the diffusion intensity on the different regions of the image while preventing smoothing artefacts. For the proposed model, conservation properties (intensity and non-negativity) are proved and a variational interpretation is showed for specific choices of g. Upon suitable spatial discretisation, both an explicit and a semi-implicit iterative scheme are considered, for which convergence restrictions and unconditional stability are proved, respectively. To validate the proposed modelling and the computational speed of the numerical schemes considered, we report several results and comparisons for the problem of shadow/light-spot removal and compact data representation, showing that artefact-free and computationally efficient results are obtained in comparison to standard linear and anisotropic models, and state-of-the art approaches., Comment: 25 pages, 14 figures
Published: 2023

8. Modeling Under Location Uncertainty: A Convergent Large-Scale Representation of the Navier-Stokes Equations

Author: Debussche, Arnaud, Hug, Berenger, Mémin, Etienne, École normale supérieure - Rennes (ENS Rennes), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Océan Dynamique Observations Analyse (ODYSSEY), Université de Bretagne Occidentale - UFR Sciences et Techniques (UBO UFR ST), Université de Brest (UBO)-Université de Brest (UBO)-Université de Rennes (UR)-Institut Français de Recherche pour l'Exploitation de la Mer (IFREMER)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-IMT Atlantique (IMT Atlantique), and Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)
Subjects: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract: We construct martingale solutions for the stochastic Navier-Stokes equations in the framework of the modelling under location uncertainty (LU). These solutions are pathwise and unique when the spatial dimension is 2D. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier-Stokes equation. This warrants that the LU Navier-Stokes equations can be interpreted as a large-scale model of the deterministic Navier-Stokes equation.
Published: 2023

9. Analysis of the Rolling Carpet Strategy to Eradicate an Invasive Species

Author: Luis Almeida, Alexis Léculier, Nicolas Vauchelet, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), 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), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, European Project: 740623,H2020 Pilier ERC,ADORA(2017), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Subjects: Comparison principle, Population dynamics, comparison principle AMS subject classifications: 35K57, Applied Mathematics, Reaction-diusion equations, 35C07, 92D25, Computational Mathematics, Mathematics - Analysis of PDEs, Reaction-diffusion equations, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], comparison principle AMS subject classications: 35K57, AMS subject classifications: 35K57, 92D25, 35C07, Analysis, Analysis of PDEs (math.AP)
Abstract: International audience; In order to prevent the propagation of human diseases transmitted by mosquitoes (such as dengue or zika), one possible solution is to act directly on the mosquito population. In this work, we consider an invasive species (the mosquitoes) and we study two strategies to eradicate the population in the whole space by a local intervention. The dynamics of the population is modeled through a bistable reaction diffusion equation in an one dimensional setting and both strategies are based on the same idea : we act on a moving interval. The action of the first strategy is to kill as many individuals as we can in this moving interval. The action of the second strategy is to release sterile males in this moving interval. For both strategies, we manage to generate traveling waves that propagate in the opposite direction relative to the one of the natural invasive traveling wave. These cases correspond to succeeding in eradicating the invasive species. Furthermore, for the first strategy, we fully characterize the minimal size of the interval. All the results are illustrated by numerical simulations.
Published: 2023

10. On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

Author: Cancès, Clément, Venel, Juliette, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Matériaux Céramiques et de Mathématiques (CERAMATHS), Université Polytechnique Hauts-de-France (UPHF)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), Université Polytechnique Hauts-de-France (UPHF), ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019), ANR-19-CE40-0012,MICMOV,Description microscopique des interfaces mobiles(2019), European Project: 847593,EURAD(2019), and Laboratoire Paul Painlevé - UMR 8524 (LPP)
Subjects: energy dissipation, Mathematics - Analysis of PDEs, convergence, Nonlinear convection diffusion, General Mathematics, FOS: Mathematics, 65M08, 65M12, 35K51, 35Q92, 92D25, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], finite volume method, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.
Published: 2023

11. On the vanishing rigid body problem in a viscous compressible fluid

Author: Marco Bravin, Šárka Nečasová, Bravin, Marco, Basque Center for Applied Mathematics (BCAM), and Basque Center for Applied Mathematics
Subjects: Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Applied Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Analysis
Abstract: In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on the influence of a vanishing obstacle in a compressible fluid for $ \gamma \geq 6$ .
Published: 2023

12. Partially dissipative systems in the critical regularity setting, and strong relaxation limit

Author: Danchin, Raphaël, UMR 8050, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), and Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics - Analysis of PDEs, 2022. 2020 Mathematics Subject Classification. 35Q35, critical regularity, September 19, General Mathematics, 76N10 Hyperbolic systems, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], partially dissipative, September 26, Analysis of PDEs (math.AP), relaxation limit
Abstract: Many physical phenomena may be modelled by first order hyperbolic equations with degenerate dissipative or diffusive terms. This is the case for example in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term, or, in numerical simulations, of conservation laws by relaxation schemes. Such so-called partially dissipative systems have been first pointed out by S.K. Godunov in a short note in Russian in 1961. Much later, in 1984, S. Kawashima highlighted in his PhD thesis a simple criterion ensuring the existence of global strong solutions in the vicinity of a linearly stable constant state. This criterion has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima's results and to establish global existence results in some situations that were not covered before. These notes originate essentially from the PhD thesis of T. Crin-Barat that was initially motivated by an earlier observation of the author in a Chapter of the handbook coedited by Y. Giga and A. Novotn{\'y}. Our main aim is to adapt the method of Beauchard and Zuazua to a class of symmetrizable quasilinear hyperbolic systems (containing the compressible Euler equations), in a critical regularity setting that allows to keep track of the dependence with respect to e.g. the relaxation parameter. Compared to Beauchard and Zuazua's work, we exhibit a 'damped mode' that will have a key role in the construction of global solutions with critical regularity, in the proof of optimal time-decay estimates and, last but not least, in the study of the strong relaxation limit. For simplicity, we here focus on a simple class of partially dissipative systems, but the overall strategy is rather flexible, and adaptable to much more involved situations.
Published: 2023

