Abstract: Publication date: September 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 6Author(s): Casey Kelleher, Jeffrey StreetsAbstractWe study singularity structure of Yang–Mills flow in dimensions n≥4. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang–Mills connections or solitons as blowup limits at any point in the singular set.

Abstract: Publication date: Available online 31 August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Piero Montecchiari, Paul H. RabinowitzAbstractFor about 25 years, global methods from the calculus of variations have been used to establish the existence of chaotic behavior for some classes of dynamical systems. Like the analytical approaches that were used earlier, these methods require nondegeneracy conditions, but of a weaker nature than their predecessors. Our goal here is study such a nondegeneracy condition that has proved useful in several contexts including some involving partial differential equations, and to show this condition has an equivalent formulation involving stable and unstable manifolds.

Abstract: Publication date: Available online 31 August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Grégory Faye, Matt HolzerAbstractWe study invasion fronts and spreading speeds in two component reaction-diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.

Abstract: Publication date: Available online 27 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Mostafa Fazly, Yannick SireAbstractWe pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p−Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.

Abstract: Publication date: Available online 27 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Hiroshi Matano, Yoichiro Mori, Mitsunori NaraAbstractWe consider the Cauchy problem for the anisotropic (unbalanced) Allen-Cahn equation on Rn with n≥2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow Vγ=κγ+c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen-Cahn equation plays a key role.RésuméNous considérons le problème de Cauchy pour l'équation d'Allen-Cahn (de moyenne non nulle) anisotropique dans Rn avec n≥2, et étudions le comportement en temps grand des solutions propageantes. Nous montrons, sous des hypothèses assez faibles sur la donnée initiale, que la solution développe un véritable front de propagation dont la position peut être approchée d'assez près, en temps grand, par une forme de Wulff en expansion. Un tel comportement peut être attendu formellement, et il existe aussi dans la littérature certaines études rigoureuses sur des problèmes analogues. Le principal objectif de cet article est d'établir des résultats d'approximation plus fins que ce qui était connu auparavant. Plus précisément, la distance de Hausdorff entre un ensemble de niveau de la solution et la forme de Wulff en expansion reste bornée uniformément en temps grand. De plus, chaque ensemble de niveau devient en temps fini une hypersurface régulière, quelque soit l'irrégularité de sa configuration initiale, et le mouvement de cette hypersurface est régi (approximativement) par le flot de courbure moyenne anisotropique Vγ=κγ+c, avec une marge d'erreur petite. Nous prouvons aussi la rigidité asymptotique du profil de la solution, c'est-à-dire qu'il converge, à proximité du front et quand le temps tend vers l'infini, vers le profil de l'onde progressive. Une extension au cas anisotropique d'un théorème de type Liouville de Berestycki et Hamel (2007), portant sur les solutions entières de l'équation d'Allen-Cahn, joue un rôle clé dans la preuve de ce dernier résultat, ainsi que de la régularité des ensembles de niveau.

Abstract: Publication date: Available online 26 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Benjamin Gess, Xavier LamyAbstractWe prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.

