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The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.

International audience; Fat cells, called adipocytes, are designed to regulate energy homeostasis by storing energy in the form of lipids. Adipocyte size distribution is assumed to play a role in the development of obesity-related diseases. This population of cells that do not have a characteristic size, indeed a bimodal size distribution is observed in adipose tissue. We propose a model based on a partial differential equation to describe adipocyte size distribution. The model includes a description of the lipid fluxes and the cell size fluctuations and using a formulation of a stationary solution fast computation of bimodal distribution is achieved. We investigate the parameter identifiability and estimate parameter values with CMA-ES algorithm. We first validate the procedure on synthetic data, then we estimate parameter values with experimental data of 32 rats. We discuss the estimated parameter values and their variability within the population, as well as the relation between estimated values and their biological significance. Finally, a sensitivity analysis is performed to specify the influence of parameters on 1 cell size distribution and explain the differences between the model and the measurements. The proposed framework enables the characterization of adipocyte size distribution with four parameters and can be easily adapted to measurements of cell size distribution in different health conditions.

We address a problem of stabilization of a scalar conservation laws with fixed flux pointwiseconstraints. The PDE models the impact of two toll gates on the traffic flow and the feedbacklaws is designed to limit the number of cars passing through the two toll gates. The existenceof solutions is proved using a Schauder’s fixed point theorem, together with a coupled PDEODE. To prove stabilization results, we study the structure of the solutions using an extensionof backward generalized characteristics.

In this work, we analyse the metastability of non-reversible diffusion processes $$dX_t=\boldsymbol{b}(X_t)dt+\sqrt h\,dB_t$$ on a bounded domain $\Omega$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell})$ and $\nabla f \cdot \mathbf{\ell}=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t)_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partial\Omega$ of $\Omega$ is exponentially close to the inverse of the mean exit time from $\Omega$, uniformly in the initial conditions $X_0=x$ within the compacts of $\Omega$. The asymptotic behavior of the law of the exit time in this limit is also obtained. The main novelty of these first results follows from the consideration of non-reversible elliptic diffusions whose associated dynamical systems $\dot X=\mathbf{b}(X)$ admit equilibrium points on $\partial\Omega$. In a second time, when in addition $\div \mathbf{\ell} =0$, we derive a new sharp asymptotic equivalent in the limit $h\to 0$ of the principal eigenvalue of the generator of the process and of its mean exit time from $\Omega$. Our proofs combine tools from large deviations theory and from semiclassical analysis, and truly relies on the notion of quasi-stationary distribution.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classification all possible behaviour for $p_n(e, e)$ on such groups.; Nous étudions les marches aléatoires sur des groupes relativement hyperboliques dont la loi est convergente, au sens où la dérivée de sa fonction de Green est finie au rayon spectral.Lorsque les sous-groupes paraboliques sont virtuellement abéliens, nous montrons qu'une telle marche vérifie un théorème de la limite locale de la forme $p_n(e, e)\sim CR^{-n}n^{-d/2}$, où $p_n(e, e)$ est la probabilité de retour à l'origine au temps $n$, $R$est l'inverse du rayon spectral de la marche et $d$ est le rang minimal d'un sous-groupe parabolique le long duquel la marche aléatoire est spectralement dégénérée.

International audience; We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

International audience; Building on the idea of Tolman and Ehrenfest that heat has weight, Luttinger established a deep connection between gravitational fields and thermal transport. However, this relation does not include anomalous quantum fluctuations that become paramount in strongly curved spacetime. In this work, we revisit the celebrated Tolman-Ehrenfest and Luttinger relations and show how to incorporate the quantum energy scales associated with these fluctuations, captured by gravitational anomalies of quantum field theories. We point out that such anomalous fluctuations naturally occur in the quantum atmosphere of a black hole. Our results reveal that analogous fluctuations are also observable in thermal conductors in flat-space time provided local temperature varies strongly. As a consequence, we establish that the gravitational anomalies manifest themselves naturally in non-linear thermal response of a quantum wire. In addition, we propose a systematic way to identify thermal analogues of black hole's anomalous quantum fluctuations associated to gravitational anomalies. We identify their signatures in propagating energy waves following a thermal quench, as well as in the energy density of heating Floquet states induced by repeated thermal quenches.