13. A PDE-ODE model for traffic control with autonomous vehicles

Author: Liard, Thibault, Stern, Raphael, Delle Monache, Maria Laura, Universidad de Deusto [Bilbao] (DEUSTO), Universidad de Deusto (DEUSTO), Department of Civil and Environmental Engineering [Urbana], University of Illinois at Urbana-Champaign [Urbana], and University of Illinois System-University of Illinois System
Subjects: Statistics and Probability, Traffic control, Applied Mathematics, MathematicsofComputing_NUMERICALANALYSIS, Autonomous vehicles, General Engineering, Computer Science Applications, Computer Science::Robotics, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], PDE-ODE systems, [MATH]Mathematics [math], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We consider a partial differential equation - ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model, the bulk flow of human drivers is represented by a scalar conservation law, while each autonomous vehicle is described by an ordinary differential equation. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. Next, we propose a control strategy for the speeds of the autonomous vehicles to minimize the average fuel consumption of the entire traffic flow. Existence of solutions for the optimal control problem is proved, and we numerically show that a reduction in average fuel consumption is possible with an AV acting as a moving bottleneck.
Published: 2023

14. Analyticity of the semigroup corresponding to a strongly damped wave equation with a Ventcel boundary condition

Author: Badra, Mehdi, Takahashi, Takéo, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)
Subjects: Applied Mathematics, analytic semigroups, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ventcel boundary condition, structurally damped wave equation, Analysis, 2020 Mathematics Subject Classification. 35B65, 35L05, 35K90
Abstract: International audience; We consider a wave equation with a structural damping coupled with an undamped wave equation located at its boundary. We prove that, due to the coupling, the full system is parabolic. In order to show that the underlying operator generates an analytical semigroup, we study in particular the effect of the damping of the "interior" wave equation on the "boundary" wave equation and show that it generates a structural damping.
Published: 2023

15. Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects

Author: Alphonse, Paul, Bernier, Joackim, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Multi-scale numerical geometric schemes (MINGUS), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), École normale supérieure - Rennes (ENS Rennes)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Recherche Mathématique de Rennes (IRMAR), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: Subelliptic estimates, Mathematics - Analysis of PDEs, Mathematics::Operator Algebras, 35B65, 35H20, 65P10, 47A60, General Mathematics, Fourier integral operators, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Splitting method, Mathematics::Spectral Theory, Polar decomposition, Quadratic operators, Analysis of PDEs (math.AP)
Abstract: We characterize geometrically the regularizing effects of the semigroups generated by accretive non-selfadjoint quadratic differential operators. As a byproduct, we establish the subelliptic estimates enjoyed by these operators, being expected to be optimal. These results prove conjectures by M. Hitrik, K. Pravda-Starov and J. Viola. The proof relies on a new representation of the polar decomposition of these semigroups. In particular, we identify the selfadjoint part as the evolution operator generated by the Weyl quantization of a time-dependent real-valued nonnegative quadratic form for which we prove a sharp anisotropic lower bound., Comment: arXiv admin note: text overlap with arXiv:1510.01992, arXiv:1703.02797 by other authors
Published: 2023

16. Direct computation of nth-order correlations of the solution of a non-linear stochastic equation

Author: Saidi, Abdelkader, Dušek, Jan, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, and Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mechanics of Materials, Applied Mathematics, Mechanical Engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Condensed Matter Physics
Abstract: SummaryA system of nonlinear stochastic equations excited by a Gaussian random term leading to a statistically stationary solution is considered. The Carleman linearization is used to handle the nonlinearity and the statistical characterization of the solution is formulated in terms of a sequence of correlations of increasing order. Truncated in a finite but arbitrary order, the problem leads to a linear system of equations yielding directly the correlations. It is demonstrated in two examples that, if the excitation is not too strong, the solution converges as a function of the truncation order and provides an alternative to the Monte Carlo approach consisting in ensemble averaging a large number of time-dependent random solutions. For a low-dimensional system, it is shown that replacing the tensor indexing by a numbering accounting for redundancies makes it possible to keep the total problem dimension within reasonable limits even if relatively high-order correlations are accounted for. A model reduction based on the singular value decomposition of the second-order correlation matrix is tested with success for a case of a partial differential equation (Burgers’ equation), showing that the method can be potentially applied even to high-dimensional systems originating in partial differential equations.
Published: 2022

17. Sharp large time behaviour in N-dimensional reaction-diffusion equations of bistable type

Author: Jean-Michel Roquejoffre, Violaine Roussier-Michon, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Mathematics - Analysis of PDEs, Bistable reaction-diffusion equation, 35K57, Applied Mathematics, 35B40, FOS: Mathematics, Long-time asymptotics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B35, Analysis, Travelling fronts, Analysis of PDEs (math.AP)
Abstract: We study the large time behaviour of the reaction-diffsuion equation $\partial_t u=\Delta u +f(u)$ in spatial dimension $N$, when the nonlinear term is bistable and the initial datum is compactly supported. We prove the existence of a Lipschitz function $s^\infty$ of the unit sphere, such that $u(t,x)$ converges uniformly in $\mathbb{R}^N$, as $t$ goes to infinity, to $U_{c_*}\bigg(|x|-c_*t + \frac{N-1}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg)$, where $U_{c*}$ is the unique 1D travelling profile. This extends earlier results that identified the locations of the level sets of the solutions with $o_{t\to+\infty}(t)$ precision, or identified precisely the level sets locations for almost radial initial data.
Published: 2022

18. Propagation and blocking in a two-patch reaction-diffusion model

Author: Hamel, François, Lutscher, Frithjof, Zhang, Mingmin, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), University of Ottawa [Ottawa], and University of Science and Technology of China [Hefei] (USTC)
Subjects: Patchy landscapes, Applied Mathematics, General Mathematics, 35B40, Mathematics::Analysis of PDEs, 35C07, Blocking, Mathematics - Analysis of PDEs, Reaction-diffusion equations, 35K57, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Propagation, Analysis of PDEs (math.AP), Spreading speeds
Abstract: This paper is concerned with propagation phenomena for the solutions of the Cauchy problem associated with a two-patch one-dimensional reaction-diffusion model. It is assumed that each patch has a relatively well-defined structure which is considered as homogeneous. A coupling interface condition between the two patches is involved. We first study the spreading properties of solutions in the case when the per capita growth rate in each patch is maximal at low densities, a configuration which we call the KPP-KPP case, and which turns out to have some analogies with the homogeneous KPP equation in the whole line. Then, in the KPP-bistable case, we provide various conditions under which the solutions show different dynamics in the bistable patch, that is, blocking, virtual blocking (propagation with speed zero), or spreading with positive speed. Moreover, when propagation occurs with positive speed, a global stability result is proved. Finally, the analysis in the KPP-bistable frame is extended to the bistable-bistable case.
Published: 2022