Abstract: Publication date: Available online 17 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Daniel Coutand, Steve ShkollerAbstractIn fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for d-dimensional flows, d=2 or 3, the free-surface of a viscous water wave, modeled by the incompressible Navier-Stokes equations with moving free-boundary, has a finite-time splash singularity for a large class of specially prepared initial data. In particular, we prove that given a sufficiently smooth initial boundary (which is close to self-intersection) and a divergence-free velocity field designed to push the boundary towards self-intersection, the interface will indeed self-intersect in finite time.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Paulo BrandãoAbstractWe study the non-wandering set of contracting Lorenz maps. We show that if such a map f doesn't have any attracting periodic orbit, then there is a unique topological attractor. Furthermore, we classify the possible kinds of attractors that may occur.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Eduard Feireisl, Václav Mácha, Šárka Nečasová, Marius TucsnakAbstractWe consider a system modelling the motion of a piston in a cylinder filled by a viscous heat conducting gas. The piston is moving longitudinally without friction under the influence of the forces exerted by the gas. In addition, the piston is supposed to be thermally insulating (adiabatic piston). This fact raises several challenges which received a considerable attention, essentially in the statistical physics literature. We study the problem via the methods of continuum mechanics, specifically, the motion of the gas is described by means of the Navier–Stokes–Fourier system in one space dimension, coupled with Newton's second law governing the motion of the piston. We establish global in time existence of strong solutions and show that the system stabilizes to an equilibrium state for t→∞.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Inwon Kim, Olga TuranovaAbstractWe study a model introduced by Perthame and Vauchelet [19] that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We adopt the viscosity solution approach to establish an optimal uniform convergence result of the tumor density as well as the pressure in the incompressible limit. The system lacks standard maximum principle, and thus modification of the usual approach is necessary.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Mitia Duerinckx, Julian FischerAbstractWe establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in [24] as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Andrei V. FaminskiiAbstractInitial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are established.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Denis SerreAbstractWe consider d×d tensors A(x) that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of (detA)1d−1. We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc. We deduce an a priori estimate for a new quantity, namely the space–time integral of ρ1np, where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Christoph Scheven, Thomas SchmidtAbstractWe investigate the Dirichlet minimization problem for the total variation and the area functional with a one-sided obstacle. Relying on techniques of convex analysis, we identify certain dual maximization problems for bounded divergence-measure fields, and we establish duality formulas and pointwise relations between (generalized) BV minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of BV minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as BV obstacles, thin obstacles, and boundary obstacles, and they include information on exceptional sets and up to the boundary. As a side benefit, in some cases we also obtain assertions on the limit behavior of p-Laplace type obstacle problems for p↘1.On the technical side, the statements and proofs of our results crucially depend on new versions of Anzellotti type pairings which involve general divergence-measure fields and specific representatives of BV functions. In addition, in the proofs we employ several fine results on (BV) capacities and one-sided approximation.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Wei Zhang, Jiguang BaoAbstractWe classify all solutions to−utdetD2u=f(x) in R−n+1, where f∈Cα(Rn) is a positive periodic function in x. More precisely, if u is a parabolically convex solution to above equation, then u is the sum of a convex quadratic polynomial in x, a periodic function in x and a linear function of t. It can be viewed as a generalization of the work of Gutiérrez and Huang in 1998. And along the line of approach in this paper, we can treat other parabolic Monge–Ampère equations.RésuméNous classifions toutes les solutions à−utdetD2u=f(x) in R−n+1, où f∈Cα(Rn) est une fonction périodique positive en x. Plus précisément, si u est une solution paraboliquement convexe de l'équation ci-dessus, alors u est la somme d'un polynôme quadratique convexe en x, une fonction périodique en x et une fonction linéaire de t. Cela peut être considéré comme une généralisation du travail de Gutiérrez et Huang en 1998. Et le long de la ligne d'approche dans cet article, nous pouvons traiter d'autres équations paraboliques Monge–Ampère.

Abstract: Publication date: August 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 5Author(s): Haïm Brezis, Petru MironescuAbstractWe investigate the validity of the Gagliardo–Nirenberg type inequality(1)‖f‖Ws,p(Ω)≲‖f‖Ws1,p1(Ω)θ‖f‖Ws2,p2(Ω)1−θ, with Ω⊂RN. Here, 0≤s1≤s≤s2 are non negative numbers (not necessarily integers), 1≤p1,p,p2≤∞, and we assume the standard relationss=θs1+(1−θ)s2,1/p=θ/p1+(1−θ)/p2 for some θ∈(0,1).By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s1,s2,s are integers. It turns out that (1) holds for “most” of values of s1,…,p2, but not for all of them. We present an explicit condition on s1,s2,p1,p2 which allows to decide whether (1) holds or fails.

Abstract: Publication date: Available online 5 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Matt McGonagle, Chong Song, Yu YuanAbstractWe derive Hessian estimates for convex solutions to quadratic Hessian equation by compactness argument.RésuméNous dérivons des estimations de Hessian pour des solutions convexes á l'équation de Hessian quadratique par argument de compacité.

