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Report of the ANR project “Mathematical Analysis of
Topological Singularities in some physical problems”
(MAToS), 2015-2019
Rapha¨el Cˆote ∗Radu Ignat †Stefan Le Coz ‡Mihai Maris §
Vincent Millot ¶Didier Smets ‖
1 Introduction
This is the final report of the ANR project 14-CE25-0009-01 entitled “Mathematical Anal-
ysis of Topological Singularities in some physical problems” (MAToS) that was developed
by the authors between January 2015-December 2019. The central theme of this project
lied in the area of nonlinear analysis (nonlinear partial differential equations and calculus of
variations). We focused on the structure and dynamics of topological singularities arising in
some variational physical models driven by the Landau-Lifshitz equation (in micromagnet-
ics) and the Gross-Pitaevskii equation (in superconductivity, Bose-Einstein condensation,
nonlinear optics). These included vortex singularities, traveling waves and domain walls in
magnetic thin films. These structures are observed experimentally and in numerical simu-
lations and play an important role in the dynamics of the corresponding physical systems.
We made significant progress in the mathematical analysis of these structures (both at the
stationary and dynamical level) that gives more insight into the physical phenomena.
∗IRMA, Universit´e de Strasbourg, CNRS UMR 7501, Inria, 7 rue Ren´e Descartes, 67084 Strasbourg,
France. Email: cote@math.unistra.fr
†Institut de Math´ematiques de Toulouse & Institut Universitaire de France, UMR 5219, Universit´e de
Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. Email: Radu.Ignat@math.univ-toulouse.fr
‡Institut de Math´ematiques de Toulouse, UMR 5219, Universit´e de Toulouse, CNRS, UPS IMT, F-31062
Toulouse Cedex 9, France. Email: stefan.le-coz@math.univ-toulouse.fr
§Institut de Math´ematiques de Toulouse & Institut Universitaire de France, UMR 5219, Universit´e de
Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. Email: Mihai.Maris@math.univ-toulouse.fr
¶LAMA, Universit´e Paris Est Cr´eteil, Universit´e Gustave Eiffel, UPEM, CNRS, F-94010, Cr´eteil, France.
Email: vincent.millot@u-pec.fr
‖Sorbonne Universit´e, Laboratoire Jacques-Louis Lions (UMR CNRS 7598), F-75005 Paris, France. E-
mail: didier.smets@sorbonne-universite.fr
1
2 State of art
We tackled new and difficult mathematical problems on the structure and dynamics of
topological singularities arising in condensed matter physics that are described by contin-
uum variational principles. The Landau-Lifshitz-Gilbert equation and the Gross-Pitaevskii
(GP) equation are two paradigms for the models we addressed. Each problem involves
topological information, as order parameters tend (by a penalty) to take values into a non-
trivial manifold (for instance the unit sphere). The models are also multiscale and can
be analyzed for large or small parameter regimes by means of asymptotic analysis. The
formation of singularities in certain limits is due either to topological constraints or to the
failure of some critical functional embeddings, and there is substantial overlap in some of
the analytic difficulties (although magnetic and superconducting materials obey different
dynamical laws).
Our scientific program was composed by three major parts:
(A) Pattern formation in micromagnetics. Micromagnetics, a continuum model for
the behavior of ferromagnetic materials introduced by Landau-Lifshitz in the 30’ and devel-
oped by Brown in the 60’, can be seen as a typical example of multiscale problems. Indeed,
ferromagnetic materials exhibit complex microstructures such as magnetic domains, domain
walls and vortices. We studied the formation of such structures in order to predict the mor-
phology and to determine the properties of patterns at different scales. These questions are
of vital importance for a number of key technological applications.
(B) Dynamics of micromagnetic singularities. The fundamental dynamic law in
ferromagnetism, given by the Landau-Lifshitz-Gilbert (LLG) equation, is a combination
of precession and relaxation dynamics (thus neither a hamiltonian system nor a gradient
flow). The balance of energetic stray-field forces and dynamic gyro-forces becomes singular
in certain regimes. The resulting effects include complex oscillatory phenomena such as spin
waves and resonances. Important ingredients for the evolution of micromagnetic patterns
within a complex energy landscape, are effective motion laws for micromagnetic singularities
that we investigated in specific situations.
