Journées ANR HEAD

13 avr. 2026, 15:00 → 16 avr. 2026, 18:00 Europe/Paris

Amphithéâtre Laurent Schwartz (Institut de Mathématiques de Toulouse)

Présentation

Ces journées sont organisées dans le cadre du projet ANR "Hyperbolic Equations, Approximations and Dynamics". Elles se tiendront du lundi 13 au jeudi 16 avril 2026 à l'Institut de Mathématiques de Toulouse. Le programme prévisionnel est le suivant.

 

Mini-cours de 3h chacun :

Antoine BENOIT (Calais, France)

Jean-François COULOMBEL (Toulouse, France)

Björn DE RIJK (Karlsruhe, Allemagne)

 

Exposés :

Corentin AUDIARD (Paris, France)

Junsik BAE (Kyungpook National University, Corée du Sud)

Gilles BELLON (Toulouse, France)

Marianne BESSEMOULIN (Nantes, France)

Timothée CRIN-BARAT (Toulouse, France)

Maria KAZAKOVA (Chambéry, France)

Sam KRUPA (Paris, France)

Xavier LAMY (Toulouse, France)

Corrado MASCIA (Rome, Italie)

Jose Manuel VALDOVINOS (Toulouse, France)

Marie-Hélène VIGNAL (Lyon, France)

 

Informations pratiques

La rencontre se déroulera au sein de l'Institut de Mathématiques de Toulouse, dans l'amphithéâtre Laurent Schwartz (bâtiment 1R3). Un plan détaillé du campus se trouve ici : plan du campus. L'amphithéâtre se situe au bâtiment 1R3, rue Sébastienne Guyot.

Depuis le centre-ville, le plus commode est de prendre la ligne de métro B en direction de Ramonville et de descendre à l'arrêt "Université Paul Sabatier". Plus de renseignements sur le site de Tisséo.

Les premiers exposés se dérouleront le lundi 13 avril à partir de 16h et les journées se termineront le jeudi 16 avril vers 18h.

lun. 13 avril

15:00 → 16:00 Arrival - welcome

16:00 → 17:00 Large-time stability of partially dissipative hyperbolic systems

In this talk, we consider hyperbolic systems with dissipative effects arising from viscosity or friction. We start by reviewing recent results on the stability of perturbations around constant equilibria. Then, we discuss how the stability analysis changes when passing from constant states to space-periodic traveling waves. In this setting, we introduce a space-averaged Shizuta–Kawashima-type condition and show that it characterizes high-frequency spectral stability for one-dimensional partially diffusive hyperbolic–parabolic systems with space-periodic coefficients. This criterion further enables us to establish nonlinear stability results for sufficiently small initial perturbations.

Orateur: Timothée Crin-Barat

17:00 → 18:00 A hyperbolic dispersive model for coastal waves

A shallow water model for the propagation and breaking of surface waves is proposed, under the form of a hyperbolic system of conservation laws, with dispersive effects introduced through a relaxation term and with a localized dissipative term. The latter is activated in regions where a breaking criterion is met. The objective is to get a simple mathematical and numerical structure while capturing the main features of wave breaking.

The governing equations, the associated breaking criterion, and the numerical strategy used for their approximation are presented. Particular attention is paid to the persistence of the dissipation once activated, and to its influence on the behaviour of solutions. Several test cases illustrate the behaviour of the model.

This is joint work with G. Richard, J. Chauchat and Y.-C. Hung.

Orateur: Maria Kazakova (LAMA, USMB)

mar. 14 avril

09:00 → 10:30 Mini-course 1: on discrete integration by parts methods

The course will review several aspects of discrete integration by parts methods. The ultimate goal is to construct finite difference approximations of the first order derivative that satisfy a similar integration by parts formula as in the continuous setting on a half-line. Basic questions (and partial answers) include existence, uniqueness and non-existence results. We shall connect the theory with various problems in matrix theory or discrete mathematics (Hankel determinants, Vandermonde matrices, Bernoulli polynomials etc.). Several open questions will be listed.

Orateur: Jean-François Coulombel (Institut de Mathématiques de Toulouse)

10:30 → 10:45 Break

10:45 → 12:15 Mini-course 1: on discrete integration by parts methods

The course will review several aspects of discrete integration by parts methods. The ultimate goal is to construct finite difference approximations of the first order derivative that satisfy a similar integration by parts formula as in the continuous setting on a half-line. Basic questions (and partial answers) include existence, uniqueness and non-existence results. We shall connect the theory with various problems in matrix theory or discrete mathematics (Hankel determinants, Vandermonde matrices, Bernoulli polynomials etc.). Several open questions will be listed.

Orateur: Jean-François Coulombel (Institut de Mathématiques de Toulouse)

12:15 → 14:00 Lunch at Esplanade

14:00 → 15:00 Asymptotic-preserving schemes

In this presentation, I will begin by explaining the objectives and underlying principles of asymptotic-preserving schemes. I will then focus on two specific cases: schemes that preserve the low Mach number limit for the Euler equations, and schemes that preserve the quasi-neutral limit for the Vlasov–Poisson equations. I will discuss the main challenges in simulating these problems numerically, and show how asymptotic-preserving schemes can overcome them. Such schemes are uniformly stable in the considered limit allowing time steps that do not tend to zero in the asymptotic regime. Moreover, they preserve the asymptotic limit, giving an accurate approximation of the corresponding reduced model in the limit.

Orateur: Marie-helene Vignal (Institut de Mathématiques de Toulouse, Université Toulouse 3 - Paul Sabatier)

15:00 → 16:00 Are $L^\infty$ solutions to hyperbolic systems of conservation laws unique?

For hyperbolic systems of conservation laws in 1-D, fundamental questions about uniqueness and blow up of weak solutions still remain even for the apparently “simple” systems of two conserved quantities such as isentropic Euler and the p-system. Similarly, in the multi-dimensional case, a longstanding open question has been the uniqueness of weak solutions with initial data corresponding to the compressible vortex sheet. We address all of these questions by using the lens of convex integration, a general method of constructing highly irregular and non-unique solutions to PDEs. Our proofs involve computer-assistance. This talk is based on joint work with László Székelyhidi, Jr.

Orateur: Sam Krupa (Ecole Normale Supérieure)

16:00 → 16:30 Break

16:30 → 17:30 Dispersionless limit in the Euler-Korteweg system

The Euler-Korteweg equations are a modification of the Euler equations which include in the momentum equation a term modelling capillary forces. Mathematically, this supplementary term is of dispersive nature, and after a reformulation the system looks like a degenerate Schrödinger equation. We consider here the behaviour of smooth solutions when the capillary coefficient is very small. When the problem is posed on the full space, we prove that the solutions converge to a solution of the Euler equation. On the half space, we obtain a formal WKB expansion which indicates the presence of a boundary layer. We shall also discuss the question of the limiting problem if the initial data exhibit a phase transition across a layer whose thinness depends on the capillary coefficient.

Orateur: Corentin Audiard (Sorbonne Universite)

mer. 15 avril

09:00 → 10:30 Mini-course 2: orbital stability of periodic waves in Hamiltonian systems under localized perturbations

In Hamiltonian systems, periodic waves often correspond to coherent structures: recurrent, robust patterns that persist over time. Notable examples include water waves, periodic sequences of light pulses in nonlinear optical fibers, and soliton trains in Bose-Einstein condensates. To date, nonlinear stability results for periodic standing or traveling waves in Hamiltonian systems have primarily addressed co-periodic perturbations. A longstanding open problem concerns their stability with respect to localized perturbations: a natural setting in many physical applications. We begin this minicourse by reviewing classical stability methods for Hamiltonian systems with symmetry. These approaches characterize stable solutions as constrained minimizers of an appropriate Lagrangian functional, which is built from conserved quantities of the system and is positive definite on a finite-codimensional constraint space. We then explain why this framework breaks down for periodic waves under localized perturbations and introduce a novel approach that combines variational methods, Floquet-Bloch theory, and Duhamel-based estimates with a modulational ansatz. This alternative approach yields orbital stability results for periodic waves in key Hamiltonian models, such as the Korteweg-de Vries, Klein-Gordon, and nonlinear Schrödinger equations, with respect to $L^2$-localized perturbations.

Orateur: Björn de Rijk (Karlsruher Institut für Technologie)

10:30 → 10:45 Break

10:45 → 12:15 Mini-course 2: orbital stability of periodic waves in Hamiltonian systems under localized perturbations

In Hamiltonian systems, periodic waves often correspond to coherent structures: recurrent, robust patterns that persist over time. Notable examples include water waves, periodic sequences of light pulses in nonlinear optical fibers, and soliton trains in Bose-Einstein condensates. To date, nonlinear stability results for periodic standing or traveling waves in Hamiltonian systems have primarily addressed co-periodic perturbations. A longstanding open problem concerns their stability with respect to localized perturbations: a natural setting in many physical applications. We begin this minicourse by reviewing classical stability methods for Hamiltonian systems with symmetry. These approaches characterize stable solutions as constrained minimizers of an appropriate Lagrangian functional, which is built from conserved quantities of the system and is positive definite on a finite-codimensional constraint space. We then explain why this framework breaks down for periodic waves under localized perturbations and introduce a novel approach that combines variational methods, Floquet-Bloch theory, and Duhamel-based estimates with a modulational ansatz. This alternative approach yields orbital stability results for periodic waves in key Hamiltonian models, such as the Korteweg-de Vries, Klein-Gordon, and nonlinear Schrödinger equations, with respect to $L^2$-localized perturbations.

Orateur: Björn de Rijk (Karlsruher Institut für Technologie)

12:15 → 14:00 Lunch at Esplanade

14:00 → 15:00 Simple non-linear behaviors in models of climate phenomena

Although the climate system is inherently chaotic, many idealized models employed in climate science display behaviors characteristic of low-dimensional nonlinear dynamical systems. Even models incorporating more comprehensive and detailed physical processes may reproduce similar behaviors in idealized configurations. Such models provide valuable conceptual frameworks for understanding modes of atmospheric or oceanic variability, cloud regimes, and fundamental geophysical processes. This presentation will provide selected examples of multiple equilibria and limit cycles arising in these systems, and will further detail one recent interdisciplinary collaboration between mathematicians and geophysical fluid dynamicists aimed at elucidating a peculiar spatial organization of convective clouds.

Orateur: Gilles Bellon (Centre National de Recherches Météorologiques)

15:00 → 16:00 Global well-posedness and asymptotic behavior of an inviscid non-equilibrium radiation hydrodynamics system

We consider the one-dimensional diffusion approximation, non-equilibrium model of radiation hydrodynamics derived by Buet and Després (J. Quant. Spectrosc. Radiat. Transf. 85 (2004), no. 3-4, 385–418). This system describes a non-relativistic inviscid fluid subject to a radiative field under the non-equilibrium hypothesis, that is, when the temperature of the fluid is different from the radiation temperature, and it is a non-conservative parabolic balance law system. We are interested in the global existence and asymptotic behavior of small perturbation of constant equilibrium states. The approach we take can be divided into three steps: first, we study the local well-posedness of the system; second, the decay properties of the linear system around a constant state are studied under the framework of Sizhuta and Kawashima (Hokkaido Math. J. 14 (1985), no. 2, 249–275); and third, we perform the nonlinear energy estimate based on the linear results, which will give us the a priori energy estimate as well as the decay rate of solutions needed to conclude the global existence and asymptotic behavior. For this last step we introduce a notion of entropy for the system that allows us to recast it in a form such that the nonlinear estimate can be closed. This talk is based on a joint work with C. Lattanzio and R. G. Plaza.

Orateur: José Manuel Valdovinos (Institut de Mathématiques de Toulouse)

16:00 → 16:30 Break

16:30 → 17:30 Variations on the Gatenby-Gawlinski model for acid-mediated tumour growth

For evident reasons, Cancer Biology is one of the most challenging topics of current medical research and understanding the mechanism behind its uncontrolled growth is a crucial issue. Among other explanations of the process, the Warburg effect posits that a pivotal role is played by the so-called aerobic glycolysis, i.e. the fact that, even in presence of oxygen, lactic acid fermentation can be favoured by tumour cells, enhancing their metabolism and, as a consequence, the invasive features.

The aim of the talk is to present a mathematical model for such a phenomenon, in the form proposed in 1996 by Robert A. Gatenby and Edward T. Gawlinski, based on a system of reaction-diffusion equations and discusses some of its most significant properties, including the computational evidence of the existence of propagation fronts. Time permitting, I will also discuss other complementary items such as the existence of fronts for simplified models, stability (dynamic and structural), heterogeneity, homogenisation...

Collaborations with Irene Anello, Thierry Gallay, Pierfrancesco Moschetta, Donato Pera, Elisa Scanu, Chiara Simeoni (variable subgroups…).

Orateur: Corrado Mascia (Sapienza Università di Roma)

19:30 → 21:30 Social diner

Restaurant Emile, 13 place Saint-Georges, Toulouse

jeu. 16 avril

09:00 → 10:30 Mini-course 3: symmetrizers

The aim of this lecture is to present new results concerning the well-posedness of hyperbolic systems defined in a domain with a corner. In the canonical half-space geometry, Kreiss’s theory [1970] characterizes well-posed problems in terms of an algebraic condition, the so-called uniform Kreiss–Lopatinskii condition. The main contribution of Kreiss’s work is the construction of a so-called Kreiss symmetrizer, which reduces the proof of the a priori energy estimate to a simple integration-by-parts argument.

Althought it seems to be a natural extension, the well-posedness issue for hyperbolic corner problems is a rather open question since the seminal work of Osher [1973].

Here, we introduce a new notion of symmetrizer adapted to corner problems, which makes it possible to characterize well-posed problems in terms of a non-intersection condition. In this way, Kreiss’s half-space theory is extended to corner domains.

The lecture will be divided into four sections of increasing technical difficulty, all relying on the same fundamental ideas. Section 1 is devoted to the Cauchy problem. Section 2 extends the ideas of Section 1 to the half-space problem. Section 3 considers the strip problem, which serves as a toy model for studying interactions arising from multiple boundaries. Finally, Section 4 is devoted to corner domains and relies heavily on the toy-model analysis developed in Section 3.

Orateur: Antoine Benoit (ULCO)

10:30 → 10:45 Break

10:45 → 12:15 Mini-course 3: symmetrizers

The aim of this lecture is to present new results concerning the well-posedness of hyperbolic systems defined in a domain with a corner. In the canonical half-space geometry, Kreiss’s theory [1970] characterizes well-posed problems in terms of an algebraic condition, the so-called uniform Kreiss–Lopatinskii condition. The main contribution of Kreiss’s work is the construction of a so-called Kreiss symmetrizer, which reduces the proof of the a priori energy estimate to a simple integration-by-parts argument.

Althought it seems to be a natural extension, the well-posedness issue for hyperbolic corner problems is a rather open question since the seminal work of Osher [1973].

Here, we introduce a new notion of symmetrizer adapted to corner problems, which makes it possible to characterize well-posed problems in terms of a non-intersection condition. In this way, Kreiss’s half-space theory is extended to corner domains.

The lecture will be divided into four sections of increasing technical difficulty, all relying on the same fundamental ideas. Section 1 is devoted to the Cauchy problem. Section 2 extends the ideas of Section 1 to the half-space problem. Section 3 considers the strip problem, which serves as a toy model for studying interactions arising from multiple boundaries. Finally, Section 4 is devoted to corner domains and relies heavily on the toy-model analysis developed in Section 3.

Orateur: Antoine Benoit (ULCO)

12:15 → 14:00 Lunch at Esplanade

14:00 → 15:00 Discrete hypocoercivity for a nonlinear kinetic reaction model

In this talk, I will present a finite volume discretization of a 1D nonlinear kinetic reaction model, which describes a two-species recombination-generation process. More specifically, we establish the long-time convergence of the approximate solutions to equilibrium, at an exponential rate. To do this, we adapt the proof proposed in [Favre, Pirner, Schmeiser, ARMA 2023], based on an adaptation of the hypocoercivity method of [Dolbeault, Mouhot, Schmeiser, Trans. Amer. Math. Soc. 2015]. As in the continuous setting, this result is valid for bounded initial data and requires establishing a maximum principle, which necessitates the use of monotonic numerical fluxes.

This is a joint work with Tino Laidin (Univ. Brest) and Thomas Rey (Univ. Nice).

Orateur: Marianne Bessemoulin-Chatard (Laboratoire de Mathématiques Jean Leray)

15:00 → 16:00 Emergence of peaked singularities in the Euler-Poisson system

We consider the one-dimensional Euler-Poisson system equipped with the Boltzmann relation. We provide the exact asymptotic behavior of the peaked solitary wave solutions near the peak. This enables us to study the cold ion limit of the peaked solitary waves with the sharp range of Holder exponents. Furthermore, we provide numerical evidence for $C^1$ blow-up solutions to the pressureless Euler-Poisson system, whose blow-up profiles are asymptotically similar to its peaked solitary waves and exhibit a different form of blow-up compared to the Burgers-type (shock-like) blow-up. This is a joint work with Sang-Hyuck Moon (Pusan National University) and Kwan Woo(University of Basel).

Orateur: Junsik Bae (Kyungpook National University)

16:00 → 16:30 Break

16:30 → 17:30 Hyperbolic regularization effects for degenerate elliptic equations

Among weak solutions of Burgers' equation, a single strictly convex entropy is sufficient to characterize the sign of all entropy productions. In particular, if that entropy production vanishes, then the solution must be continuous. It turns out that this fact can be interpreted as a regularity result for a degenerate elliptic equation in the plane, and generalized to prove partial regularity results for a large class of planar nonlinear equations $\mathrm{div}\: G(\nabla u)=0$ which are only qualitatively elliptic. This is joint work with Riccardo Tione.

Orateur: Xavier Lamy (Institut de Mathématiques de Toulouse)