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Diffusion non linéaire et non locale : analyse variationnelle et numérique
No full textThis thesis focuses on partial differential problems related to porous medium and fast diffusion equations. A significant part of the thesis is dedicated to the Cauchy-Dirichlet problem for fractional porous medium and fast diffusion equations, where the Laplacian is replaced by a fractional one. Using energy methods, we establish the well-posedness of the problem within a class of so-called energy solutions that satisfy certain energy inequalities. The proofs are based on a variational approach and do not require any sign restriction. In the porous medium case, we show that solutions decay at an algebraic rate, while in the fast diffusion case, a finite-time extinction phenomenon is established. Moreover, we show the convergence of energy solutions to asymptotic profiles after a suitable rescaling, using a Łojasiewicz-Simon inequality for the fractional Laplacian. Finally, the numerical analysis of the problem is addressed. We propose a numerical scheme and show that it preserves the main properties of the continuous equation, such as the algebraic decay rate in the fractional porous medium case and the finite time extinction phenomenon in the fractional fast diffusion case. Additionally, we develop a method to accurately compute the extinction time for the fractional fast diffusion equation. The convergence of rescaled solutions to asymptotic profiles near the extinction time is investigated numerically, along with the convergence rate.The second problem addressed in this thesis is a Keller-Segel system where both species diffuse according to a porous medium or fast diffusion equation, with a sensitivity depending on the concentration of the chemical signal. This system has been extensively studied in the case of constant sensitivity and linear diffusion for the chemical signal, where a critical mass phenomenon emerges for certain values of the exponent associated with the degenerate diffusion of the first species. By combining optimal transport arguments and energy methods, we show that the critical mass phenomenon extends to the case of nonlinear diffusion for the chemical species and non-constant sensitivity, for specific parameter choices.The final problem concerns a nonlinear diffusion equation posed on the boundary of a domain. Using the Dirichlet-to-Neumann operator, this problem can be viewed as a fractional porous medium or fast diffusion equation on a manifold. Using the energy methods developed for the previous problems, we establish the well-posedness of the equation within a class of energy solutions without any sign restriction.Le travail de cette thèse porte sur divers problèmes au dérivées partielles en lien avec les équations des milieux poreux et de diffusion rapide. Une large partie de la thèse se concentre sur le problème de Cauchy-Dirichlet des équations des milieux poreux et de diffusion rapide fractionnaire, dans lesquelles le Laplacien est remplacé par un Laplacien fractionnaire. En s'appuyant des méthodes d'énergie, nous montrons le caractère bien posé du problème dans une classe de solutions dites énergétiques, satisfaisant certaines inégalités d'énergie. Les démonstrations reposent sur une approche variationnelle et ne nécessitent aucune restriction de signe. Dans le cas des milieu poreux, nous montrons la décroisse des solutions à un taux algébrique, tandis que dans le cas de la diffusion rapide, le phénomène d'extinction en temps fini est démontré. Nous montrons ensuite la convergence des solutions énergétiques vers des profils asymptotiques après un redimensionnement approprié, en se basant sur une inégalité de Łojasiewicz-Simon pour le Laplacien fractionnaire. Enfin, l'analyse numérique du problème est abordée. Nous proposons un schéma numérique et montrons qu'il préserve les principales propriétés de l'équation continue, telles que le taux de décroissance algébrique dans le cas du milieu poreux fractionnaire et le phénomène d'extinction en temps fini dans le cas de la diffusion rapide fractionnaire. De plus, nous proposons une méthode permettant de calculer avec précision le temps d'extinction pour l'équation de diffusion rapide fractionnaire. La convergence des solutions redimensionnées vers des profils asymptotiques près du temps d'extinction est illustrée numériquement, ainsi que le taux de convergence.Le second problème abordé dans cette thèse est un système de Keller-Segel dans lequel les deux espèces diffusent selon une équation des milieux poreux ou de diffusion rapide, et avec une sensitivité dépendante de la concentration du signal chimique. Ce système a fait l'objet de nombreuses études dans le cas d'une sensitivité constante et pour une diffusion linéaire du signal chimique, où apparait un phénomène de masse critique pour une certaine valeur de l'exposant associé à la diffusion dégénérée de la première espèce. En combinant des arguments de transport optimal et des méthodes d'énergie, nous montrons que le phénomène de masse critique s'étend au cas d'une diffusion non linéaire de l'espèce chimique et d'une sensitivité non constante, pour certains choix de paramètres.Le dernier problème abordé est une équation de diffusion non linéaire posée sur la frontière d'un domaine. Par l'intermédiaire de l'opérateur de Dirichlet-Neumann, elle peut être vue comme une équation des milieux poreux ou de diffusion rapide fractionnaire posée une variété. En se basant sur les méthodes d'énergies mise en œuvre sur les problèmes précédant, nous montrons le caractère bien posé de l'équation dans une classe de solutions énergétiques sans restriction de signe