19. A Controllability Method for Maxwell's Equations

Author: T. Chaumont-Frelet, M. J. Grote, S. Lanteri, J. H. Tang, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Department of Computer Science (University of Basel), University of Basel (Unibas), Institut des Sciences de la Terre (ISTerre), Institut national des sciences de l'Univers (INSU - CNRS)-Institut de recherche pour le développement [IRD] : UR219-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel-Université Grenoble Alpes (UGA), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and Laboratoire Jean Alexandre Dieudonné (JAD)
Subjects: Computational Mathematics, Mathematics - Analysis of PDEs, time-harmonic scattering, Maxwell's equations, Applied Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], exact controllability, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP), discontinuous Galerkin
Abstract: International audience; We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.
Published: 2022

20. Geometrically consistent aerodynamic optimization using an isogeometric Discontinuous Galerkin method

Author: Stefano Pezzano, Régis Duvigneau, Mickaël Binois, Analysis and Control of Unsteady Models for Engineering Sciences (ACUMES), Inria Sophia Antipolis - Méditerranée (CRISAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: Computational Mathematics, Computational Theory and Mathematics, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph]
Abstract: International audience; The objective of the current work is to define a design optimization methodology in aerodynamics, in which all numerical components are based on a unique geometrical representation, consistent with Computer-Aided Design (CAD) standards. In particular, the design is parameterized by Non-Uniform Rational B-Splines (NURBS), the computational domain is automatically constructed using rational Bézier elements extracted from NURBS boundaries without any approximation and the resolution of the flow equations relies on an adaptive Discontinuous Galerkin (DG) method based on rational representations. A Bayesian framework is used to optimize NURBS control points, in a single-or multi-objective, constrained, global optimization framework. The resulting methodology is therefore fully CAD-consistent, high-order in space and time, includes local adaption and shock capturing capabilities, and exhibits high parallelization performance. The proposed methods are described in details and their properties are established. Finally, two design optimization problems are provided as illustrations: the shape optimization of an airfoil in transonic regime, for drag reduction with lift constraint, and the multi-objective optimization of the control law of a morphing airfoil in subsonic regime, regarding the time-averaged lift, the minimum instantaneous lift and the energy consumption.
Published: 2022

21. Trajectorial hypocoercivity and application to control theory

Author: Dietert, Helge, Hérau, Frédéric, Hutridurga, Harsha, Mouhot, Clément, Université Paris Cité (UPCité), Sorbonne Université (SU), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Nantes université - UFR des Sciences et des Techniques (Nantes univ - UFR ST), Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ)-Nantes Université - pôle Sciences et technologie, Nantes Université (Nantes Univ)-Nantes Université (Nantes Univ), Indian Institute of Technology Bombay (IIT Bombay), University of Cambridge [UK] (CAM), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), and European Project: 279600,EC:FP7:ERC,ERC-2011-StG_20101014,MATKIT(2011)
Subjects: Hypocoercivity, Mathematics - Analysis of PDEs, divergence inequality, spectral gap, kinetic theory, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General Medicine, Fokker-Planck, controllability, Bogovosiǐ operator, Analysis of PDEs (math.AP)
Abstract: We present the quantitative method of the recent work arXiv:2209.09340 in a simple setting, together with a compactness argument that was not included in arXiv:2209.09340 and has interest per se. We are concerned with the exponential stabilization (spectral gap) for linear kinetic equations with degenerate thermalization, i.e. when the collision operator vanishes on parts of the spatial domain. The method in arXiv:2209.09340 covers both scattering and Fokker-Planck type operators, and deals with external potential and boundary conditions, but in these notes we present only its core argument and restrict ourselves to the kinetic Fokker-Planck in the periodic torus with unit velocities and a thermalization degeneracy. This equation is not covered by the previous results of Bernard and Salvarani (2013), Han-Kwan and L\'eautaud (2015), Evans and Moyano (arXiv:1907.12836)., Comment: 10 pages, 1 figure
Published: 2022

22. Singular perturbation of manifold-valued maps with anisotropic energy

Author: Contreras, Andres, Lamy, Xavier, New Mexico State University, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Numerical Analysis, Mathematics - Analysis of PDEs, 35J47, Applied Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis, Analysis of PDEs (math.AP)
Abstract: We establish small energy H\"{o}lder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_\Omega W(\nabla u)+ \frac{1}{\varepsilon^2} \int_\Omega f(u),\] where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathcal N$. This implies that, up to subsequence, $u_\varepsilon$ converges locally uniformly to an $\mathcal N$-valued $W$-harmonic map, away from its singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. Similar results are known in the isotropic case $W(\nabla u)=\vert \nabla u\vert^2$ and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider, all of these ingredients are absent. In particular, the lack of monotonicity formula is an important reason why optimal estimates on the singular set of $W$-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than $\varepsilon$: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of $W$-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open., Comment: The initial proof of the energy improvement lemma 2.2 contained a gap and has been corrected in this new version
Published: 2022

23. Small perturbations in the type of boundary conditions for an elliptic operator

Author: Bonnetier, Eric, Dapogny, Charles, Vogelius, Michael, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Department of Mathematics - Rutgers School of Arts and Sciences, Rutgers, The State University of New Jersey [New Brunswick] (RU), Rutgers University System (Rutgers)-Rutgers University System (Rutgers), ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018), and ANR-17-CE40-0029,MultiOnde,Problèmes Inverses Multi-Onde(2017)
Subjects: Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Applied Mathematics, General Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, Analysis of PDEs (math.AP)
Abstract: In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this ``background'' situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a ``small'' subset $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon $ in the former case, we introduce a notion of ``Neumann capacity'' to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general $\omega_\varepsilon $, under the sole assumption that it is ``small'' in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset $\omega_\varepsilon $ is a vanishing surfacic ball.
Published: 2022

