Abstract: Publication date: Available online 5 December 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Gung-Min Gie, James P. Kelliher, Milton C. Lopes Filho, Anna L. Mazzucato, Helena J. Nussenzveig Lopes The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier-Stokes and the Euler solutions. Using properties of these correctors, we establish convergence of the Navier-Stokes solution to the Euler solution as viscosity vanishes with optimal rates of convergence. In addition, we investigate vorticity production on the boundary in the limit of vanishing viscosity. Our work significantly extends prior work in the literature.

Abstract: Publication date: Available online 23 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Moshe Marcus Consider operators of the form LγV:=Δ+γV in a bounded Lipschitz domain Ω⊂RN. Assume that V∈C1(Ω) satisfies V(x) ≤a¯dist(x,∂Ω)−2 for every x∈Ω and γ is a number in a range (γ−,γ+) described in the introduction.The model case is V(x)=dist(x,F)−2 where F is a closed subset of ∂Ω and γ

Abstract: Publication date: Available online 23 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Mattia Fogagnolo, Lorenzo Mazzieri, Andrea Pinamonti We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani.

Abstract: Publication date: Available online 23 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Christian Brennecke, Phan Thành Nam, Marcin Napiórkowski, Benjamin Schlein We consider a system of N bosons interacting through a singular two-body potential scaling with N and having the form N3β−1V(Nβx), for an arbitrary parameter β∈(0,1). We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose-Einstein condensation in terms of a cubic nonlinear Schrödinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.

Abstract: Publication date: Available online 16 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Ryosuke Hyakuna This paper is concerned with the Cauchy problem for the Hartree equation on Rn,n∈N with the nonlinearity of type ( ⋅ −γ⁎ u 2)u,0

Abstract: Publication date: Available online 16 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Claudio Muñoz, José M. Palacios In this article we prove that 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem H1×L2. The solutions that we study are the 2-kink, kink-antikink and breather of SG. In order to prove this result, we will use Bäcklund transformations implemented by the Implicit Function Theorem. These transformations will allow us to reduce the stability of the three solutions to the case of the vacuum solution, in the spirit of previous results by Alejo and the first author [3], which was done for the case of the scalar modified Korteweg-de Vries equation. However, we will see that SG presents several difficulties because of its vector valued character. Our results improve those in [5], and give the first rigorous proof of the nonlinear stability in the energy space of the SG 2-solitons.

Abstract: Publication date: Available online 16 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Céline Grandmont, Matthieu Hillairet, Julien Lequeurre We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

Abstract: Publication date: Available online 15 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Filippo Cagnetti, Gianni Dal Maso, Lucia Scardia, Caterina Ida Zeppieri We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of u.We obtain three main results: compactness with respect to Γ-convergence, representation of the Γ-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.

Abstract: Publication date: Available online 15 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Simon Zugmeyer We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, as well as a sharp weighted trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.

Abstract: Publication date: Available online 14 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): L. Caffarelli, S. Patrizi, V. Quitalo, M. Torres We show the existence of a Lipschitz viscosity solution u in Ω to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface ∂{u>0}∩Ω and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition uν++=uν−−, and hence u is a solution to a two-phase free boundary problem. We show that we can apply the classical method of sup-convolutions developed by the first author in [5], [6], and generalized by Wang [20], [21] and Feldman [11] to fully nonlinear operators, to conclude that the regular points in ∂{u>0}∩Ω form an open set of class C1,α. A novelty in our problem is that we have different operators, F+ and F−, on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators M+ and M−, our results provide an alternative approach to obtain the stationary limit of a segregation model of populations with nonlinear diffusion in [19].

Abstract: Publication date: Available online 9 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Olivier Pinaud This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al in [7]. The model is obtained as the diffusive limit of the quantum Liouville-BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium.