International audience; Enteric pathogens colonize the digestive tract of farm livestock, such as chickens or pigs and potentially infect food products, which results in a threat for human health and important economic losses. It has been shown that the ability to excrete the pathogen in the environment and contaminate other animals is variable. This heterogeneity in pathogen carriage and shedding results from interactions between the host’s immune response, the pathogen and the commensal intestinal microbiota and plays and important role in epidemic propagation and control. The objective here is to propose a theoretical model of pathogen excretion and transmission linking within and between host dynamic.

International audience; The Gauss map g of a surface ⌃ in R 4 takes its values in the Grassmannian of oriented 2-planes of R 4 : G + (2, 4). We give geometric criteria of stability for minimal surfaces in R 4 in terms of g. We show in particular that if the spherical area of the Gauss map |g(⌃)| of a minimal surface is smaller than 2⇡ then the surface is stable by deformations which fix the boundary of the surface. This answers the question of [BDC3] in R 4 .

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $\Omega \subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the saddle points of $f$ on $\partial \Omega$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function., 42 pages, 8 figures

We present a spatial model describing the growth of a photosynthetic microalgae biofilm. In this 2D-model we consider photosynthesis, cell carbon accumulation, extracellular matrix excretion, and mortality. The rate of each of these mechanisms is given by kinetic laws regulated by light, nitrate, oxygen and inorganic carbon. The model is based on mixture theory and the behaviour of each component is defined on one hand by mass conservation, which takes into account biological features of the system, and on the other hand by conservation of momentum, which expresses the physical properties of the components. The model simulates the biofilm structural dynamics following an initial colonization phase. It shows that a 75 μm thick active region drives the biofilm development. We then determine the optimal harvesting period and biofilm height which maximize productivity. Finally, different harvesting patterns are tested and their effect on biofilm structure are discussed. The optimal strategy differs whether the objective is to recover the total biofilm or just the algal biomass.

In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the ZigZag process and the Bouncy Particle Sampler process) with refreshment rate $\alpha$ on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate $\alpha$ vanishes on T is also considered.

We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions., 45 pages, 3 figures

The paper is concerned with a new approach for the recurrence property of the oscillating process on Z in Kemperman's sense. In the case when the random walk is ascending on Z − and descending on Z + , we determine the invariant measure of the embedded process of successive crossing times and then prove a necessary and sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating process is recurrent under some Hölder-typed moment assumptions.

This work is dedicated to the analysis of a class of energy stable and linearly well-balanced numerical schemes dedicated to the non-linear Shallow Water equations with Coriolis force. The proposed algorithms rely on colocated finite volume approximations formulated on cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semi-discrete energy estimates and numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes.

43 pages, 4 fig.; We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

We present new exact results for the complex generalized integral means spectrum (in the sense of [DHLZ18]) for two kinds of whole-plane Loewner evolutions driven by a Lévy process: (1) The case of a Lévy process with continuous trajectories, which corresponds to Schramm-Loewner evolution SLE κ with a drift term in the Brownian driving function. There is no known result for its standard integral means spectrum, and we show that a natural path to access it goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. (2) The case of symmetric Lévy processes for which we generalize results by Loutsenko and Yermolayeva ([Lou12, LY13, LY14, LY19]).

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate \emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior.As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support., 26 pages

Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra., 26 pages

International audience; We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a Uq(sln) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their Uq(sl3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the effective central charge reveals that for q=−eiγ, in the range γ∈[0,π/3], the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense q=−eiπ/4 case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a Z3 spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the q=−eiπ/3 case, which is the n=3 analogue of the polymer limit in the loop model.

We consider the first exit point distribution from a bounded domain $$\Omega $$ of the stochastic process $$(X_t)_{t\ge 0}$$ solution to the overdamped Langevin dynamics $$\begin{aligned} d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t \end{aligned}$$ starting from deterministic initial conditions in $$\Omega $$ , under rather general assumptions on f (for instance, f may have several critical points in $$\Omega $$ ). This work is a continuation of the previous paper [14] where the exit point distribution from $$\Omega $$ is studied when $$X_0$$ is initially distributed according to the quasi-stationary distribution of $$(X_t)_{t\ge 0}$$ in $$\Omega $$ . The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

Within the Russo-Susskind-Thorlacius (RST) two-dimensional model that includes a scalar (dilaton) field we address the important question of how the classical black hole geometry is modified in a semiclassical gravitational theory. It is the principle goal of this paper to analyze what is the back-reacted geometry that corresponds to a given quantum state. The story is shown to be dramatically different for the Hartle-Hawking state (HH) and for the Boulware state. In the HH case the back-reacted geometry is a modification of the classical black hole metric that still has a smooth horizon with a regular curvature. On the other hand, for the Boulware state the classical horizon is replaced by a throat in which the $(tt)$ component of the metric (while non-zero) is extremely small. The value of the metric at the throat is bounded by the inverse of the classical black hole entropy. On the other side of the throat the spacetime is ended at a null singularity. More generally, we identify a family of quantum states and their respective back-reacted geometries. We also identify a certain duality in the space of states. Finally, we study a hybrid set-up where both physical and non-physical fields, such as the ghosts, could be present. We suggest that it is natural to associate ghosts with the Boulware state, while the physical fields can be in any quantum state. In particular, if the physical fields are in the HH state, then the corresponding semiclassical geometry is horizonless. Depending on the balance between the number of physical fields and ghosts, it generically has a throat that may join with another asymptotically flat region on the other side of the throat., Comment: 38 pages, 7 figures. Version 2. Published version. Minor typos corrected

Recent experiments [Grinenko $\textit{et al.}~\href{http://doi.org/10.1038/s41567-021-01350-9}{\text{Nature Physics}~\textbf{17},~1254\!-\!1259~(2021)}$] reported the observation of a condensate of four-fermion composites. This is a resistive state that spontaneously breaks the time-reversal symmetry, leading to unconventional magnetic properties, detected in muon spin rotation experiments and by the appearance of a spontaneous Nernst effect. In this work, we derive an effective model that describes the observed spontaneous magnetic fields in this state. We show that this model, which is alike to the Faddeev-Skyrme model for the four-fermion order parameter, hosts skyrmions: magnetic-flux-carrying topological excitations.

Consider an Galton Watson tree of height $m$: each leaf has one of $k$ opinions or not. In other words, for $i \in \{1, . . . , k\}$, $x$ at generation $m$ thinks $i$ with probability $p_i$ and nothing with probability $p_0$. Moreover the opinions are independently distributed for each leaf. Opinions spread along the tree based on a specific rule: the majority wins! In this paper, we study the asymptotic behavior of the distribution of the opinion of the root when $m \to +\infty$., Comment: 30 pages, 13 figures, 1 table

We establish a large deviation principle for time dependent trajectories (paths) of the empirical density of $N$ particles with long range interactions, for homogeneous systems. This result extends the classical kinetic theory that leads to the Balescu--Guernsey--Lenard kinetic equation, by the explicit computation of the probability of typical and large fluctuations. The large deviation principle for the paths of the empirical density is obtained through explicit computations of a large deviation Hamiltonian. This Hamiltonian encodes all the cumulants for the fluctuations of the empirical density, after time averaging of the fast fluctuations. It satisfies a time reversal symmetry, related to the detailed balance for the stochastic process of the empirical density. This explains in a very simple way the increase of the macrostate entropy for the most probable states, while the stochastic process is time reversible, and describes the complete stochastic process at the level of large deviations., Comment: arXiv admin note: text overlap with arXiv:2101.04455

We discuss the prediction of critical behavior of lattice observables in SU(2) and SU(3) gauge theories. We show that feed-forward neural network, trained on the lattice configurations of gauge fields as input data, finds correlations with the target observable, which is also true in the critical region where the neural network has not been trained. We have verified that the neural network constructs a gauge-invariant function and this property does not change over the entire range of the parameter space., 7 pages, 2 figures, 1 table, contribution to the proceedings of XXXIII International (ONLINE) Workshop on High Energy Physics "Hard Problems of Hadron Physics: Non-Perturbative QCD & Related Quests" November 8-12, 2021, Submission to SciPost

The relation between thermal transport and gravity was highlighted in the seminal work by Luttinger in 1964, and has been extensively developed to understand thermal transport, most notably the thermal Hall effect. Here we review the novel concepts that relate thermal transport, the geometry of space-time and quantum field theory anomalies. We give emphasis to the cross-pollination between emergent ideas in condensed matter, notably Weyl and Dirac semimetals, and the understanding of gravitational and scale anomalies stemming from high-energy physics. We finish by relating to recent experimental advances and presenting a perspective of several open problems., 68 pages, 13 figures; comments welcome!

Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits and we show that each $Z$-orbit has a structure of algebraic affine bundle. In particular, in type $A$, we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type $A$, we show that the Bruhat order of the $Z$-orbits can be described in this way., 42 pages, 2 figures

We perform the analysis of the focusing nonlinear Schr\"odinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of B\"acklund transformations. The difficulty arising from having such time-dependent boundary conditions at $x=0$ is overcome by changing the viewpoint of the method and fixing the B\"acklund transformation at infinity as well as relating its value at $x=0$ to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line. The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions., Comment: 38 pages. 5 figures. Authors' version of the published article. Accessiblefrom https://onlinelibrary.wiley.com/doi/10.1111/sapm.12456

We study properties of nonnegative functions satisfying (E)$\;-\Delta u+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1, Comment: 83 pages, 5 figures

In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(\mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(\mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.

We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A_2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their U_q(sl_3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the central charge reveals that for each q $\in \mathbb{C}$ in the critical regime, |q|=1, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $q=-e^{i \pi/4}$ case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a $\mathbb{Z}_3$ spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $q=-e^{i \pi/3}$ case, which is the n=3 analogue of the polymer limit in the loop model.

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

We study the variation of the mean cross section with the density of the samples in the quantum scattering of a particle by a disordered target. The target consists of a set of pointlike scatterers, each having an equal probability of being anywhere inside a sphere whose radius may be modified. We first prove that scattering by a pointlike scatterer is characterized by a single phase shift ${\delta}$ which takes on its values in $]0 \, , {\pi}[$ and that the scattering by ${\rm N}$ pointlike scatterers is described by a system of only ${\rm N}$ equations. We then show with the help of numerical calculations that there are two stages in the variation of the mean cross section as the density of the samples (the radius of the target) increases (decreases). Depending on the value of ${\delta}$, the mean cross section first either increases or decreases, each one of the two behaviours being originated by double scattering; it decreases uniformly for any value of ${\delta}$ as the density increases further on, a behaviour which results from multiple scattering and which follows that of the cross section for diffusion by a hard sphere potential of decreasing radius. The expression of the mean cross section is derived in the particular case of an unlimited number of contributions of successive scatterings., Comment: 26 pages, 6 figures

We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switched environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some K $\in$ N * , and the number of infectives vanishes almost surely in finite time. We prove that, as K $\rightarrow$ $\infty$, the process composed of the proportions of infectives of each type X^K and the state of the environment $\Xi$^K , converges to a piecewise deterministic Markov process (PDMP) given by a system of randomly switched ODEs. The long term behaviour of this PDMP has been previously investigated by Bena{\"i}m and Strickler, and depends only on the sign of the top Lyapunov exponent $\Lambda$ of the linearised PDMP at 0: if $\Lambda$ < 0, the proportion of infectives in each group converges to zero, while if $\Lambda$ > 0, the disease becomes endemic. In this paper, we show that the large population asymptotics of X^K also strongly depend on the sign of $\Lambda$: if negative, then from fixed initial proportions of infectives the disease disappears in a time of order at most log(K), while if positive, the typical extinction time grows at least as a power of K. We prove that in the situation where the origin is accessible for the linearised PDMP, the mean extinction time of X^K is logarithmically equivalent to K^p * , where p * > 0 is fully characterised. We also investigate the quasi-stationary distribution $\mu$^K of (X^K , $\Xi$^K) and show that, when $\Lambda$ < 0, weak limit points of ($\mu$^K), K>0 are supported by the extinction set, while when $\Lambda$ > 0, limit points belong to the (non empty) set of stationary distributions of the limiting PDMP which do not give mass to the extinction set.