24. Mathematical modelling of a pest in an age-structured crop model: The coffee berry borer case

Author: Yves Fotso Fotso, Suzanne Touzeau, Berge Tsanou, Frédéric Grognard, Samuel Bowong, Faculté des Sciences - Université de Dschang [Cameroun], Université de Dschang, Biological control of artificial ecosystems (BIOCORE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire d'océanographie de Villefranche (LOV), Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut de la Mer de Villefranche (IMEV), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Laboratoire International de Recherche en Informatique et Mathématiques Appliquées (LIRIMA), Université de Yaoundé I-Université Badji Mokhtar Annaba (UBMA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Joseph Ki-Zerbo [Ouagadougou] (UJZK)-Université d'Antananarivo-Université Gaston Bergé Sénégal-Centre National de la Recherche Scientifique et Technologique (CNRST), Institut Sophia Agrobiotech (ISA), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Université Côte d'Azur (UCA), Department of Mathematics and Applied Mathematics [Pretoria], University of Pretoria [South Africa], Unité de modélisation mathématique et informatique des systèmes complexes [Bondy] (UMMISCO), Université de Yaoundé I-Institut de la francophonie pour l'informatique-Université Cheikh Anta Diop [Dakar, Sénégal] (UCAD)-Université Gaston Bergé (Saint-Louis, Sénégal)-Université Cadi Ayyad [Marrakech] (UCA)-Sorbonne Université (SU)-Institut de Recherche pour le Développement (IRD [France-Nord]), Université Côte d'Azur (UCA), Dpt of Mathematics and Computer Science [Douala], Université de Douala, EPITAG, LIRIMA, College doctoral regional de Afrique Centrale et des Grands Lacs, EPITAG, Centre National de la Recherche Scientifique et Technologique (CNRST)-Université Gaston Bergé Sénégal-Université d'Antananarivo-Université Joseph Ki-Zerbo [Ouagadougou] (UJZK)-Université Badji Mokhtar - Annaba [Annaba] (UBMA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Yaoundé I, Université Nice Sophia Antipolis (... - 2019) (UNS), and Institut de Recherche pour le Développement (IRD [France-Nord])-Institut de la francophonie pour l'informatique-Université Cheikh Anta Diop [Dakar, Sénégal] (UCAD)-Université Gaston Bergé (Saint-Louis, Sénégal)-Université Cadi Ayyad [Marrakech] (UCA)-Université de Yaoundé I-Sorbonne Université (SU)
Subjects: [SDV.SA]Life Sciences [q-bio]/Agricultural sciences, Applied Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [SDV.SA.AGRO]Life Sciences [q-bio]/Agricultural sciences/Agronomy, Semigroup theory, Hypothenemus, Epidemiological model, MSC: 92D30, 35Q92, 35A01, 35B40, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Modeling and Simulation, Hypothenemus hampei, Numerical simulations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Plant-pest interaction, Stability
Abstract: International audience; Coffee production is an important agriculture activity which contributes significantly to the economic growth of many countries in the world. One of the major constraints to coffee production throughout the world is the damage caused by the coffee berry borer (CBB), Hypothenemus hampei (Coleoptera: Scolytidae). It is the most damaging insect pest to coffee production in the world. These insects infest coffee berries, preferably mature berries, and spend most of their life cycle inside the berries, which make them quite difficult to control. In this work, we introduce and analyse a berry age-structured model describing the infestation dynamics of coffee berries by CBB. Using the semigroup theory in the case of Lipschitz perturbation, we show that the model has a unique nonnegative and bounded solution. We derive an explicit formula of the basic reproduction number, R0, using the next generation approach and the biological interpretation of this threshold in a specific case. Numerical simulations are carried out to illustrate the theoretical results.
Published: 2022

25. The Cauchy problem for Laplace's equation via a modified conjugate gradient method and energy space approaches

Author: Saber Amdouni, Amel Ben Abda, National Engineering School of Tunis = Ecole Nationale d'Ingénieurs de Tunis [University of Tunis El Manar] (ENIT), University of Tunis El Manar, and Amdouni, Saber
Subjects: Cauchy problem, Finite element method, General Mathematics, General Engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Energy space approaches, Laplace problem, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Conjugate gradient method
Abstract: In the present work, we focus on the resolution of the Cauchy Laplace problem using an energetic variational minimization approach [18] in the framework of a finite element method. A new strategy of regularization, called a filtering procedure regularization, is developed. The advantage of using this new regularization is that it does not require a regularization parameter and is easy to implement. An optimal a priori error estimate is proven, for the first time up to our knowledge, in the context of the finite element method. Some numerical results are presented to illustrate the performance of our approach.
Published: 2022

26. Sharp reachability results for the heat equation in one space dimension

Author: KELLAY, Karim, NORMAND, Thomas, TUCSNAK, Marius, Institut de Mathématiques de Bordeaux (IMB), 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)
Subjects: Numerical Analysis, Applied Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Analysis
Abstract: This paper gives a complete characterization of the reachable space for a system described by the 1D heat equation with L2 (with respect to time) Dirichlet boundary controls at both ends. More precisely, we prove that this space coincides with the sum of two spaces of analytic functions (of Bergman type). These results are then applied to give a complete description of the reachable space via inputs which are n-times differentiable functions of time. Moreover, we establish a connection between the norm in the obtained sum of Bergman spaces and the cost of null controllability in small time. Finally we show that our methods yield new complex analytic results on the sums of Bergman spaces in infinite sectors.
Published: 2022

27. Global quasi-neutral limit for a two-fluid Euler-Poisson system in one space dimension

Author: Yue-Jun Peng, Cunming Liu, Université Clermont Auvergne (UCA), and Qufu Normal University
Subjects: Applied Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS, Analysis
Abstract: International audience
Published: 2022

28. Estimations de stabilité améliorées pour la résolution du problème de Stokes avec les éléments finis de Fortin-Soulie

Author: Jamelot, Erell, Service de Thermo-hydraulique et de Mécanique des Fluides (STMF), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Université Paris-Saclay, and CEA SIMU/SITHY project
Subjects: nonconforming finite elements, Stokes problem, éléments finis mixtes de Fortin-Soulie, opérateur de Fortin, Fortin-Soulie mixed finite elements, éléments finis non conformes, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], T-coercitivity, Problème de Stokes, T-coercitivité, Fortin operator, 2020 Mathematics Subject Classification 65N30, 35J57, 76D07
Abstract: We propose to analyse the discretization of the Stokes problem with nonconforming finite elements in light of the T-coercivity. First we explicit the stability constants with respect to the shape regularity parameter for order 1 in 2 or 3 dimension, and order 2 in 2 dimension. In this last case, we improve the result of the original Crouzeix-Raviart paper. Second, we illustrate the importance of using a divergence-free velocity reconstruction on some numerical experiments.; Nous proposons d’analyser la discrétisation du problème de Stokes avec des éléments finis nonconformes à la lumière de la T-coercitivité . Tout d'abord, nous explicitons les constantes de stabilité par rapport au paramètre de stabilité pour l'ordre 1 en dimensions 2 et 3; et pour l'ordre 2 en dimension 2. Dans ce dernier cas, on améliore le résultat obtenu dans le papier original de Carouzeix-Raviart. Enfin, nous donnons des résultats numériques illustrant l’importance d’utiliser une méthode de reconstruction de vitesse à divergence nulle.
Published: 2023