Abstract: Publication date: Available online 8 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Matthias Ruf We propose a new Γ-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension.RésuméNous proposons une nouvelle discrétisation de la fonctionnelle de Mumford-Shah convergente au sense de la Γ-convergence. Les fonctionnelles discrètes agissent sur des fonctions définies sur des réseaux aléatoires stationnaires et dépendent de différences finies générales via un potentiel non-convexe. Dans ce cadre, la géométrie du réseau aléatoire influence grandement l'anisotropie de la fonctionnelle limite. Ainsi, en utilisant des réseaux aléatoires statistiquement isotropes, on démontre par des techniques d'homogénéisation le résultat d'approximation de la fonctionnelle vectorielle de Mumford-Shah en toutes dimensions.

Abstract: Publication date: Available online 8 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Z. Badreddine This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.

Abstract: Publication date: Available online 7 November 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Rémy Rodiac Let Ω be a bounded open set in R2. The aim of this article is to describe the functions h in H1(Ω) and the Radon measures μ which satisfy −Δh+h=μ and div(Th)=0 in Ω, where Th is a 2×2 matrix given by (Th)ij=2∂ih∂jh−( ∇h 2+h2)δij for i,j=1,2. These equations arise as equilibrium conditions satisfied by limiting vorticities and limiting induced magnetic fields of solutions of the magnetic Ginzburg-Landau equations. This was shown by Sandier-Serfaty in [32], [33]. Let us recall that they obtained that ∇h is continuous in Ω. We prove that if z0 in Ω is in the support of μ and is such that ∇h (z0)≠0 then μ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a C1-curve near z0 whereas μ⌊{ ∇h =0}=h { ∇h =0}L2. We also prove that if Ω is smooth bounded and star-shaped and if h=0 on ∂Ω then h≡0 in Ω. This rules out the possibility of having critical points of the Ginzburg-Landau energy with a number of vortices much larger than the applied magnetic field hex in that case.

Abstract: Publication date: Available online 4 October 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Claudia Bucur, Luca Lombardini, Enrico Valdinoci In this paper, we consider the asymptotic behavior of the fractional mean curvature when s→0+. Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter s∈(0,1) is small, in a bounded and connected open set with C2 boundary Ω⊂Rn. We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω, fill all Ω, or possibly develop a wildly oscillating boundary.Also, we prove the continuity of the fractional mean curvature in all variables, for s∈[0,1]. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.

Abstract: Publication date: Available online 29 September 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Marco Bonacini, Barbara Niethammer, Juan J.L. Velázquez We consider Smoluchowski's coagulation equation in the case of the diagonal kernel with homogeneity γ>1. In this case the phenomenon of gelation occurs and solutions lose mass at some finite time. The problem of the existence of self-similar solutions involves a free parameter b, and one expects that a physically relevant solution (i.e. nonnegative and with sufficiently fast decay at infinity) exists for a single value of b, depending on the homogeneity γ. We prove this picture rigorously for large values of γ. In the general case, we discuss in detail the behavior of solutions to the self-similar equation as the parameter b changes.

Abstract: Publication date: Available online 28 September 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): I-Kun Chen, Chun-Hsiung Hsia, Daisuke Kawagoe We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.

Abstract: Publication date: September 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 6Author(s): Casey Kelleher, Jeffrey Streets We 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. Rabinowitz For 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 Holzer We 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 Sire We 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 Nara We 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 Lamy We 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 Shkoller In 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: Available online 5 July 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Matt McGonagle, Chong Song, Yu Yuan We 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 Iurlano We 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 Schaftingen Given 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 Zhu For 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 Sousa The 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 Méhats In 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 Sigalotti We 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 Wang For 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 Tentarelli We 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. Titi Based 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 Zaag We 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 Schlenker We 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 Rousset We 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 Dávila We 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. Iandoli We 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 Wang We 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 Zhu Let 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. Pimentel In 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 Xiang This 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 Zhang Through 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. Tzirakis We 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 Yang We 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 Gravejat It 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 Kocić We 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 Tournus We 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 Zaccur The 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 Robert In 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 Vento We 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 Mira We 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 Herr For 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 Iacobelli In 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 Yang We 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 Müller We 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. Mészáros In 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 Novotný We 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 Parviainen We 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.