(C) Vortex and traveling waves in the Gross-Pitaevskii equation. The Gross-
Pitaevskii equation occurs in many areas in Physics from superfluidity and Bose-Einstein
condensates to nonlinear optics. Although it was intended to be a simple model and was
extensively studied in the last three decades, it is still far from being well understood.
The balance between nonlinearity and dispersion gives traveling waves, while the nonzero
condition at infinity and the penalty constraint in the energy generate vortices. It can
be approximated by Euler’s equations in some regime, or by the Kadomtsev-Petviashvili I
equation in another regime.
2
3 Scientific approach
In this project, we developed new mathematical tools for the analysis of Gross-Pitaevskii and
Landau-Lifshitz equations. In the mathematical models driven by these two equations, the
interplay of only a very small number of basic parameters creates a wealth of phenomena,
including ones that are experimentally and numerically largely inaccessible due to their
multiscale nature. With the help of mathematical analysis, we rigorously derived results
that explain the physical observations by relating them to the few basic effects that are the
foundation of the model - thereby seconding (or disproving) its validity. In particular, to
understand micromagnetics, we tested two new approaches:
(a) identifying the scaling law of the minimum energy, and the character of magnetization
patterns that achieve it;
(b) identifying simpler models, valid in appropriate regimes, whose behavior is easier
to understand or simulate (for example, the reduced model derived for asymmetric
domain walls).
These new approaches are both based on asymptotic analysis (such as the Gamma-
convergence method). We were able to take advantage of the presence of small nondi-
mensional ratios, for example exchange length or film thickness divided by diameter of
transversal section. For the asymptotic analysis, we assumed that some or all of these ra-
tios tend to zero, with specified relations between them. The existence of more than one
nondimensional parameter makes the problem rich, due to the presence of several distinct
regimes. Previous research had established a firm mathematical foundation for many micro-
magnetic or superconducting phenomena, including the formation and dynamics of domain
walls in magnetic thin-films, vortices / vortex lines for superconductors or dark solitons in
optics. By exporting modern techniques from the calculus of variations and partial differ-
ential equations (PDEs), we were able to investigate some of the many subtle features of
the theory remaining unelucidated such as the interaction energy between the N´eel walls
and their effective dynamic equation, the structure of asymmetric domain walls or the in-
teraction between different modes in the Gross-Pitaevskii equation. We highlight that the
mathematical methods we developed here have also their own interest by their richness, the
variety of tools and the links they weave between different domains: analysis, PDEs and
geometry.
4 Results
We summarize now our results following the items (A),(B),(C) presented in Section 2.
(A) Pattern formation in micromagnetics. It is an intriguing question how the simple,
yet subtle, micromagnetic energy functional describes a rich variety of patterns on very
different scales. Our aim was to develop mathematical methods to analyse this multiscale
problem and characterize the formation of domain walls.
3
Symmetric domain walls. A first example is the (symmetric) N´eel wall, a smooth transition
layer characterized by a one-dimensional in-plane rotation connecting two (opposite) direc-
tions of the magnetization. Ignat-Moser [38, 40] succeeded to solve an open problem in the
physical literature concerning the interaction energy between the N´eel walls; more precisely,
we computed the renormalized energy that governs the position of these walls. In two other
articles (Ignat-Moser [34, 21]) we studied composite N´eel walls that have a prescribed wind-
ing number; we succeeded to fully describe the diagram of existence / non-existence of such
minimizing composite N´eel walls in terms of the applied magnetic field and the prescribed
winding number. We also mention here the papers Ignat-Monteil [14, 12] where we devel-
oped a general calibration/entropy method in order to prove the symmetry of domain walls
(such as Bloch walls). More precisely, we determined a large class of anisotropies for which
calibrations exist, thus yielding the 1D structure of these domain walls. Similar symmetry
results can be given in the context of a phase-field-crystal model (see Ignat-Zorgati [16]).
Interior vortices and boundary vortices. Next to N´eel walls, interior and boundary vortices
play an important role as topological point defects of the magnetization. Recent progress
in the study of boundary vortices has been done by Ignat-Kurzke [5] in a thin film regime
where boundary vortices are energetically less expensive than interior vortices. More pre-
cisely, we developed a theory for the “global Jacobian” and the Gamma-convergence method
in order to prove that the micromagnetic energy and the vorticity measure concentrate at
the boundary and to determine the interaction energy between the boundary vortices that
governs their optimal position. An important ingredient here is based on a global estimate
for the modulus of solutions to a 2D Ginzburg-Landau equation proved by Ignat-Lamy-
Kurzke [7]. In a recent work, Ignat-Jerrard [33, 4] studied a thin film regime of magnetic
shells where vortices appear naturally on the limiting 2D surface. We determined the
renormalized energy between vortex points as a Gamma-limit (at the second order); the in-
teraction energy governing the optimal location of vortices depends on the Gauss curvature
of the surface as well as on a quantized flux. The coupling between flux quantization con-
straints and vorticity, and its impact on the renormalized energy, are new phenomena in the
theory of Ginzburg-Landau type models. We also mention here the works of Ignat-Nguyen-
Slastikov-Zarnescu [44, 39, 28, 15, 23, 8] on vortex defects in nematic liquid crystals and
superconductors, many features of these defects (stability, symmetry, uniqueness results)
being similar to micromagnetic vortices. In the article [10], Goldman-Merlet-Millot studied
defect patterns in a variational problem that is a combination of the complex Ginzburg-
Landau functional and the Mumford-Shah functional allowing for line discontinuities in the
orientation of the order parameter. They proved that minimizers of such functional develop
finitely many point singularities of vortex type. In addition, those vortices are connected
by line singularities to form clusters of entire topological degree, and each cluster is given
by a Steiner tree connecting its own vortices. A phase field regularization of the classical
2D Steiner problem was studied by Bonnivard-Lemenant-Millot [24] which deals with the
asymptotic behavior of minimizers of a Ginzburg-Landau type functional. A consistency
result shows that level sets of minimizers converge to solutions of the Steiner problem in
the small parameter limit. Let us mention that we also developed a more general theory on
4
the structure and properties of maps with values into manifolds that may satisfy divergence
or curl constraints: the vortex structure in the eikonal equation (see De Lellis-Ignat [43],
Bochard-Ignat [30]) as well as the lifting structure for BV maps with values into projective
spaces (see Ignat-Lamy [22]).
Asymmetric domain walls. A puzzling question concerns the cross-over from symmetric
to asymmetric domain walls. D¨oring-Ignat [37] succeeded to analyse the bifurcation phe-
nomenon in a regime where a symmetry breaking occurs and the (symmetric) N´eel wall loses
stability. The derived reduced model allows to capture asymmetric domain walls including
their extended tails (which were previously inaccessible to brute-force numerical simula-
tion). In the paper Ignat-Otto [18], we studied another asymmetric domain wall, called the
magnetization ripple that is a microstructure formed by the magnetization in a thin film
ferromagnet. It is triggered by the random orientation of the grains in the polycrystalline
material. We developed a small-data well-posedness theory, taking inspiration from the
recent rough-path approach to singular stochastic PDEs.
Singularities in nonlocal variationnal problems. In the articles [17, 27, 6, 11], Millot and co-
authors have studied nonlocal fractional versions of the Ginzburg-Landau equation and the
harmonic map system involving the fractional Laplacian. A main objective was to study the
influence of such type of operator for the singularities induced by topological constraints. In
[17], it is shown the convergence of solutions in the small (Ginzburg-Landau) parameter limit
towards stationary nonlocal minimal surfaces; a regularity result for such nonlocal minimal
surfaces is also obtained, and a connection with fractional harmonic maps is established.
Articles [27, 6] address the regularity issue for fractional harmonic maps into spheres in
the stationary or minimizing setting. In each case, a partial regularity result is obtained
showing smoothness away from a closed singular set whose Hausdorff dimension is estimated
in an optimal way. In the article [11], a classification result for all possible singularities is
established in 2D for a critical exponent, a case of specific interest in the study of boundary
singularities arising in micromagnetics or liquid crystals models.
(B) Dynamics of micromagnetic singularities.
Traveling waves in the Landau-Lifshitz-Gilbert equation. In the article Cˆote-Ignat [1], we
describe precessing domain walls, and prove that they are asymptotically stable under an
applied magnetic field which is sufficiently small. The method we developed also opens a
line of research and provides tools to understand the interaction of several domain walls
and the description of general magnetizations for large times. Periodic structures have
been the object of the article Gustafson-Le Coz-Tsai [31]. The authors proved the stabil-
ity of periodic waves in the framework of the nonlinear Schr¨odinger equation instead of
the Landau-Lifshitz-Gilbert equation (which can be the object of further study using the
tools developed in the framework of the Schr¨odinger equation). Also in the context of a
nonlinear Schr¨odinger system, Delebecque-Le Coz-Weisha¨upl [36] studied numerically and
theoretically solitary waves. This study is continued in a work in progress by De Bi`evre-
Genoud-Le Coz-Rota Nodari on the Manakov system and its stationary waves. We mention
the work Le Coz-Martel-Rapha¨el [42] where the authors constructed a blowing up solution
5
for a nonlinear Schr¨odinger equation with double power nonlinearity. This solution has the
peculiarity to be a blowing up solution at minimal mass whose speed does not correspond
to any of the blowup speeds that were known before.
Vortex dynamics. The evolution of Ginzburg-Landau vortices was studied by Cˆote-Cˆote
[35] and is given by a weak form of the mean curvature flow. For this, we extended the
articles by Bethuel-Orlandi-Smets, the main contribution is to work in a physically relevant
framework (i.e., locally finite energy density). We mention that the evolution of boundary
vortices under the flow of the Landau-Lifshitz-Gilbert equation is a work in progress by
Ignat-Kurzke.
(C) Vortex and traveling waves in the Gross-Pitaevskii equation.
Vorticity tubes. The evolution of vorticity tubes for the Gross-Pitaevskii equation (in its 3D
axisymmetric version) is studied by Jerrard-Smets [26]; more precisely, they give a math-
ematical proof of the leap-frogging phenomenon, i.e., an interaction phenomenon between
vortex tubes which was foreseen already by Helmholtz’ in 1857 (in the case of incompressible
perfect fluids), but for which a mathematical confirmation was not yet available. The same
two authors then started the study of the evolution of vorticity tubes under no symmetry
assumption. Very recently, they proved the validity of the Klein-Majda-Damodaran model
in the limit of small curvature tubes. This work (still in progress) seems to be the first truly
3D mathematical result regarding the evolution of vortex tubes. In the articles [20, 13],
Gallay-Smets studied the evolution of vortex tubes in the context of classical incompress-
ible non viscous fluids (3D Euler equation). Although this was not explicitly mentioned in
the ANR proposal, it is a very natural extension of the framework for the Gross-Pitaevskii
equation which will surely benefit from it. The situation is notably more complex than
for the Gross-Pitaveskii equation, due to co-existence of several scales. The authors proved
under general assumptions the 3D spectral stability of a straight columnar vortex. By doing
so, they answered positively a classical problem dating back to Kelvin, and for which no
progress had been made since the 1960s.
Stability of localized structures. In a Gross-Pitaevskii-Schr¨odinger system, if the Schr¨odinger
component becomes highly concentrated, it is natural to approximate the system as a Gross-
Pitaevskii (GP) equation perturbed by a Dirac mass. This equation has been the subject of
an in depth investigation (Cauchy problem and stability of stationary waves) in the article
of Ianni-Le Coz-Royer [32]. Also, Le Coz-Wu [29] established the stability of multi-solitons
for the derivative nonlinear Schr¨odinger equation. This result is one of the rare results
concerning stability of multi-solitons; it is due to the particular structure of the equation,
which is at the same time critical and admits stable solitons. In the article Cˆote-Martel [25],
we constructed excited multi-solitons for the nonlinear Klein-Gordon equation: this is the
first result of existence for progressive waves based on excited states without any restriction
like high relative speeds. The main point is that the linearized flow around these states is
linearly unstable, and beyond this, little is known (to the contrary of the case of ground
states). On a similar topic, we described completely 2-soliton solutions to the damped
nonlinear Klein-Gordon equation in Cˆote-Martel-Yuan-Zhao [3]: construction, description
6
of the profile for large times, and smoothness of the set of initial data leading to 2-solitons
(essentially, a Lipschitz manifold of codimension 2).
Interaction between modes. In the preprints by Correia-Cˆote-Vega [9, 2], we studied a dis-
persive model for the formation and evolution of vortex filaments making a corner. We
studied the self-similar solutions to the modified Korteweg-de Vries equation (which corre-
sponds to the curvature of the filaments), and we found a (critical) functional framework in
which the flow can be studied. We mention that Maris [41] developed a very general profile
decomposition method and used it for a broad family of minimization problems. Although
it was initially motivated by the study of special solutions of the (GP) and (LLG) equa-
tions, it has been applied to many other problems (including solitary waves for dispersive
equations, optimal functions for the Gagliardo-Nirenberg inequalities, liquid crystals...). In
a work in progress, Alhelou-Maris study traveling waves for a Gross-Pitaevskii-Schr¨odinger
system describing the movement of an impurity in a Bose-Einstein condensate. They proved
the existence of such solutions with large momentum, small mass and speed close to zero.