Scattering theory and an index theorem on the radial part of SL(2,R)
No full textInternational audienceWe present the spectral and scattering theory of the Casimir operator acting on the radial part of SL(2,R). After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Moeller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality links various asymptotic behaviors of the hypergeometric function
Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
No full textInternational audienceWe consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator is known to be essentially self-adjoint. We define complex powers by functional calculus, and show that the trace density exists as a meromorphic function of . We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine-Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments operate in Lorentzian signature directly and do not rely on a transition from the Euclidean setting. The results hold also true in the case of ultrastatic spacetimes
About Berge-Füredi's conjecture on the chromatic index of hypergraphs
No full textInternational audienceWe show that the chromatic index of a hypergraph satisfies{Berge-F\"uredi} conjectured bound under certain hypotheses on the antirank or on the maximum degree . This provides sharp information in connection with {Erd\H{o}s-Faber-Lov\'asz} Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex
A duality operators/Banach spaces
No full textInternational audienceGiven a set of operators between subspaces of spaces, we characterize the operators between subspaces of spaces that remain bounded on the -valued space for every Banach space on which elements of the original class are bounded. This is a form of the bipolar theorem for a duality between the class of Banach spaces and the class of operators between subspaces of spaces, essentially introduced by Pisier. The methods we introduce allow us to recover also the other direction --characterizing the bipolar of a set of Banach spaces--, which had been obtained by Hernandez in 1983
Analytical derivation of delayed prey-predator model with hunting-and-resting delay
No full textInternational audienceWe investigate a prey-predator model based on a general Gause type system. We take for the predator two phases into account, the hunting phase and the resting one. We suppose that the predators stop hunting after they catch the prey. Then they enter the resting phase where they stay for a fixed limited time. The resulting mathematical model is a system of two age-structured partial differential equations. By integrating this system over age and using the characteristics method, we reduce it to a delay differential system, and we investigate the existence and stability of the steady states. In particular, we have shown that the introduction of the delay (the duration of the resting phase) stabilizes the coexistence equilibrium
Optimal control of an impulsive VS-EIAR epidemic model with applications to COVID-19
No full textInternational audienceIn this paper, we investigate a VS-EIAR epidemiological model that incorporates vaccinated individuals \{V_i : i = 1, \ldots, n\}, where n\in\mathbb{N}^{*}. The dynamics of the VS-EIAR model are governed by a system of ordinary differential equations describing the evolution of vaccinated, susceptible, exposed, infected, asymptomatic, and deceased population groups. Our primary objective is to minimize the number of susceptible, exposed, infected, and asymptomatic individuals by administering vaccination doses to susceptible individuals and providing treatment to the infected population. To achieve this, we employ optimal control theory to regulate the epidemic dynamics within an optimal terminal time \tau^{*}. Using Pontryagin’s Maximum Principle (PMP), we establish the existence of an optimal control pair (v^{*}(t), u^{*}(t)). Additionally, we extend the model to an impulsive VS-EIAR framework, with particular emphasis on the impact of immigration and population movement. Finally, we present numerical simulations to validate the theoretical results and demonstrate their practical applicability
One-arm exponents of the high-dimensional Ising model
No full textWe study the probability that the origin is connected to the boundary of the box of size n (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality:- For the FK-Ising measure in a box of size n with wired boundary conditions, we prove that this probability decays as 1/n in dimensions d>4, and as 1/n1+o(1) when d=4.- For the infinite volume FK-Ising measure, we prove that this probability decays as 1/n2 in dimensions d>6, and as 1/n2+o(1) when d=6.- For the sourceless double random current measure, we prove that this probability decays as 1/nd−2 in dimensions d>4, and as 1/n2+o(1) when d=4.Additionally, for the infinite volume FK-Ising measure, we show that the one-arm probability is 1/n1+o(1) in dimension d=4, and at least 1/n3/2 in dimension d=5. This establishes that the FK-Ising model has upper-critical dimension equal to 6, in contrast to the Ising model, where it is known to be less or equal to 4, thus solving a conjecture of Chayes, Coniglio, Machta, and Shtengel
Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction
No full textThis paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the -dimensional persistent homology (PH) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH for . Based on this observation, we introduce a novel generalization of TopoAE to -dimensional persistent homology (PH), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the -chains filling persistent -cycles, hence resulting in more faithful geometrical reconstructions of the -cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus