The motion of a compressible viscous barotropic fluid is described by the Navier-Stokes system. It is a system of hyperbolic-parabolic mixed-type PDEs. In this talk, we will study the so-called density patch problem: If we are given a density that is initially discontinuous across a C^(1+\alpha) curve alpha and alpha- Hölder continuous on the two disjoint components delimited by gamma, is this structure preserved in time?
An important quantity in the mathematical analysis of this system is the so-called effective flux, which was discovered by Hoff and Smoller in 1985. More precisely, the mathematical properties of this quantity play a crucial role in the study of the propagation of oscillations in compressible fluids (Serre, 1991), in the construction of weak solutions (P-L Lions 1996) or the propagation of discontinuity surfaces (Hoff 2002), to cite just a few examples. In the case of density-dependent viscosities, the behavior of the effective flux degenerates, which renders the analysis more subtle.
In the first part of this talk we will give an overview of our past and present research activity, highlighting the different fields of applied mathematics that have been considered so far. In the second part of the talk, I present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC) models introduced by Godunov in 1961. Specifically, our numerical method discretizes the equations for the conser- vation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified mathematical formalism for a wide range of physical phenomena in continuum mechanics, spanning from ideal and viscous fluids to hyperelastic solids. By means of two conservative corrections directly embedded in the definition of the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is satisfied by introducing the new concepts of general potential and generalized Gibbs relation. The new potential is nothing but the determinant of the distortion tensor, and the associated Gibbs relation is derived by introducing a set of dual or thermodynamic variables such that the GCL is retrieved by dot multiplying the original system with the new dual variables. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete level as well.
Wave breaking is a challenging subject that is not encompassed in the usual mathematical description of water waves. This is the consequence of the impossibility to represent the water-air interface as the graph of a function. In the first part of this presentation, we shall exhibit the strong non-linear nature of the breaking phenomena through the mathematical study of two water waves models: (1) KdV, whose solutions do not break and (2) Camassa-Holm, whose non-global solutions do break at some point. Next we shall discuss the way to incorporate multi-valued interfaces in the usual water waves problem before discussing whether or not this model efficiently describes the breaking phenomenon. In a third, ultimate, part we will present new numerical results that have been obtained using a finite-element code to solve the free-surface Navier-Stokes equations for an initial condition leading to a plunging jet. We compare these results with those obtained solving the Euler equation in the exact same configuration and conclude about the convergence of the two methods whenever the Reynolds number increases. We also discuss the influence of viscous dissipation on the overall shape of the wave.
L'imagerie haute résolution de la Terre, et en particulier de la croûte, est fondamentale pour la transition énergétique: pour la massiffication du stokage de CO2, une technologie mise en avant par le GIEC pour lutter contre le réchauffement climatique, mais aussi pour l'exploitation des ressources nécessaires pour la construction des infrastructures énergétiques éoliennes et salaires, et les batteries électriques. L'état de l'art pour l'imagerie haute résolution de la croûte repose sur une méthode appelée "inversion des formes d'ondes complètes". D'un point de vue mathématique, ceci revient à un problème d'estimation de paramètres d'une équation aux dérivées partielles (EDP) modélisant la propagation d'ondes dans le sous-sol à partir de données collectées ponctuellement en surface. Dans cette présentation, on introduit les bases géophysiques et mathématiques autour de cette méthode, avant de passer en revue des travaux de recherche menés au sein du projet SEISCOPE. Ces travaux recoupent les thèmes suivants: méthode d'optimisation de second-ordre basées sur des méthodes adjointes d'ordre deux, utilisation de distances transport optimal pour lutter contre le caractère mal posé du problème inverse, modélisation numérique 3D de la propagation d'onde dans l'approximation élastique, théorie des milieux équivalents (homogénisation) pour la propagation d'ondes en milieux élastiques, estimation des incertitudes pour les problèmes inverse de grande taille en se basant sur une méthode de filtre de Kalman d'ensemble.
Après un rappel sur les équations d'Euler 2D, je parlerai des tourbillons concentrés. J’exposerai les arguments principaux pour montrer la persistance de la concentration vers des points vérifiant le système des points vortex. Dans la seconde partie, je présenterai les équations des lacs qui peuvent se voir comme une généralisation d’Euler 3D axisymétrique sans swirl. Je montrerai que les points vortex se déplacent selon une loi de type « courbure binormale ». Ce travail est en collaboration avec Lars Eric Hientzsch et Evelyne Miot.
Geophysical phenomenon such as magnetic field reversal are a challenge to observe numerically. They are quite demanding in terms of numerical resources, and with the upcoming generation of exascale computers, it becomes necessary to ensure an efficient and full usage of such clusters. In geophysical and astrophysical flows, the classical method for parallelism is data parallelism. The physical domain is distributed across a large number of cores. The scaling of such a distribution can quickly saturate when the number of cores increases. Introducing an additional pipeline parallelism, through a distribution of a time interval across a number of cores, is a potential solution to use a larger number of cores, and perform numerical simulations of magnetic field reversals. When used by parallel in time schemes, time-steppers need to validate a few of criteria. We extracted 18 time-steppers from literature, from second to eighth order of accuracy. We will compare the accuracy and efficiency of those time steppers in the context of liquid planetary cores, in order to identify potential candidates to build parallel in time schemes.