Abstract: Publication date: Available online 5 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Sergio Conti, Matteo Focardi, Flaviana IurlanoAbstractWe consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization of the decay estimate by De Giorgi, Carriero, and Leaci to the vectorial situation. This is based on replacing the coarea formula by a method to approximate SBDp functions with small jump set by Sobolev functions and is restricted to two dimensions. The other two ingredients are contained in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy GSBDp functions by SBVp ones.

Abstract: Publication date: Available online 20 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Antonin Monteil, Jean Van SchaftingenAbstractGiven a connected Riemannian manifold N, an m-dimensional Riemannian manifold M which is either compact or the Euclidean space, p∈[1,+∞) and s∈(0,1], we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space Ws,p(M,N) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach–Steinhaus uniform boundedness principle in linear Banach spaces.

Abstract: Publication date: Available online 19 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Jürgen Jost, Lei Liu, Miaomiao ZhuAbstractFor a sequence of coupled fields {(ϕn,ψn)} from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.

Abstract: Publication date: Available online 19 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Emanuel Carneiro, Diogo Oliveira e Silva, Mateus SousaAbstractThe L2→Lp adjoint Fourier restriction inequality on the d-dimensional hyperboloid Hd⊂Rd+1 holds provided 6≤p

Abstract: Publication date: Available online 15 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Marine Fontaine, Mohammed Lemou, Florian MéhatsAbstractIn this paper we prove the nonlinear orbital stability of a large class of steady state solutions to the Hamiltonian Mean Field (HMF) system with a Poisson interaction potential. These steady states are obtained as minimizers of an energy functional under one, two or infinitely many constraints. The singularity of the Poisson potential prevents from a direct run of the general strategy in [16], [19] which was based on generalized rearrangement techniques, and which has been recently extended to the case of the usual (smooth) cosine potential [17]. Our strategy is rather based on variational techniques. However, due to the boundedness of the space domain, our variational problems do not enjoy the usual scaling invariances which are, in general, very important in the analysis of variational problems. To replace these scaling arguments, we introduce new transformations which, although specific to our context, remain somehow in the same spirit of rearrangements tools introduced in the references above. In particular, these transformations allow for the incorporation of an arbitrary number of constraints, and yield a stability result for a large class of steady states.

Abstract: Publication date: Available online 7 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Francesco Boarotto, Mario SigalottiAbstractWe consider in this paper the regularity problem for time-optimal trajectories of a single-input control-affine system on a n-dimensional manifold. We prove that, under generic conditions on the drift and the controlled vector field, any control u associated with an optimal trajectory is smooth out of a countable set of times. More precisely, there exists an integer K, only depending on the dimension n, such that the non-smoothness set of u is made of isolated points, accumulations of isolated points, and so on up to K-th order iterated accumulations.

Abstract: Publication date: Available online 6 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Maarten de Hoop, Gunther Uhlmann, Yiran WangAbstractFor scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.

Abstract: Publication date: Available online 6 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Raffaele Carlone, Michele Correggi, Lorenzo TentarelliAbstractWe consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.

Abstract: Publication date: Available online 6 June 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Animikh Biswas, Ciprian Foias, Cecilia F. Mondaini, Edriss S. TitiAbstractBased on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier–Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier–Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier–Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.

Abstract: Publication date: Available online 31 May 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Tej-Eddine Ghoul, Van Tien Nguyen, Hatem ZaagAbstractWe consider the following parabolic system whose nonlinearity has no gradient structure:{∂tu=Δu+ v p−1v,∂tv=μΔv+ u q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:{u(x,t)∼Γ[(T−t)(1+b x−a 2(T−t) log(T−t) )]−(p+1)pq−1,v(x,t)∼γ[(T−t)(1+b x−a 2(T−t) log(T−t) )]−(q+1)pq−1, with b=b(p,q,μ)>0 and

Abstract: Publication date: Available online 18 May 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Qiyu Chen, Jean-Marc SchlenkerAbstractWe prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.