This result holds in any space dimension greater or equal than two. No existence result was
available before, except for the dimension 1.
5 Discussion
The objectives proposed in our project have advanced very well. Our results made a sig-
nificant progress in the theoretical study of the Landau-Lifshitz equation and the Gross-
Pitaevskii equation. We succeeded to prove several conjectures in the physical and mathe-
matical literature (such as determining the energy interaction between N´eel walls, determin-
ing the evolution of vortex tubes in the Gross-Pitaevskii equation, the existence of traveling
waves with large momentum, small mass and speed close to zero for a Gross-Pitaevskii-
Schr¨odinger system). We developed new mathematical techniques that will have a strong
impact in the study of singular structures in PDEs and Calculus of Variations (such as the
calibration / entropy method to prove the symmetry of domain walls, or the decomposition
method to prove stability or uniqueness of symmetric solutions). We opened new perspec-
tives in the analysis of domain walls in micromagnetics. For example, we started the study
of the structure of asymmetric domain walls in [37]; the next step consists in analyzing
the bifurcation phenomenon for asymmetric type walls when a change of topological degree
creates a new symmetry breaking (i.e., a vortex defect becomes favored by the system).
This question will rely on the following new-insight issues:
1. the study of harmonic unimodular maps satisfying a divergence constraint and a
prescribed degree;
2. a concentration-compactness principle for sign-changing functions.
Also, the study of vortices at the surface of thin magnetic shells in [4] highlighted new
phenomena in the theory of Ginzburg-Landau type models: the coupling between flux
quantization constraints and vorticity, and its impact on the renormalized energy. Our
7
project was very ambitious, some of the many questions we raised in the project are still
work in progress. For example, the dynamics of N´eel walls for which the analysis we
developed in [38, 21] is fundamental, or the dynamics of boundary vortices for the Landau-
Lifshitz-Gilbert equation (for which the study at the stationary level of boundary vortices
we did in [5] is essential). Also, we succeeded to describe precessing domain walls in [1],
and proved that they are asymptotically stable under an applied magnetic field which is
sufficiently small; the method we developed opens a new line of research and provides tools to
understand the interaction of several domain walls and the description of the magnetization
for large times.
6 Conclusion
We believe that our project was a truly success. We succeeded to make a significant progress
in the analysis of topological singular phenomena in the Landau-Lifshitz equation and the
Gross-Pitaevskii equation. We developed new mathematical methods that enabled us to
prove important open questions in the physical and mathematical literature (as explained
in the above sections). More than that, the development of these new tools and methods in
nonlinear analysis can be used for a much larger class of problems than we expected in the
original proposal. Therefore, they open new perspectives and will have a strong impact in
the study of singular phenomena in PDEs and Calculus of Variations. Our project was very
productive: 45 articles and reports-proceedings. Our papers are published in top journals
in PDEs and Calculus of Variations (Annals of PDEs, ARMA, CalcVarPDEs, AnnIHP,
Analysis &PDEs, CMP...) as well as top journals in Pure and Applied Mathematics (CPAM,
AnnSciENS, Adv. Math, JMPA...). We were invited speakers at more than 45 national
and international conferences to present our results. We also invited international experts
as guests in our departments for several research visits or for the several conferences we
organised in Toulouse; those scientific discussions turned in well established collaborations
today. Also, during the project, 3 members of the project defended their habilitation
HDR, 2 members became full professors, and 2 members became junior members at Institut
Universitaire de France (IUF) with two research projects very related to our ANR project.
We also mention that 6 of our students obtained their PhD degree during the project by
working on related topics.
References
[1] Raphael Cˆote and Radu Ignat. Asymptotic stability of precessing domain walls for the
Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction.
Comm. Math. Phys. 401 (2023), 2901-2957.
[2] Sim˜ao Correia, Rapha¨el Cˆote, and Luis Vega. Self-similar dynamics for the modified
Korteweg-de Vries equation. Int. Math. Res. Not. IMRN 2021, no. 13, 9958-10013.
8
[3] Rapha¨el Cˆote, Yvan Martel, Xu Yuan, and Lifeng Zhao. Description and classification
of 2-solitary waves for nonlinear damped Klein-Gordon equations. Comm. Math. Phys.
388 (2021), no. 3, 1557-1601.
[4] Radu Ignat and Robert L. Jerrard. Renormalized energy between vortices in some
Ginzburg-Landau models on 2-dimensional Riemannian manifolds. Arch. Ration. Mech.
Anal. 239 (2021), 1577-1666.
[5] Radu Ignat and Matthias Kurzke. Global Jacobian and Γ-convergence in a two-
dimensional Ginzburg-Landau model for boundary vortices. J. Funct. Anal. 280 (2021),
66pp.
[6] Vincent Millot, Marc Pegon, and Armin Schikorra. Partial regularity for fractional
harmonic maps into spheres. Arch. Ration. Mech. Anal. 242 (2021), no. 2, 747-825.
[7] Radu Ignat, Matthias Kurzke, and Xavier Lamy. Global uniform estimate for the mod-
ulus of two-dimensional Ginzburg-Landau vortexless solutions with asymptotically in-
finite boundary energy. SIAM J. Math. Anal. 52 (2020), 524-542.
[8] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Symmetry and mul-
tiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals.
Arch. Ration. Mech. Anal. 237 (2020), 1421-1473.
[9] Sim˜ao Correia, Raphael Cˆote, and Luis Vega. Asymptotics in Fourier space of self-
similar solutions to the modified Korteweg-de Vries equation. J. Math. Pures Appl. (9)
137 (2020), 101-142.
[10] Mickael Goldman, Benoit Merlet, and Vincent Millot. A Ginzburg-Landau model with
topologically induced free discontinuities. Ann. Inst. Fourier (Grenoble) 70 (2020), no.
6, 2583-2675.
[11] Vincent Millot and Marc Pegon. Minimizing 1/2-harmonic maps into spheres. Calc.
Var. Partial Differential Equations 59 (2020), no. 2, Paper No. 55, 37 pp.
[12] Radu Ignat and Antonin Monteil. A De Giorgi-type conjecture for minimal solutions
to a nonlinear Stokes equation. Comm. Pure Appl. Math. 73 (2020), 771-854.
[13] Thierry Gallay and Didier Smets. Spectral stability of inviscid columnar vortices. Anal.
PDE 13 (2020), no. 6, 1777-1832.
[14] Radu Ignat and Antonin Monteil. A necessary condition in a De Giorgi type conjecture
for elliptic systems in infinite strips. Volume dedicated to Haim Brezis on the occasion
of his 75th birthday, Pure and Applied Functional Analysis 5 (2020), 981-999.
[15] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. On the uniqueness
of minimisers of Ginzburg-Landau functionals. Ann. Sci. ´
Ec. Norm. Sup´er. 53 (2020),
589-613.
9
[16] Radu Ignat and Hamdi Zorgati. Dimension reduction and optimality of the uniform
state in a Phase-Field-Crystal model involving a higher order functional, J. Nonlinear
Sci. 30 (2020), 261-282.
[17] Vincent Millot, Yannick Sire, and Kelei Wang. Asymptotics for the fractional Allen-
Cahn equation and stationary nonlocal minimal surfaces. Arch. Ration. Mech. Anal.
231 (2019), no. 2, 1129-1216.
[18] Radu Ignat and Felix Otto. The magnetization ripple: a nonlocal stochastic PDE
perspective. J. Math. Pures Appl.(9) 130 (2019), 157-199.
[19] Philippe Gravejat and Didier Smets. Smooth travelling-wave solutions to the inviscid
surface quasi-geostrophic equation. Int. Math. Res. Not. IMRN 2019, no. 6, 1744-1757.
[20] Thierry Gallay and Didier Smets. On the linear stability of vortex columns in the
energy space. J. Math. Fluid Mech. 21 (2019), no. 4, Paper No. 48, 27 pp.
[21] Radu Ignat and Roger Moser. Energy minimisers of prescribed winding number in
an S1-valued nonlocal Allen-Cahn type model. Advances in Mathematics 357 (2019),
106819.
[22] Radu Ignat and Xavier 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, Calc. Var. Partial Differential Equations 58 (2019), Art. 68, 26 pp.
[23] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Some uniqueness
results for minimisers of Ginzburg-Landau functionals, Oberwolfach Reports, Volume
20/2018 (Nonlinear Data: Theory and Algorithms), 2018.
[24] Matthieu Bonnivard, Antoine Lemenant, and Vincent Millot. On a phase field ap-
proximation of the planar Steiner problem: existence, regularity, and asymptotic of
minimizers. Interfaces Free Bound., 20(1):69-106, 2018.
[25] Raphael Cˆote and Yvan Martel. Multi-travelling waves for the nonlinear Klein-Gordon
equation. Trans. Amer. Math. Soc., 370(10), 7461-7487, 2018.
[26] Robert L. Jerrard and Didier Smets. Leapfrogging vortex rings for the three dimensional
Gross-Pitaevskii equation. Ann. PDE, 4(1):Art. 4, 48, 2018.
[27] Vincent Millot, Yannick Sire, and Hui Yu. Minimizing fractional harmonic maps on
the real line in the supercritical regime. Discrete and Continuous Dynamical Systems
- A, 38:6195, 2018.
[28] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Uniqueness of degree-
one Ginzburg-Landau vortex in the unit ball in dimensions N≥7. C. R. Math. Acad.
Sci. Paris 356, 922-926, 2018.
10
[29] Stefan Le Coz and Yifei Wu. Stability of multi-solitons for the derivative nonlinear
Schr¨odinger equation. International Mathematics Research Notices, 2018(13), 4120-
4170, 2018.
[30] Pierre Bochard and Radu Ignat. Kinetic formulation of vortex vector fields. Anal. PDE
10, 729-756, 2017.
[31] Stephen Gustafson, Stefan Le Coz, and Tai-Peng Tsai. Stability of periodic waves of 1D
cubic nonlinear Schr¨odinger equations. Appl. Math. Res. Express. AMRX 2, 431-487,
2017.
[32] 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,
1060-1099, 2017.
[33] Radu Ignat and Robert L. Jerrard. Interaction energy between vortices of vector fields
on Riemannian surfaces. Comptes Rendus Mathematique 355, 515-521, 2017.
[34] Radu Ignat and Roger Moser. N´eel walls with prescribed winding number and how a
nonlocal term can change the energy landscape. J. Differential Equations 263, 5846-
5901, 2017.
[35] Delphine Cˆote and Raphael Cˆote. Limiting motion for the parabolic Ginzburg-Landau
equation with infinite energy data. Comm. Math. Phys. 350, 507-568, 2017.
[36] Fanny Delebecque, Stefan Le Coz, and Rada Maria Weish¨aupl. Multi-speed solitary
waves of nonlinear Schr¨odinger systems: theoretical and numerical analysis. Comm.
Math. Sci. 14, 1599-1624, 2016.
[37] Lukas D¨oring and Radu Ignat. Asymmetric domain walls of small angle in soft ferro-
magnetic films. Arch. Rational Mech. Anal. 220, 889-936, 2016.
[38] 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.
[39] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Stability of point
defects of degree ±1
2in a two-dimensional nematic liquid crystal model. Calc. Var.
Partial Differential Equations 55 (2016), 33pp.
[40] Radu Ignat. Interaction energy of domain walls of logarithmically decaying tails in a
nonlocal variational model. Oberwolfach Reports, Volume 13/2016 (Calculus of Varia-
tions: S. Brendle, A. Figalli, R. Jerrard, N. Wickramasekera), 2016.
[41] Mihai Maris. Profile decomposition for sequences of Borel measures. preprint
arxiv:1410.6125, 2016.
11
[42] Stefan Le Coz, Yvan Martel, and Pierre Raphael. Minimal mass blow up solutions for
a double power nonlinear Schr¨odinger equation. Rev. Mat. Iberoam. 32, 795-833, 2016.
[43] Camillo De Lellis and Radu Ignat. A regularizing property of the 2D-eikonal equation.
Comm. Partial Differential Equations 40, 1543-1557, 2015.
[44] Radu Ignat, Luc Nguyen, Valeriy Slastikov, and Arghir Zarnescu. Instability of point
defects in a two-dimensional nematic liquid crystal model. Ann. Inst. H. Poincar´e Anal.
Non Lin´eaire 33, 1131-1152, 2015.
[45] Mihai Maris. On some minimization problems in RN. Proceedings of the 8th Congress
of Romanian Mathematicians, pages 216-231, 2015.
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