29. Stable schemes for second-moment turbulent models for incompressible flows

Author: Ferrand, Martin, Hérard, Jean-Marc, Norddine, Thomas, Ruget, Simon, Centre d'Enseignement et de Recherche en Environnement Atmosphérique (CEREA), École des Ponts ParisTech (ENPC)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), EDF R&D (EDF R&D), EDF (EDF), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and École des Ponts ParisTech (ENPC)
Subjects: second-moment turbulent closure, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], hyperbolic systems, Riemann problem, incompressible turbulent flows
Abstract: International audience; A stable scheme is proposed in this paper in order to obtain approximate solutions of second-moment turbulent models for incompressible flows with or without thermal transport equation. The analysis of the convective terms, which includes the solution of the associated Riemann problem, enables to propose a standard projection scheme, and to get rid of spurious oscillations.
Published: 2023

30. Convergence rate of entropy-regularized multi-marginal optimal transport costs

Author: Nenna, Luca, Pegon, Paul, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), 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)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011)
Subjects: Primary: 49Q22, Secondary: 49N15, 94A17, 49K40, Entropic regularization, Schrödinger problem, Multi-marginal optimal transport, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Functional Analysis (math.FA), Convex analysis, Mathematics - Functional Analysis, Mathematics - Analysis of PDEs, Optimization and Control (math.OC), Optimal transport, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Mathematics - Optimization and Control, 49Q22, 49N15, 94A17, 49K40, Analysis of PDEs (math.AP)
Abstract: We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter $\varepsilon$ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon\log(1/\varepsilon)+O(\varepsilon)$ for some explicit dimensional constants $C$ depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for $\mathcal{C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.
Published: 2023

31. Local exact controllability to the steady states of a parabolic system with coupled nonlinear boundary conditions

Author: Bhandari, Kuntal, Boyer, Franck, Institute of Mathematics of the Czech Academy of Science (IM / CAS), Czech Academy of Sciences [Prague] (CAS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), and Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.)
Subjects: method of moments, Parabolic systems, fixed-point argument, boundary controllability, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], source term method
Abstract: In this article, we study the boundary local exact controllability to any steady state of a one-dimensional parabolic system with coupled nonlinear boundary conditions by means of only one control. The significant point is that the state components are interacting only at the boundary points in terms of some nonlinear terms. We consider two cases : either the control function is acting through a mixed nonlinear boundary condition on the first component or through a Neumann condition on the second component. The results are slightly different in the two cases. To study this problem, we first consider the associated linearized systems around the given steady state. The method of moments let us to prove its controllability and to obtain a suitable estimate of the control cost of the form $Me^{M/T}$. To this end, we need to develop a precise spectral analysis of a non self-adjoint operator. Thanks to those preliminary results, we can use the source term method developed in [Liu-Takahashi-Tucsnak 2013], followed by the Banach fixed point argument, to obtain the small-time local boundary exact controllability to the steady state for the original system.
Published: 2023

32. Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system

Author: Cauvin-Vila, Jean, Ehrlacher, Virginie, Marino, Greta, Pietschmann, Jan-Frederik, École des Ponts ParisTech (ENPC), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut für Mathematik [Augsburg], Universität Augsburg [Augsburg], The authors acknowledge support from project COMODO (ANR-19-CE46-0002).GM acknowledges the support of Deutsche Forschungsgemeinschaft (DFG) via grant GZ: MA 10100/1-1 project number 496629752. GM is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitae le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). JFP thanks the DFG for support via the Research Unit FOR 5387 POPULAR, Project No. 461909888., and ANR-19-CE46-0002,COMODO,Systèmes de diffusion croisée sur des domaines en mouvement(2019)
Subjects: Mathematics - Analysis of PDEs, Cahn-Hilliard, Cross-diffusion, FOS: Mathematics, Degenerate Ginzburg-Landau, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Finite volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions.
Published: 2023

33. Scattering, random phase and wave turbulence

Author: Faou, Erwan, Mouzard, Antoine, Multi-scale numerical geometric schemes (MINGUS), École normale supérieure - Rennes (ENS Rennes)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and Simons collaboration on wave turbulence
Subjects: Scattering, Mathematics - Analysis of PDEs, FOS: Mathematics, Random initial data, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Wave turbulence, Analysis of PDEs (math.AP)
Abstract: We start from the remark that in wave turbulence theory, exemplified by the cubic twodimensional Schr{\"o}dinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converges towards the kinetic operator but with different form of convergence.
Published: 2023

34. Modelling, simulation and mathe-matical analysis of drone swarms viaentropy methods

Author: Negulescu, Claudia, Maupoux, Axel, Lehman, Etienne, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH]Mathematics [math], [NLIN.NLIN-AO]Nonlinear Sciences [physics]/Adaptation and Self-Organizing Systems [nlin.AO]
Abstract: The mathematical modelling and analysis of the collective behaviour of a cloud of N interacting particles or agents has attracted a lot of interest in the last years in several communities, such as biologists, physicists, mathematicians, computer scientists etc. This is motivated not only by fundamental reasons, such as the understanding of the natural phenomena occurring around us, but also by the wide applications of this field in several domains, such as collective robotics, unmanned areal vehicles, ...Several mathematical models appeared in literature in the last years, such as for ex. the Viscek model, the Kuramoto model, the Cucker-Smale model etc, each one being specifically adapted for a particular situation, and several mathematical and numerical studies have been performed, the literature being constantly growing. The basic models have been fully understood today, what is still open in our opinion is the design of more realistic models, permitting to get closer to reality, and the corresponding mathematical and numerical analysis. In fact, a truly good model must on one hand recreate the real-life behaviour one is investigating, and on the other hand it must be simple enough to enable a detailed mathematical and numerical study. So in our particular case of a drone swarm, all the specificities mentionned in Section 3.2 shall be step by step included in a realistic drone model and efficient, multi-scale numerical schemes designed to be proposed to the industrials
Published: 2023

35. Dynamic Modeling of Carbon Dioxide Transport through the Skin Using a Capnometry Wristband

Author: Grangeat, Pierre, Duval Comsa, Maria-Paula Duval, Koenig, Anne, Phlypo, Ronald, Commissariat à l'énergie atomique et aux énergies alternatives - Laboratoire d'Electronique et de Technologie de l'Information (CEA-LETI), Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), project CAPNO2 CEA C38753 L3Médical, and European Project: 800945,NUMERICS
Subjects: state-space Markovian model, capnometry, wearable health device, convection diffusion equation, augmented dynamical system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], blood carbon dioxide concentration, [SDV.IB]Life Sciences [q-bio]/Bioengineering, Kalman filter, carbon dioxide transcutaneous pressure, compartmental model, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, continuous monitoring
Abstract: International audience; The development of a capnometry wristband is of great interest for monitoring patients at home. We consider a new architecture in which a non-dispersive infrared (NDIR) optical measurement is located close to the skin surface and is combined with an open chamber principle with a continuous circulation of air flow in the collection cell. We propose a model for the temporal dynamics of the carbon dioxide exchange between the blood and the gas channel inside the device. The transport of carbon dioxide is modeled by convection–diffusion equations. We consider four compartments: blood, skin, the measurement cell and the collection cell. We introduce the state-space equations and the associated transition matrix associated with a Markovian model. We define an augmented system by combining a first-order autoregressive model describing the supply of carbon dioxide concentration in the blood compartment and its inertial resistance to change. We propose to use a Kalman filter to estimate the carbon dioxide concentration in the blood vessels recursively over time and thus monitor arterial carbon dioxide blood pressure in real time. Four performance factors with respect to the dynamic quantification of the CO2 blood concentration are considered, and a simulation is carried out based on data from a previous clinical study. These demonstrate the feasibility of such a technological concept.
Published: 2023

36. Fine asymptotic expansion of the ODE's flow

Author: Briane, Marc, Hervé, Loïc, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
Subjects: asymptotic expansion, 37C10, incommensurable vector, ODE's flow asymptotic expansion rotation number incommensurable vector Diophantine number Liouville's number Mathematics Subject Classification: 34E05 34E10 37C10 37C40, 37C40, Liouville's number Mathematics Subject Classification: 34E05, Diophantine and Liouville numbers, rotation number, Diophantine number, 34E10, 34E05, 34E10, 37C10, 37C40, Mathematics - Analysis of PDEs, Euler flow, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ODE's flow, Analysis of PDEs (math.AP)
Abstract: In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the nonlinear ODE: X (t, x) = b X(t, x) with X(0, x) = x $\in$ R d , where b is a regular Z dperiodic vector field in R d. More precisely, we provide various conditions on b to obtain a "fine" asymptotic expansion of X of the type: |X(t, x) -- x -- t $\zeta$(x)| $\le$ M < $\infty$, which is uniform with respect to t $\ge$ 0 and x $\in$ R d (or at least in a subset of R d), and where $\zeta$(x) for x $\in$ R d , are the rotation vectors induced by the flow X. On the one hand, we give a necessary and sufficient condition on the vector field b so that the expansion X(t, x) -- x -- t $\zeta$(x) reads as $\Phi$ X(t, x) -- $\Phi$(x), which yields immediately the desired expansion when the vector-valued function $\Phi$ is bounded. In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on Y d and of vector-valued functions $\Phi$. On the other hand, assuming that the two-dimensional Kolmogorov theorem and some extension in higher dimension hold, we establish different regimes depending on the commensurability of the rotation vectors of the flow X for which the fine estimate expansion of X is valid or not. It turns out that for any two-dimensional flow X associated with a non vanishing smooth vector field b and inducing a unique incommensurable rotation vector $\xi$, the fine asymptotic expansion of X holds in R 2 if, and only if, $\xi$ 1 /$\xi$ 2 is a Diophantine number. This result seems new in the setting of the ODE's flow. The case of commensurable rotation vectors $\zeta$(x) is investigated in a similar way. Finally, several examples and counterexamples illustrate the different results of the paper, including the case of a vanishing vector field b which blows up the asymptotic expansion in some direction.
Published: 2023

37. Electromagnetic waves propagation in thin heterogenous coaxial cables. Comparaison between 3D and 1D models

Author: Beck, Geoffrey, Beni Hamad, Akram, Multi-scale numerical geometric schemes (MINGUS), École normale supérieure - Rennes (ENS Rennes)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)
Subjects: [PHYS.PHYS.PHYS-CLASS-PH]Physics [physics]/Physics [physics]/Classical Physics [physics.class-ph], [SPI.ELEC]Engineering Sciences [physics]/Electromagnetism, Maxwell's equations, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Asymptotic analysis, Finite elements, Coaxial cables, Telegrapher's models, Numerical simulations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Model's derivation, MSC: 35, 65 68
Abstract: International audience; This work deals with wave propagation into a coaxial cable, which can be modeled by the 3D Maxwell equations or 1D simplified models. The usual one, called the telegrapher's model, is a 1D wave equation on the electrical voltage and current. We derived a more accurate model from the Maxwell equations that takes into account dispersive effects. These two models aim to be a good approximation of the 3D electromagnetic fields in the case where the thinness of the cable is small. We perform some numerical simulations of the 3D Maxwell equations and of the 1D simplified models in order to validate the usual model and the new one. Moreover, we show that while the usual telegrapher model is of order one with respect to the thinness of the cable, the dispersive 1D model is of order two.
Published: 2023

38. GROWTH OF SOBOLEV NORMS AND STRONG CONVERGENCE FOR THE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

Author: Chauleur, Quentin, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Systèmes de particules et systèmes dynamiques (Paradyse), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011)
Subjects: Growth of Sobolev norms, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Continuum limit, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP), Discrete nonlinear Schrödinger equation, 35Q55, 37K60
Abstract: We show the strong convergence in arbitrary Sobolev norms of solutions of the discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the nonlinear Schr{\"o}dinger equation on the whole space. We restrict our attention to the one and two-dimensional case, with a set of parameters which implies global well-posedness for the continuous equation. Our proof relies on the use of bilinear estimates for the Shannon interpolation as well as the control of the growth of discrete Sobolev norms that we both prove.
Published: 2023

39. Stability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems: Supplementary material

Author: Chambolle, Antonin, Mazari-Fouquer, Idriss, Privat, Yannick, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), 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)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), and ANR-18-CE40-0013,SHAPO,Optimisation de forme(2018)
Subjects: Quantitative inequalities, 49M05, 49M41, 49N05, 49Q10, Thresholding scheme AMS classification: 49M05, Convergence analysis, Numerical algorithms in optimal control, PDE constrained optimisation, Shape optimisation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: We address the convergence of thresholding schemes for spectral optimisation problems and linear control problems.
Published: 2023

40. Controllability with one scalar control of a system of interaction between the Navier-Stokes system and a damped beam equation

Author: Buffe, Rémi, Takahashi, Takéo, Equations aux dérivées partielles (EDP), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)
Subjects: Navier-Stokes systems, Null controllability, fluid-structure interactions systems, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Carleman estimates
Abstract: We consider the controllability of a fluid-structure interaction system, where the fluid is modeled by the Navier-Stokes system and where the structure is a damped beam located on a part of its boundary. The motion of the fluid is bi-dimensional whereas the deformation of the structure is one-dimensional and we use periodic boundary conditions in the horizontal direction. Our result is the local null-controllability of this free-boundary system by using only one scalar control acting on an arbitrary small part of the fluid domain. This improves a previous result obtained by the authors where three scalar controls were needed to achieve the local null-controllability. In order to show the result, we prove the final-state observability of a linear Stokes-beam interaction system in a cylindrical domain. This is done by using a Fourier decomposition, proving Carleman inequalities for the corresponding system for the low-frequencies solutions and in the case where the observation domain is an horizontal strip. Then we conclude this observability result by using a Lebeau-Robbiano strategy for the heat equation and a uniform exponential decay for the high-frequencies solutions. Then, the result on the nonlinear system can be obtained by a change of variables and a fixed-point argument.
Published: 2023

41. Monotonic diamond and DDFV type finite-volume schemes for 2D elliptic problems

Author: Blanc, Xavier, Hermeline, Francois, Labourasse, Emmanuel, Patela, Julie, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire en Informatique Haute Performance pour le Calcul et la simulation (LIHPC), DAM Île-de-France (DAM/DIF), Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and CEA DAM ILE-DE-FRANCE - Bruyères-le-Châtel [Arpajon] (CEA DAM IDF)
Subjects: Order 2, Monotonicity, Elliptic problem, Finite volume method, DDFV, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract: The DDFV (Discrete Duality Finite Volume) method is a finite volume scheme mainly dedicated to diffusion problems, with some outstanding properties. This scheme has been found to be one of the most accurate finite volume methods for diffusion problems. In the present paper, we propose a new monotonic extension of DDFV, which can handle discontinuous tensorial diffusion coefficient. Moreover, we compare its performance to a diamond type method with an original interpolation method relying on polynomial reconstructions. Monotonicity is achieved by adapting the method from [44, 19, 49, 18] to our schemes. Such a technique does not require the positiveness of the secondary unknowns. We show that the two new methods are second-order accurate and are indeed monotonic on some challenging benchmarks as a Fokker-Planck problem.
Published: 2023

42. REGULARITY OF THE STRESS FIELD FOR DEGENERATE AND/OR SINGULAR ELLIPTIC PROBLEMS

Author: Lledos, Benjamin, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Regularity, Degenerate problem, Elliptic equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Calculus of variations
Abstract: We investigate the regularity of the solutions of degenerate and/or singular ellipticequations. We prove the continuity of G(∇u) where u is a locally Lipschitz solution of div G(∇u) =λ ∈ R in dimension two under some growth assumptions on G. We also present a result true in anydimension stating that the distance between ∇u and the degeneracy set of G is continuous.
Published: 2023

43. On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

Author: Boutin, Benjamin, Barbenchon, Pierre Le, Seguin, Nicolas, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), and Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers
Subjects: Kreiss-Lopatinskii determinant, GKS stability, 65M12, 65M06, GKS-stability, finite-difference methods, boundary conditions, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], inverse Lax-Wendroff, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: International audience; In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss-Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss-Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to elate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam-Warming scheme together with the simplified inverse Lax-Wendroff procedure at the boundary.
Published: 2023

44. Mathematical modelling of cancer growth and drug treatments: taking into account cell population heterogeneity and plasticity

Author: Clairambault, Jean, Modelling and Analysis for Medical and Biological Applications (MAMBA), 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), Silviu-Iulian Niculescu, Clairambault, Jean, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Subjects: [SDV] Life Sciences [q-bio], MSC 35B40, 35G92, 49J20, 92C50, [SDV.CAN] Life Sciences [q-bio]/Cancer, [SDV]Life Sciences [q-bio], [SDV.BA]Life Sciences [q-bio]/Animal biology, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [SDV.CAN]Life Sciences [q-bio]/Cancer, [MATH] Mathematics [math], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH]Mathematics [math]
Abstract: Version v2 après acceptation du document par la conférence de la prépublication hal-03938311; International audience; Mathematical models of cancer growth and evolution of cancer cell characteristics, aka traits or phenotypes, together with optimisation and optimal control methods to contain them, in the framework of adaptive dynamics of cell populations, are presented. They take into account the heterogeneity of cancer cell populations, i.e., their biological variability, and their intrinsic plasticity, i.e., their nongenetic instability that allows them to quickly adapt to changing environments. The presented vision of the cancer disease, which is specific to multicellular organisms, relies on a relatively novel vision, consistent with a billion-year evolutionary perspective. Based on recent contributions from philosophy of cancer, these mathematical models aim at designing theoretical therapeutic strategies to simultaneously contain tumour progression and limit adverse events of drugs to healthy cell populations.
Published: 2023

45. On the limit problem arising in the kinetic derivation of the Cahn-Hilliard equation

Author: Elbar, Charles, Perthame, Benoît, Skrzeczkowski, Jakub, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), 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), Institute of Mathematics [Warsaw], Faculty of Mathematics, Informatics, and Mechanics [Warsaw] (MIMUW), and University of Warsaw (UW)-University of Warsaw (UW)
Subjects: Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 2020 MSC: 35B40, 35D30, 35K25, 35K55, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B40, 35D30, 35K25, 35K55, Analysis of PDEs (math.AP)
Abstract: The non-local degenerate Cahn-Hilliard equation is derived from the Vlasov equation with long-range attraction. We study the local limit as the delocalization parameter converges to 0. The difficulty arises from the degeneracy which requires compactness estimates, but all necessary a priori estimates can be obtained only on the nonlocal quantities yielding almost no information on the limiting solution itself. We introduce a novel condition on the nonlocal kernel which allows us to exploit the available nonlocal a priori estimates. The condition, satisfied by most of the kernels appearing in the applications, can be of independent interest. Our approach is flexible and systems can be treated as well., 15 pages + Appendix
Published: 2023

46. Flatness approach for the boundary controllability of a system of heat equations

Author: COLLE, Blaise, Lohéac, Jérôme, Takahashi, Takéo, Centre de Recherche en Automatique de Nancy (CRAN), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Equations aux dérivées partielles (EDP), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-20-CE40-0009,TRECOS,Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux(2020)
Subjects: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: We study the boundary controllability of $2 × 2$ system of heat equations by using a flatness approach. According to the relation between the diffusion coefficients of the heat equation, it is known that the system can be not null-controllable or null-controllable for any $T > T_0$ where $T_0 \in [0, \infty]$. Here we recover this result in the case that $T_0 \in [0, \infty)$ by using the flatness method, and we obtain an explicit formula for the control and for the corresponding solutions. In particular, the state and the control have Gevrey regularity in time and in space.
Published: 2023

47. About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations

Author: Marino, Lorenzo, Menozzi, Stéphane, Priola, Enrico, Institut of Mathematics - Polish Academy of Sciences (PAN), Polska Akademia Nauk = Polish Academy of Sciences (PAN), Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Dipartimento di Matematica 'Felice Casorati' Department of Mathematics [Univ Pavia] (UNIPV), and Università degli Studi di Pavia = University of Pavia (UNIPV)
Subjects: Schauder estimates, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Analysis of PDEs, Degenerate Ornstein-Uhlenbeck operators, Probability (math.PR), FOS: Mathematics, Lp estimates, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Poisson process, Mathematics - Probability, non-local parabolic equations, Analysis of PDEs (math.AP), MSC: 35K10, 35K15, 60J76
Abstract: We study here the effects of a time-dependent second order perturbation to a degenerate Ornstein-Uhlenbeck type operator whose diffusive part can be either local or non-local. More precisely, we establish that some estimates, such as the Schauder and Sobolev ones, already known for the non-perturbed operator still hold, and with the same constants, when we perturb the Ornstein-Uhlenbeck operator with second order diffusions with coefficients only depending on time in a measurable way. The aim of the current work is twofold: we weaken the assumptions required on the perturbation in the local case which has been considered already in [KP17] and we extend the approach presented therein to a wider class of degenerate Kolmogorov operators with non-local diffusive part of symmetric stable type., Comments are welcome!
Published: 2023

48. Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness

Author: Chamorro, Diego, Llerena, David, Vergara-Hermosilla, Gastón, Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), and Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)
Subjects: Micropolar fluid equations, Mathematics - Analysis of PDEs, Liouville type theorems, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Analysis of PDEs (math.AP)
Abstract: We consider here the stationary Micropolar fluid equations which are a particular generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one based mainly in the velocity field u and the other one based in the microrotation field $\omega$. We will study in this work some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establish, under some suitable decay at infinity conditions for the velocity field only, the uniqueness of the trivial solution.
Published: 2023

49. IsoGeometric LaTIn method for non-conformal coupling with non-linear interface behaviour

Author: Lapina, Evgeniia, Oumaziz, Paul, Bouclier, Robin, Institut Clément Ader (ICA), Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-IMT École nationale supérieure des Mines d'Albi-Carmaux (IMT Mines Albi), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
Subjects: [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract: International audience; With the development of volume imaging techniques, mechanical models of a material at the microstructure scale can be created from 3D images of a sample. However, the computation of the image-based models still seems unattainable for real mechanical applications due to massive data and complex constitutive behaviours. The aim of this work is to develop an efficient numerical approach for the simulation of solids with multiple non-linear, non-conformal interfaces. Of special interest are fiber-reinforced composite materials, where local models of fibers are coupled with the global matrix model. Immersed boundary methods are used in the framework of IsoGeometric Analysis (IGA) to tackle the complex geometry of multi-phase objects, resulting in improved per-degree-of-freedom accuracy by using spline basis functions and alleviating meshing process. Since the non-linearities are located at the interfaces, we aim to take advantage of the Large Time Increment (LaTIn) method. Here, the LaTIn method separates the non-linear local and the global linear equations and applies an iterative scheme between both. LaTIn-based solvers have been applied, for instance, with a FE immersed approach (CutFEM), to model multiple contacts. The present work extends the LaTIn to higher-order immersed IGA. The non-conformal coupling is treated by Nitsche’s approach, which has shown optimal convergence behaviour, particularly for global/local analysis. In the proposed method, two conformal layers are built on the matrix/fibre interface, thus separating the difficulties regarding non-linearity and non-conformity. The LaTIn method is used to couple the layers, incorporating the non-linear behaviour, and Nitsche's method is then used to couple the layers with the geometrically simple, non-conformal matrix or fibre models. The method has been tested through multiple numerical examples, including unilateral and friction contacts, and examples with cohesive interfaces will also be presented.
Published: 2023

50. Optimal control for a two-sidedly degenerate aggregation equation

Author: Bendahmane, Mostafa, Karami, Fahd, Erraji, Elmahdi, Atlas, Abdelghafour, Afraites, Abdelkbir, Modélisation et calculs pour l'électrophysiologie cardiaque (CARMEN), 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)-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)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-IHU-LIRYC, Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-CHU Bordeaux [Bordeaux], 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), Ecole Supérieure de Technologie d'Essaouira, Université Cadi Ayyad [Marrakech] (UCA), Ecole Nationale des Sciences Appliquées [Marrakech] (ENSA), and Université Sultan Moulay Slimane (USMS )
Subjects: Nonlocal models, Aggregation equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Degenerate diffusion, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Finite volume, Adjoint problem, Optimal control
Abstract: International audience; In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the well-posedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo-Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and first order necessary conditions. The main novelty of this work concern the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations.
Published: 2023

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