Abstract: Publication date: Available online 7 May 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Pierre Germain, Fabio Pusateri, Frédéric RoussetAbstractWe consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.

Abstract: Publication date: Available online 25 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Luis Caffarelli, Gonzalo DávilaAbstractWe study the regularity of solutions of elliptic fractional systems of order 2s, s∈(0,1), where the right hand side f depends on a nonlocal gradient and has the same scaling properties as the nonlocal operator. Under some structural conditions on the system we prove interior Hölder estimates in the spirit of [1]. Our results are stable in s allowing us to recover the classic results for elliptic systems due to S. Hildebrandt and K. Widman [11] and M. Wiegner [19].

Abstract: Publication date: Available online 19 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): R. Feola, F. IandoliAbstractWe prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.

Abstract: Publication date: Available online 13 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Yinshan Chang, Yiming Long, Jian WangAbstractWe consider a continuously differentiable curve t↦γ(t) in the space of 2n×2n real symplectic matrices, which is the solution of the following ODE:dγdt(t)=J2nA(t)γ(t),γ(0)∈Sp(2n,R), where J=J2n=def[0Idn−Idn0] and A:t↦A(t) is a continuous path in the space of 2n×2n real matrices which are symmetric. Under a certain convexity assumption (which includes the particular case that A(t) is strictly positive definite for all t∈R), we investigate the dynamics of the eigenvalues of γ(t) when t varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein-Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland: {t∈R:γ(t) has a Krein indefinite eigenvalue of modulus 1} is a discrete set.

Abstract: Publication date: Available online 12 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Jiayu Li, Xiangrong ZhuAbstractLet u be a map from a Riemann surface M to a Riemannian manifold N and α>1, the α energy functional is defined asEα(u)=12∫M[(1+ ▽u 2)α−1]dV.We call uα a sequence of Sacks–Uhlenbeck maps if uα are critical points of Eα andsupα>1Eα(uα)

Abstract: Publication date: Available online 9 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Edgard A. PimentelAbstractIn this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a model-problem of a non-convex operator. In addition, the usual mechanisms to access regularity of solutions fall short in addressing these equations. We approximate an Isaacs equation by a Bellman one, and make assumptions on the latter to recover information for the former. Our techniques produce results in Sobolev and Hölder spaces; we also examine a few consequences of our main findings.

Abstract: Publication date: Available online 9 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Lili Fan, Lizhi Ruan, Wei XiangAbstractThis paper is devoted to the study of the wellposedness of the radiative Euler equations. By employing the anti-derivative method, we show the unique global-in-time existence and the asymptotic stability of the solutions of the radiative Euler equations for the composite wave of two viscous shock waves with small strength. This method developed here is also helpful to other related problems with similar analytical difficulties.

Abstract: Publication date: Available online 9 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Habib Ammari, Brian Fitzpatrick, David Gontier, Hyundae Lee, Hai ZhangAbstractThrough the application of layer potential techniques and Gohberg–Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.

Abstract: Publication date: Available online 9 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): M.B. Erdoğan, T.B. Gürel, N. TzirakisAbstractWe study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on R and prove smoothing estimates by combining the restricted norm method with a normal form transformation.

Abstract: Publication date: Available online 9 April 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Sébastien Alvarez, Jiagang YangAbstractWe consider a transversally conformal foliation F of a closed manifold M endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either there is a transverse holonomy-invariant measure for F, or the foliated geodesic flow admits a finite number of physical measures, which have negative transverse Lyapunov exponents and whose basin covers a set full for the Lebesgue measure. We also give necessary and sufficient conditions for the foliated geodesic flow to be partially hyperbolic in the case where the foliation is transverse to a projective circle bundle over a closed hyperbolic surface.

Abstract: Publication date: Available online 17 March 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): André de Laire, Philippe GravejatAbstractIt is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.

Abstract: Publication date: Available online 7 March 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Konstantin Khanin, Saša KocićAbstractWe prove that, for every ε∈(0,1), every two C2+α-smooth (α>0) circle diffeomorphisms with a break point, i.e. circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, with the same irrational rotation number ρ∈(0,1) and the same size of the break c∈R+\{1}, are conjugate to each other via a conjugacy which is (1−ε)-Hölder continuous at the break points. An analogous result does not hold for circle diffeomorphisms even when they are analytic.

Abstract: Publication date: Available online 7 March 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Marie Doumic, Miguel Escobedo, Magali TournusAbstractWe consider the fragmentation equation∂f∂t(t,x)=−B(x)f(t,x)+∫y=xy=∞k(y,x)B(y)f(t,y)dy, and address the question of estimating the fragmentation parameters - i.e. the division rate B(x) and the fragmentation kernel k(y,x) - from measurements of the size distribution f(t,⋅) at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x)=αxγ and a self-similar fragmentation kernel k(y,x)=1yk0(xy), we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet (α,γ,k0) and a representation formula for k0. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

Abstract: Publication date: Available online 7 March 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Boyan Sirakov, Carlos Tomei, André ZaccurAbstractThe well-known Ambrosetti–Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non-self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.

Abstract: Publication date: Available online 7 March 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Tristan RobertAbstractIn this article, we address the Cauchy problem for the KP-I equation∂tu+∂x3u−∂x−1∂y2u+u∂xu=0 for functions periodic in y. We prove global well-posedness of this problem for any data in the energy space E={u∈L2(R×T),∂xu∈L2(R×T),∂x−1∂yu∈L2(R×T)}. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.

Abstract: Publication date: Available online 21 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Luc Molinet, Didier Pilod, Stéphane VentoAbstractWe show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin–Ono equation∂tu−Dxα∂xu=∂x(u2),0sα:=32−5α4. As a consequence, we obtain global well-posedness in the energy space Hα2(R) as soon as α2>sα, i.e. α>67.

Abstract: Publication date: Available online 15 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): José A. Gálvez, Asun Jiménez, Pablo MiraAbstractWe give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space.

Abstract: Publication date: Available online 9 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Timothy Candy, Sebastian HerrAbstractFor arbitrarily large initial data in an open set defined by an approximate Majorana condition, global existence and scattering results for solutions to the Dirac equation with Soler-type nonlinearity and the Dirac–Klein–Gordon system in critical spaces in spatial dimension three are established.

Abstract: Publication date: Available online 2 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Emanuele Caglioti, François Golse, Mikaela IacobelliAbstractIn this paper we study a perturbative approach to the problem of quantization of probability distributions in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view [10], [12], [15], we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a new mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.

Abstract: Publication date: Available online 1 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Chao Liang, Karina Marin, Jiagang YangAbstractWe consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are Cr open and there exists a Cr open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.

Abstract: Publication date: Available online 1 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Nicolas Ginoux, Olaf MüllerAbstractWe consider the Cauchy problem for massless Dirac–Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac–Higgs–Yang–Mills theories.

Abstract: Publication date: Available online 1 February 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): P. Jameson Graber, Alpár R. MészárosAbstractIn this paper we obtain Sobolev estimates for weak solutions of first order variational Mean Field Game systems with coupling terms that are local functions of the density variable. Under some coercivity conditions on the coupling, we obtain first order Sobolev estimates for the density variable, while under similar coercivity conditions on the Hamiltonian we obtain second order Sobolev estimates for the value function. These results are valid both for stationary and time-dependent problems. In the latter case the estimates are fully global in time, thus we resolve a question which was left open in [23]. Our methods apply to a large class of Hamiltonians and coupling functions.

Abstract: Publication date: Available online 31 January 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Eduard Feireisl, Antonín NovotnýAbstractWe consider the stationary compressible Navier–Stokes system supplemented with general inhomogeneous boundary conditions. Assuming the pressure to be given by the standard hard sphere EOS we show existence of weak solutions for arbitrarily large boundary data.

Abstract: Publication date: Available online 4 January 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Hannes Luiro, Mikko ParviainenAbstractWe establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations.