MAToS

Mathematical Analysis of Topological Singularities in some physical problems

Presentation

The central theme of this project lies in the area of nonlinear analysis (nonlinear partial differential equations, calculus of variations). Our main focus will be on the structure and dynamics of topological singularities arising in some variational physical models driven by the Landau-Lifshitz equation (micromagnetics) and Gross-Pitaevskii equation (superconductivity, Bose-Einstein condensates etc.). These include vortices, traveling waves and domain walls in magnetic thin films.

The mathematical challenges lie in the description of microscopic structures and of phenomena that occur at very different spatial or temporal scales. They often require recently developed mathematical tools and the introduction of new mathematical techniques (such as Gamma-convergence ....). We intend to make significant progress in challenging open mathematical problems that will give more insight into the corresponding physical phenomena.

Members

Radu Ignat

Radu Ignat (Principal Investigator of the project) is Professor at University Paul Sabatier (UPS) in Toulouse since 2013. He was student at Ecole Normale Supérieure in Paris (2000-2004) and obtained the PhD in Mathematics at University Paris 6 in 2006. He was awarded the Arconati-Visconti Prize in Sciences by the Chancellery of Universities of Paris in 2007. After a PostDoc at University of Bonn, he was Maître de Conférences in Orsay (2007-2013). He develops methods from nonlinear analysis, calculus of variations and partial differential equations to understand characteristic phenomena in variational physical models: pattern formation due to energy minimization, microstructures and formation of singularities. He has experience in leading and memberships of french / international networks (involving both mathematical and physics communities), he has an active and internationally recognized expertise in the field working with many collaborators in France and abroad and is the author of twenty papers published in high-ranked journals.

Stefan Le Coz

Stefan Le Coz is Maître de Conférences at UPS in Toulouse since 2010. He obtained the PhD in Mathematics at University of Franche-Comté in 2007, then he was PostDoc for two years at SISSA and Paris 6. He works on nonlinear dispersive PDE with a particular emphasis on the large time behavior of solitons and multi-soliton solutions.

Mihai Maris

Mihai Maris is Professor at UPS in Toulouse since 2009. He was student at Ecole Normale Supérieure in Paris and obtained his PhD in Mathematics at Orsay in 2001. He was Maître de Conférences at University of Franche-Comté (2002-2009). He is specialist in nonlinear elliptic PDEs, concentration-compactness phenomena, existence and qualitative properties of nonlinear waves for dispersive PDEs.

Raphaël Côte

Raphaël Côte is Chargé de Recherche (CNRS) at the Ecole Polytechnique since 2007. He was student at Ecole Normale Supérieure in Paris (2000-2004) and obtained the PhD in Mathematics at University of Cergy-Pontoise in 2006. His research focuses on large time dynamics for dispersive equations, the construction of multi-solitons, the formation of singularities and classification of solutions.

Vincent Millot

Vincent Millot is Maître de Conférences at Paris 7 since 2008. He obtained the PhD at Paris 6 in 2005, then he was PostDoc for two years at Carnegie Mellon University. He works on nonlinear elliptic PDEs, Ginzburg-Landau type problems, homogenization, phase transitions and free discontinuity problems.

Didier Smets

Didier Smets is Professor at Paris 6 since 2005. He obtained the PhD at University of Louvain-la-Neuve in Belgium in 2000, then he was Maître de Conférences at Paris 6 till 2005. He was awarded the Maurice Audin prize and the Eugène Catalan prize of the Royal Academy of Sciences of Belgium (2005). He is an expert in the Ginzburg-Landau energy and the related evolution equations, of which the GP equation. He was trained in topological methods in the calculus of variations and has an expertise in nonlinear PDEs, geometric flows and vortex structures.

Events

  • Upcoming event: Conference Analysis of singular patterns in variational models, June 18 to 22, 2018, Toulouse, France
  • Recurring event: Seminar on Mathematical Analysis of topological singularities, Toulouse, France
  • Second Meeting of the project MAToS, December 12 & 13, 2016, Toulouse, France.
    • Venue: salle MIP, first floor, building 1R3, IMT, University of Toulouse 3
    • Programm to be announced
  • Topological Patterns and Dynamics in Magnetic Elements and in Condensed Matter, Dresden, Germany
  • First Meeting of the project MAToS, January 27 & 28 2015, Toulouse, France
    • Tuesday, January 27, starting 2 p.m.
      • Radu Ignat (Toulouse)
        Présentation du projet ANR MAToS
        Equation de Landau-Lifshitz
      • Raphael Côte (Ecole Polytechnique)
        Parois de Néel et équation de Landau-Lifshitz-Gilbert.
      • Mihai Maris (Toulouse)
        Présentation du projet ANR MAToS
        Equation de Gross-Pitaevskii
    • Wednesday, January 28, starting 9:30 a.m.
      • Didier Smets (Paris 6)
        Leapfrogging pour les anneaux de vorticité.
      • Vincent Millot (Paris 7 + ENS)
        Allen-Cahn fractionnaire et surfaces minimales non locales

Second Meeting of the project MAToS, December 12 and 13, 2016, Toulouse, France

  • Venue: salle MIP, first floor, building 1R3, IMT, University of Toulouse 3
  • Monday, December 12
    • 11:30-12:30 Gilles Carbou (Pau)
      Murs dans les nano-fils ferromagnétiques pincés
      Les fils ferromagnétiques sont utilisés pour le stockage des données numériques. Les informations sont codées sous forme de domaines ferromagnétiques séparés par des murs. La fiabilité du stockage dépend de manière cruciale de la stabilité de la position des murs, celle ci étant assurée dans les applications par des encoches faites dans le fil. Après avoir établi un modèle 1d de tels fils avec plusieurs pincements, on étudiera l'existence et la stabilité de solutions codant une suite de bits quelconque. Il s'agit d'un travail en collaboration avec David Sanchez.
    • 14:30-15:30 Stefan Le Coz (Toulouse)
      Stability of periodic waves of 1D nonlinear Schrödinger equations
      We consider the stability of periodic waves of the 1D cubic Schrödinger equations. The profiles of these periodic waves are rescaled Jacobi elliptic functions \(\operatorname {sn}\), \(\operatorname {cn}\) and \(\operatorname {dn}\) with parameter \(0 < k < 1\). We examine their stability and instability properties, analytically and numerically, under same period and multiple period perturbations in the energy space. Highlights include the spectral stability of cn in the energy space under same period perturbations if \(0< k < 0.9\), and instability for \(0< k < 1\) under \(n\)-period perturbations for some \(n=n(k)\). The techniques used include variational methods, spectral analysis, and perturbation arguments.
    • 15:30-16:30 Raphael Côte (Strasbourg)
      Multi-solitons pour l'équation de Klein-Gordon non linéaire
      Les multi-solitons sont des solutions qui se décomposent en une somme de solitons se découplant exactement pour les temps grands. Ce sont des objets très particuliers, car dans un sens ils sont non dispersifs, et ils semblent devoir jouer un rôle clé dans l'étude de la dynamique en temps long des équations dispersives focalisantes, au vu de la conjecture de résolution en solitons. La construction de multi-solitons dans des cadres stables et instables a été étudiée dans de nombreux contextes. Dans un travail en collaboration avec Yvan Martel, nous nous intéressons à de telles solutions pour l'équation de Klein-Gordon non linéaire, dans le cas où les solitons sont fabriqués à partir d'états excités, et non plus uniquement de ground-states, et sans condition sur les vitesses.
  • Tuesday, December 13
    • 08:45-09:45 Vincent Millot (Paris 7)
      Sur une approximation champ de phase du problème de Steiner planaire
      Dans cet exposé, je vais présenter une approximation par champ de phase du problème de Steiner introduite par Bonnivard, Lemenant, et Santambrogio - ou plutôt une petite variante de celle-ci. J’expliquerai quels bénéfices tirer de cette variation en ce qui concerne l’étude de minimiseurs: existence, régularité, et étude asymptotique. Si le temps le permet, je présenterais une application au problème de compliance optimale.
    • 09:45-10:45 Mihai Maris (Toulouse)
      Autour de quelques problèmes de minimisation
      On présentera quelques améliorations récentes du principe de concentration-compacité et des applications à des problèmes plus ou moins classiques en calcul des variations. Dans chaque cas on aboutit à des conditions quasiment optimales d'existence de minimiseurs.
    • 14:00-15:00 Nicolas Godet (Toulouse)
      Solutions explosives pour NLS en géométrie courbe
      On s'intéresse à une version non linéaire de l'équation de Schrödinger dépendant du temps posée sur un domaine quelconque. Il est bien connu que la géométrie du domaine peut influencer le comportement qualitatif des solutions notamment leur tendance à se disperser au cours du temps. Après avoir rappeler les résultats euclidiens connus, je donnerai un résultat concernant les singularités des solutions explosant en temps fini quand le domaine est courbe.
    • 15:00-16:00 Didier Smets (Paris 6)
      Dynamique des tourbillons quasi 2D et modèle Klein-Majda-Damodara
      Je raconterai des travaux en cours avec Robert Jerrard concernant l'asymptotique de l'équation de Gross-Pitaveskii 3D dans un régime de filaments de vortex presque rectilignes. Le résultat principal affirme la convergence vers un système d'EDP en 1 dimension d'espace qui fut écrit formellement dans le cadre des équations d'Euler par Klein-Majda-Damodaran.

Publications

Preprints

[1] M. Goldman, B. Merlet, and V. Millot. A Ginzburg-Landau model with topologically induced free discontinuities. preprint arXiv:1711.08668, 2017. [ bib | .pdf ]
[2] P. Gravejat and D. Smets. Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation. preprint arXiv:1705.06935, 2017. [ bib | .pdf ]
[3] R. Ignat and X. Lamy. Lifting of RPd-1-valued maps in BV and applications to uniaxial Q-tensors. With an appendix on an intrinsic BV-energy for manifold-valued maps. preprint arXiv:1706.01281, 2017. [ bib | .pdf ]
[4] R. Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. On the uniqueness of minimisers of Ginzburg-Landau functionals. preprint arXiv:1708.05040, 2017. [ bib | .pdf ]
[5] Radu Ignat and Felix Otto. The magnetization ripple: a nonlocal stochastic PDE perspective. preprint arXiv:1709.01374, 2017. [ bib | .pdf ]
[6] Vincent Millot, Yannick Sire, and Hui Yu. Minimizing fractional harmonic maps on the real line in the supercritical regime. preprint arXiv:1710.04754, 2017. [ bib | .pdf ]
[7] Matthieu Bonnivard, Antoine Lemenant, and Vincent Millot. On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers. preprint arXiv:1611.07875, 2016. [ bib | .pdf ]
[8] Raphaël Côte and Yvan Martel. Multi-travelling waves for the nonlinear Klein-Gordon equation. preprint arXiv:1612.02625, 2016. [ bib | .pdf ]
[9] Robert Jerrard and Didier Smets. Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation. preprint arXiv:1606.05103, 2016. [ bib | .pdf ]
[10] Vincent Millot, Yannick Sire, and Kelei Wang. Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces. preprint arXiv:1610.07194, 2016. [ bib | .pdf ]
[11] Mihai Maris. Profile decomposition for sequences of Borel measures. preprint arxiv:1410.6125, 2016. [ bib | .pdf ]

In journals

[1] Radu Ignat and Roger Moser. Néel walls with prescribed winding number and how a nonlocal term can change the energy landscape. J. Differential Equations, 263:5846--5901, 2017. [ bib | .pdf ]
[2] Stephen Gustafson, Stefan Le Coz, and Tai-Peng Tsai. Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations. Appl. Math. Res. Express, pages 1--57, 2016. [ bib | http ]
[3] Radu Ignat and Robert L. Jerrard. Interaction energy between vortices of vector fields on Riemannian surfaces. Comptes Rendus Mathematique, 355(5):515 -- 521, 2017. [ bib | DOI | http ]
[4] Pierre Bochard and Radu Ignat. Kinetic formulation of vortex vector fields. Anal. PDE, 10(3):729--756, 2017. [ bib | .pdf ]
[5] Isabella Ianni, Stefan Le Coz, and Julien Royer. On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation. SIAM J. Math. Anal., 49(2):1060--1099, 2017. [ bib | DOI | .pdf ]
[6] Stefan Le Coz and Yifei Wu. Stability of multi-solitons for the derivative nonlinear Schrödinger equation. Int Math Res Notices, page rnx013, 2017. [ bib | .pdf ]
[7] Delphine Côte and Raphaël Côte. Limiting motion for the parabolic Ginzburg-Landau equation with infinite energy data. Comm. Math. Phys., to appear, 2016. [ bib | .pdf ]
[8] Fanny Delebecque, Stefan Le Coz, and Rada Maria Weishäupl. Multi-speed solitary waves of nonlinear Schrödinger systems: theoretical and numerical analysis. Comm. Math. Sci., 14(6):1599–1624, 2016. [ bib | .pdf ]
[9] Lukas Döring and Radu Ignat. Asymmetric domain walls of small angle in soft ferromagnetic films. Arch. Rational Mech. Anal., 220:889--936, 2016. [ bib | .pdf ]
[10] Radu Ignat. Interaction energy of domain walls of logarithmically decaying tails in a nonlocal variational model. Oberwolfach Reports, Volume XX/2016 (Calculus of Variations: S. Brendle, A. Figalli, R. Jerrard, N. Wickramasekera)., 2016. [ bib | .pdf ]
[11] Radu Ignat and Roger Moser. Interaction energy of domain walls in a nonlocal Ginzburg-Landau type model from micromagnetics. Arch. Rational Mech. Anal., 221:419--485, 2016. [ bib | .pdf ]
[12] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Stability of point defects of degree ±1/2 in a two-dimensional nematic liquid crystal model. Calc. Var. Partial Differential Equations, to appear, 2016. [ bib | .pdf ]
[13] Stefan Le Coz, Yvan Martel, and Pierre Raphaël. Minimal mass blow up solutions for a double power nonlinear Schrödinger equation. Rev. Mat. Iberoam., 32(3):795–833, 2016. [ bib | .pdf ]
[14] Camillo De Lellis and Radu Ignat. A regularizing property of the 2D-eikonal equation. Comm. Partial Differential Equations, 40:1543--1557, 2015. [ bib | .pdf ]
[15] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Instability of point defects in a two-dimensional nematic liquid crystal model. Ann. Inst. H. Poincaré Anal. Non Linéaire, 33:1131–1152, 2015. [ bib | .pdf ]
[16] Mihai Maris. On some minimization problems in R^N. Proceedings of the 8th Congress of Romanian Mathematicians, pages 216--231, 2015. [ bib | www: ]

Guests

NameFromPlace visitedArrivalDeparture
Xavier LamyUniversité Lyon 1Toulouse21-Jan-201522-Jan-2015
Pierre BochardParis-Sud UniversityToulouse20-Feb-201525-Feb-2015
Antonin MonteilParis-Sud UniversityToulouse15-Mar-201519-Mar-2015
Roger MoserUniversity of BathToulouse29-Mar-201505-Apr-2015
Matthias KurzkeUniversity of NottinghamToulouse17-May-201521-May-2015
Pierre BochardParis-Sud UniversityToulouse10-Nov-201514-Nov-2015
Antonin MonteilParis-Sud UniversityToulouse19-Nov-201524-Nov-2015
Xavier LamyMax Planck InstituteToulouse01-Feb-201606-Feb-2016
Roger MoserUniversity of BathToulouse06-Mar-201615-Mar-2016
Valeriy SlastikovUniversity of BristolToulouse31-Aug-201607-Sep-2016
Lukas DoeringUniversity of AachenToulouse09-Oct-201615-Oct-2016
Pierre-Damien ThizyUniversity of CergyToulouse19-Oct-201620-Oct-2016
Pierre BochardUniversity of LyonToulouse27-Nov-201603-Dec-2016
Antonin MonteilUniversity of Louvain-la-NeuveToulouse22-Jan-201729-Jan-2017
Matthias Kurzke University of NottinghamToulouse24-Apr-201728-Apr-2017
Roger MoserUniversity of BathToulouse08-May-201719-May-2017