La modélisation et la simulation d’écoulements diphasiques constituent un sujet de recherche important, notamment pour leurs applications en sûreté nucléaire. Dans certains scenarii d’accidents interviennent des écoulements très hétérogènes, constitués d’eau liquide et de bulles d’air et de vapeur. Afin de modéliser de tels écoulements, on privilégie des modèles moyennés, donnant une description macroscopique des écoulements, la description à l’échelle des interfaces eau-gaz étant hors portée. Cependant connaître les propriétés de l’interface, en particulier l’évolution de l’aire interfaciale et de la tension de surface, demeure important. Le but de cet exposé est de présenter deux manières de dériver des modèles moyennés d’écoulements diphasiques avec tension de surface, la première par une méthode d’homogénéisation, la seconde par un principe d’Hamilton.
We investigate the energy decay of hyperbolic system of wave-wave with generalized acoustic boundary conditions in N-dimensional space, with the equations being coupled through boundary connection. First, by spectrum approach combining with a general criteria of Arendt-Batty, we prove that our model is strongly stable. Then, after proving that this system lacks the exponential stability, we establish different type of polynomial energy decay rates provided that the coefficients of the acoustic boundary conditions satisfy some assumptions. Further, we present some appropriate examples and show that our assumptions have been set correctly. Finally, we prove that the obtained energy decay rate is optimal in particular case.
We consider a stochastic individual-based model for the evolution of a population, whose space of possible traits is given by the vertices of a finite graph. The dynamics is driven by births, deaths, competition, and mutations along the edges of the graph. We are interested in the large population limit under a mutation rate given by a negative power of the carrying capacity K of the system. This results in several mutant traits being present at the same time and competing for invading the resident population. We describe the time evolution of the orders of magnitude of each sub-population on the \log K time scale, as K tends to infinity. Using techniques developed in [Champagnat, Méléard, Tran, 2019], we show that these are piecewise affine continuous functions, whose slopes are given by an algorithm describing the changes in the fitness landscape due to the succession of new resident or emergent types. I will illustrate the theorem by examples describing surprising phenomena arising from the geometry of the graph and/or the rate of mutations. If time permits I will finish with an application to the phenomenon of evolutionary rescue.
In this talk, we introduce nonlinear diffusion equations with absorption, in the most general form
∂_t(u) = ∆u^m − |x|^σ u^p, for m > 1 and p > 0.
Looking for solutions to the Cauchy problem in a first part of the talk, we give a brief survey of general facts for the previous equation in the case of the spatially homogeneous absorption σ = 0, related to very singular solutions and finite time extinction of solutions: that is, the existence of a time Te ∈ (0, ∞) such that u(t) ≢ 0 for any t ∈ (0, Te), but u(Te) ≡ 0. In the second and more specialized part of the talk, we present some recent results including well-posedness, instantaneous shrinking of the supports of solutions, non-extinction versus extinction depending on the initial condition, and large time behavior for the general equation with σ > 0 and 0 < p < 1, emphasizing on the importance of the critical exponent σ := 2(1 − p)/(m − 1) and its influence on the dynamics of the equation.
Joint work with Philippe Laurençot (Univ. de Savoie, Chambéry) and Ariel Sánchez (Univ. Rey Juan Carlos, Madrid).
Dans cet exposé, nous présentons l'étude numérique du système type Boussinesq d'ordre supérieur/étendu décrivant la propagation des ondes de surface. Une reformulation appropriée équivalente est proposée, rendant le modèle plus approprié pour l'implémentation numérique et significativement amélioré en termes de propriétés dispersives linéaires dans les régimes à haute fréquence grâce à l'ajustement approprié d'un paramètre de correction de dispersion. De plus, nous montrons qu'un intérêt significatif se cache derrière la dérivation d'une nouvelle formulation du système de Boussinesq d'ordre supérieur/étendu qui évite le calcul des dérivées d'ordre supérieur existant dans le modèle. Nous montrons que cette formulation a l'avantage d'un domaine d'application étendu tout en restant stable. Nous développons un schéma de ``splitting'' du second ordre où la partie hyperbolique du système est traitée avec un schéma de volumes finis d'ordre élevé et la partie dispersive est traitée avec un schéma de différences finies. Des simulations numériques sont ensuite réalisées pour valider le modèle et les méthodes numériques.
La notion de consistance au sens de Lax-Wendroff (LW-consistance) est importante pour les applications pratiques en simulation d'écoulement de fluides. Dans de nombreux cas d'intérêt, des résultats plus forts de convergence sont hors de portée, et la LW-consistance permet d'aider à la conception mathématique des schémas numériques. C'est par exemple le cas pour les écoulements multidimensionnels gouvernés par des systèmes hyperboliques, tels que les équations d'eau peu profonde, les équations d'Euler ou les modèles pour les écoulements multiphasiques.
Les maillages décalés sont utilisés dans les codes de sûreté nucléaire développés par l'IRSN depuis plus de 15 ans pour la simulation numérique de problèmes d'écoulement de type hyperbolique, et sont maintenant couramment utilisés pour des applications de sécurité industrielle telles que les problèmes d'explosion d'hydrogène, pour des écoulements non visqueux ou au moins de viscosité négligeable.
Nous montrons ici comment les hypothèses de Lax et Wendroff peuvent être généralisés à des maillages décalés pour obtenir un résultat de LW consistance.
Dans l'étude du système de Boussinesq, nous allons revisiter les résultats obtenus par M. E. Schonbek concernant le problème d'existence de solutions faibles entropiques globales pour le système de Boussinesq, ainsi que l’existence et l’unicité de solution régulière globale par C. J. Amick. Il s’agit de rétablir ces résultats dans un cadre fonctionnel plus actuel et en utilisant une ``régularisation par un opérateur fractal”. Nous allons étudier le problème de Boussinesq régularisé et nous montrerons qu’on peut passer à la limite sur la solution de ce problème pour retrouver celle du système de Boussinesq. La méthode utilisée nous permet d’améliorer l’indice de régularité Sobolev pour le problème d’existence ainsi que l’obtention de la continuité des flots associés aux différents problèmes de Cauchy sous la condition du “non-zero-depth”. En même temps, on essayera d’indiquer quelques résultats en cours concernant le cas de fond non plat modilisé par le système de Boussinesq-Peregrine. Ce travail est effectué en collaboration avec L. Molinet et I. Zaïter.
Gradient estimates for solutions to parabolic backward equations based on the Laplace operator are well understood. The Laplace operator naturally extends to non-local operators, where a large class of those non-local operators has an intrinsic connection to Lévy processes. The solutions to the corresponding non-local parabolic backward equations are of interest in applications, where the difference to the classical case is that the gradients of the solutions are infinite-dimensional in general. We investigate the singularity properties of those gradients and indicate an application of the obtained estimates.
In this talk we focus on a class of singular perturbation problems arising in the study of the dynamics of geophysical flows. Given a so-called ``primitive'' system of equations, the goal is to derive reduced models, under suitable assumptions on the fluid and on the scaling regime. The presence of a Coriolis term. encoding the Earth rotation, in the primitive system is the key element of the problems under consideration. We will discuss several aspects which enter into play in this context: the difference between the compressible and incompressible fluid cases, the presence of multiple scales, the formation of the Ekman boundary layers.
We propose a new system of equations modeling Tsunamis in this work. It is a coupled system accounting for both water compressibility and viscoelasticity of the earth. Adding these latter physical effects is responsible for the closest-to-reality time arrival predictions (among existing models), capturing the negative peak before the main wave hump, and exhibiting the negative dispersion phenomena. This comes in remarkable agreement with previous experiments and studies on the topic. The system is also delivered in a relatively simple mathematical structure of equations that is easy to solve numerically.
This talk shall focus on the presentation of a (by now) well studied research topic in the field of stochastic control theory, i.e the case of optimal switching control problems. A main objective of this talk is to provide the connection with system of semilinear PDEs with obstacles which, in addition, are inter- connected. This last feature (among some others) explains why the solution is not smooth (in general). For this reason we study existence and uniqueness of solutions of these PDEs in viscosity sense. In a first part, we shall explain the relationship between the value functional associated with a stochastic control problem and the solution of an explicit semi- linear PDE. For this, we need to introduce both the stochastic framework and some advanced probabilistic tools & technics. Next and after this introductory part, we shall give the precise structure of the system of PDEs we are interested in and provide some theoretical results. If time allows, the last slides present the main steps of one of our main results. This talk is based on several joint works (with Pr. S. Hamadène (LMM), Pr. B Djehiche (KTH Stockholm) and X. Zhao former pHD student at the LMM).
Solving boundary value problems requires implementation of sufficiently robust constitutive models. Most models try to incorporate a great deal of phenomenological ingredients, but this refining often leads to overcomplicated mathematical formulations, requiring a large number of parameters to be identified. On the other hand, geomaterials are known to have an internal microstructure, made up of an assembly of interacting particles. Most of the macroscopic properties, observed on a specimen scale or even on larger scales, mainly result from the microstructural arrangement of grains. Thus, a powerful alternative can be found with micromechanical models, where the medium is described as a distribution of elementary sets of grains. The inherent complexity is not related to the local constitutive description between particles in contact, but to the basic topological complexity taking place within the assembly. This presentation discusses this issue, highlighting very recent results obtained from discrete element simulations. In particular, the so-called critical state regime that develops during localized or diffuse failure is discussed in detail from the perspective of emerging processes taking place within complex media.