This thesis is devoted to the treatment of heterogeneity in scalar conservation laws. We call heterogeneous a conservation law which is not invariant by space translation. These equations arise for instance in traffic flow dynamics modeling. The presence of traffic lights or roads that have a variable maximum speed limit are examples of mechanisms which lead to heterogeneous conservation laws. Considering such equations is a way to expand macroscopic traffic flow models. We tackle three classes of inhomogeneous problems for which we extend the mathematical framework for both the theoretical analysis and the numerical approximation. We fully investigate the treatment of heterogeneity when one or several moving interfaces are added in the classic LWR model for traffic flow. Flux constraints are attached to each interfaces. The resulting class of models can be used to take into account the presence of slow moving vehicles that reduce the road capacity and thus generates moving bottlenecks for the surrounding traffic flow. They can also describe the regulating effect of autonomous vehicles. In this framework, the interfaces and the constraints are linked in a nonlocal way to the traffic density and/or to an orderliness marker describing the state of the drivers. The description of the heterogeneity caused by the variations in the drivers' organization leads to the analysis of a so-called second order model. The numerical aspect plays a central role in the analysis of these traffic flow models. We construct robust numerical schemes and establish specific techniques to obtain compactness of the approximate solutions. Proving the convergence of these schemes lead to existence results.Finally, with the space-dependent LWR traffic flow model in mind, we theoretically analyze a class of scalar conservation laws with explicit space dependency. Classical results such as well-posedness or the link to the associated Hamilton-Jacobi equation are obtained under a set of assumptions more fitting with the modeling hypothesis. With applications that go beyond traffic modeling in mind, we aim to tackle initial data identification problems., Dans cette thèse, on traite la prise en compte de l'hétérogénéité dans les lois de conservation scalaires, c'est-à-dire les lois de conservation non invariantes par translation en espace. Ces équations apparaissent notamment dans les modèles de trafic. Par exemple, les mécanismes suivants introduisent de l'hétérogénéité : la présence de feux de circulation, des portions de route où la vitesse maximale est limitée, la variabilité de l'état de la route, etc... La prise en compte de l'hétérogénéité permet d'enrichir les modèles de trafic. On aborde trois classes de problèmes inhomogènes pour lesquelles on complète et approfondit le cadre mathématique pour l'analyse théorique et l'approximation numérique.Nous explorons en détail le cadre où l'hétérogénéité est matérialisée par l'ajout d'une ou plusieurs interfaces mobiles. Le long des interfaces, on impose une condition de majoration sur le flux de la loi de conservation. Cette classe de modèles permet de tenir compte de la présence d'un petit nombre de véhicules encombrants et lents (ou alors, de véhicules autonomes qui ont pour rôle la régulation du trafic). Dans ce cadre, l'évolution des interfaces et des contraintes est couplée de façon non locale à l'état du trafic et/ou à des paramètres spécifiant l'état du véhicule ou du conducteur. En outre, nous élaborons une description de l'hétérogénéité du trafic résultant des variations du degré d'organisation des conducteurs, dans le cadre des modèles dits "du second ordre". L'aspect numérique est prépondérant pour les modèles de trafic que nous étudions. On construit des schémas numériques robustes et on élabore des techniques de compacité spécifiques. La convergence de ces schémas conduit à des résultats d'existence.Enfin, en lien avec le modèle décrivant l'évolution d'une densité de véhicules sur une route hétérogène, on étudie théoriquement une loi de conservation dans laquelle la dépendance spatiale du flux est explicite. Des résultats classiques sur le caractère bien posé ou la correspondance avec l'équation de Hamilton-Jacobi associée sont obtenus sous des hypothèses plus en adéquation avec la modélisation que celles rencontrées dans la littérature. Les applications allant au-delà de la description du trafic, on se donne pour objectif l'analyse approfondie des problèmes d'identification de données initiales.

28 pages, 16 ref.; We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter $1-\epsilon$. These passage times induce a random pseudo-metric $T_\epsilon$ on $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_\epsilon (0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $\mu_\epsilon (z)$ called the time constant. We investigate the behavior of $\epsilon \mapsto \mu_\epsilon (z)$ near $0$, and prove that $\mu_\epsilon (z) = \| z\|_1 - C (z) \epsilon ^{1/d_1(z)} + o ( \epsilon ^{1/d_1(z)}) $, where $d_1(z)$ is the number of non null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is partially explicit.

