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 Talk 2

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 Talk 3

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 Talk 6

Orateur: Gilles Bellon (Centre National de Recherches Météorologiques)

15:00 → 16:00 Talk 7

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

16:00 → 16:30 Break

16:30 → 17:30 Talk 8

Orateur: Corrado Mascia (Sapienza Università di Roma)

19:30 → 21:30 Social diner

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 Talk 9

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 Talk 11

Orateur: Xavier Lamy (Institut de Mathématiques de Toulouse)