19 pages; The equitable presentation of the quantum algebra Uq(sl 2) is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of Uq(sl 2) of equitable type U IT,+ q and its second realization (current algebra) U T,+ q are introduced and studied. A presentation for U T,+ q is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from U T,+ q to a subalgebra of Drinfeld's second realization (current algebra) of Uq(sl 2) is derived, and the comodule algebra structure of U T,+ q is characterized. The central extension of U T,+ q and its relation with Drinfeld's second realization of Uq(gl 2) is also described using the framework of Freidel-Maillet algebras.

This thesis focuses on the study of the $\tau$-coherence of an high-dimensional $(n \times p)$-observation matrix with $p>>n$, where $n$ is the number of individuals and $p$ the number of variables. The $\tau$-coherence is defined as the largest magnitude of the entries of the empirical correlation matrix outside a central band (with a bandwith $\tau$). The first chapter is devoted to the presentation of the Chen-Stein method, which is an approximation of weakly dependent events by a Poisson distribution, and to some bibliography concerning coherence. The second and third chapter focus on the limiting behaviour of $\tau$-coherence in a case where the observations are assumed to be Gaussian with bandwise (resp.blockwise) covariance in Chapter 2 (resp. Chapter 3). In the last chapter, we propose a Monte-Carlo simulation procedure allowing us to study numerically the limiting distribution of the $\tau$-coherence for large (Big Data) matrices. We use a splitting strategy of our matrices and HPC method such as GPGPU computation in order to, from one side, being able to compute correlation matrices even if they are too large to be loaded in a computer, and on the other side, to reduce computation time. Finally the appendix is devoted to some technical results.\\; Cette thèse concerne l'étude de la $\tau$-cohérence d'une matrice d'observations aléatoires de grande taille $(n \times p)$ où $p >> n$ et avec $p$ le nombre de variables observées sur $n$ individus. La $\tau$-cohérence est alors définie comme étant le maximum, en valeur absolue, des coefficients de la matrice de corrélation empirique associée, en dehors d'une bande centrale de largeur $\tau$. Le premier chapitre est consacré à la présentation de la méthode de Chen-Stein qui permet l'approximation d'événements faiblement dépendants par une loi de Poisson et à la présentation des travaux de T. Cai et T. Jiang concernant la cohérence. Les deuxième et troisième chapitres sont consacrés à l'étude du comportement asymptotique de la $\tau$-cohérence dans le cas où les observations proviennent d'un modèle gaussien et où la matrice de covariance possède une structure par bandes (chapitre 2) ou par blocs (chapitre 3). Dans le chapitre 4, nous présentons une méthode de simulation par réplications Monte-Carlo pour étudier numériquement la distribution asymptotique de la $\tau$-cohérence. Nous utilisons des stratégies de découpage de nos matrices et des teThis thesis focuses on the study of the $\tau$-coherence of an high-dimensional $(n \times p)$-observation matrix with $p>>n$, where $n$ is the number of individuals and $p$ the number of variables. The $\tau$-coherence is defined as the largest magnitude of the entries of the empirical correlation matrix outside a central band (with a bandwith $\tau$). The first chapter is devoted to the presentation of the Chen-Stein method, which is an approximation of weakly dependent events by a Poisson distribution, and to some bibliography concerning coherence. The second and third chapter focus on the limiting behaviour of $\tau$-coherence in a case where the observations are assumed to be Gaussian with bandwise (resp.blockwise) covariance in Chapter 2 (resp. Chapter 3). In the last chapter, we propose a Monte-Carlo simulation procedure allowing us to study numerically the limiting distribution of the $\tau$-coherence for large (Big Data) matrices. We use a splitting strategy of our matrices and HPC method such as GPGPU computation in order to, from one side, being able to compute correlation matrices even if they are too large to be loaded in a computer, and on the other side, to reduce computation time. Finally the appendix is devoted to some technical results.\\chniques HPC telles que le calcul en GPU pour, d'une part pouvoir calculer des corrélations sur des matrices trop grandes pour être stockées, et d'autre part réduire le temps de calcul. Nous présentons en annexe de ce manuscrit des éléments de preuves supplémentaires.\\

International audience; This work aims at modeling transport and sedimentation of particles in a water flow. A new erosionsedimentation model has been introduced in [14] considering the concentrations of sediment suspended in the fluid or deposited on an exchange layer. Based on conservation laws, it is composed, for each class of particle, of a transport equation on the suspended material, coupled with the exchange between deposited particles and suspended material. A benefit provided by this approach is its ability to replicate other well-known erosion models and thus to model various sedimentation processes and materials. This new model has been implemented into the FullSWOF_1D (Full Shallow Water for Overland Flow in one Dimension) software [5] originally designed to solve the Shallow Water equations. The modified software is now able to simulate various erosion situations and was tested against several test cases. In order to check the model even more, we felt important to be able to replicate a laboratory experiment. We chose the sedimentation experiment described in [13] where the transport and deposition of particles by a water flow were studied. The deposited layer and the flux of material at the outlet were measured. Thanks to the physical parameters, and with a calibration limited to settling velocities, we were able to obtain results close to the laboratory measurements. We also propose some clues to improve the results and the model.

International audience; A unified framework for simulating various transport processes in the environment is presented. It consists in a single set of partial differential equations. The main feature of this model framework is its exchange layer, which allows to treat several types of transfer between the soil and the surface water. The model framework equations, termed transfer equations, is shown to reproduce three independently-published models developed for soil erosion, river bedload, and chemical transport respectively. By allowing the different processes to be represented within a single model framework, the transfer equations are therefore unifying the representation of particles and chemical fluxes in the environment. The transfer equations are implemented into the open-source software FullSWOF_1D. The code is verified against the approximation of an exact solution, assuring its proper functioning. A good adequacy is found between our numerical results and those published in the literature, attesting the capability of the transfer equations to unify modeling of soil erosion, river bedload, and chemical transport. Hence, the transfer equations can decrease the number of models to be used for simulating transfer of materials in the environment, and limit the number of computer codes to be developed and maintained. The transfer equations could also help in drawing parallels between different fields of hydrology.

International audience; We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.

15 pages, 19 ref.; International audience; We prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies with a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.

A general theory for competitive dynamics among many strains at the epidemiological level is required to understand polymorphisms in virulence, transmissibility, antibiotic resistance and other biological traits of infectious agents. Mathematical coinfection models have addressed specific systems, focusing on the criteria leading to stable coexistence or competitive exclusion, however, due to their complexity and nonlinearity, analytical solutions in coinfection models remain rare. Here we study a 2-strain SIS compartmental model with co-infection/co-colonization, incorporating multiple fitness dimensions under the same framework: variation in transmissibility, duration of carriage, pairwise susceptibilities to coinfection, coinfection duration, and transmission priority effects from mixed coinfection. Taking advantage of a singular perturbation approach, under the assumption of strain similarity, we expose how strain dynamics on a slow timescale are explicitly governed by a replicator equation which encapsulates all traits and their interplay. This allows us to predict explicitly not only the final epidemiological outcome of a given 2-player competition, but moreover, their entire frequency dynamics as a direct function of their relative variation and of strain-transcending global parameters. Based on mutual invasion fitnesses, we analyze and report rigorous results on transition phenomena in the 2-strain system, strongly mediated via coinfection prevalence. We show that coinfection is not always a promoter of coexistence; instead, its effect to favour or prevent polymorphism is non-monotonic and depends on the type and level of phenotypic differentiation between strains. This framework offers a deeper analytical understanding of 2-strain competitive games in coinfection, with theoretical and practical applications in epidemiology, ecology and evolution.

We establish sharp-in-time pointwise kernel estimates for the Schrödinger equation on noncompact symmetric spaces of general rank. A well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general, is overcome by using a spectral decomposition introduced recently by two of the authors in the study of the wave equation. We deduce the dispersive property of the Schrödinger propagator and prove global-in-time Strichartz inequalities for a large family of admissible pairs. As consequences, we extend the global well-posedness and small data scattering results previously obtained on real hyperbolic spaces to general Riemannian symmetric spaces of noncompact type.

We establish sharp-in-time kernel and dispersive estimates for the Schrödinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schrödinger group to the wave propagator and an improved Hadamard parametrix., 19 pages, major revision with more details

We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-\Delta u+u^p-M|\nabla u|^q=\mu$ in a domain $\Omega\subset\BBR^N$ where $\mu$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We also give conditions under which nonnegative solutions of $\CL^{_{^M}}_{p,q}u=0$ in $\Omega\setminus K$ where $K$ is a compact subset of $\Omega$ can be extended as a solution of the same equation in $\Omega$, Comment: 26 pages