Abstract: Publication date: Available online 15 March 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Rayssa Caju, João Marcos do Ó, Almir Silva Santos We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [12] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.

Abstract: Publication date: Available online 8 March 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Casey Kelleher, Jeffrey Streets

Abstract: Publication date: Available online 6 March 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): N. Mizoguchi, Ph. Souplet

Abstract: Publication date: Available online 4 March 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Kuijie Li, Baoxiang Wang This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions d≥4. By establishing an ϵ regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in B˙p,q−1+d/p(Rd), d

Abstract: Publication date: Available online 4 March 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Michael Winkler The system{ut=Δu−χ∇⋅(uv∇v)−uv+B1(x,t),vt=Δv+uv−v+B2(x,t),(⋆) is considered in a disk Ω⊂R2, with a positive parameter χ and given nonnegative and suitably regular functions B1 and B2 defined on Ω×(0,∞). In the particular version obtained when χ=2, (⋆) was proposed in [33] as a model for crime propagation in urban regions.Within a suitable generalized framework, it is shown that under mild assumptions on the parameter functions and the initial data the no-flux initial-boundary value problem for (⋆) possesses at least one global solution in the case when all model ingredients are radially sysmmetric with respect to the center of Ω. Moreover, under an additional hypothesis on stabilization of the given external source terms in both equations, these solutions are shown to approach the solution of an elliptic boundary value problem in an appropriate sense.The analysis is based on deriving a priori estimates for a family of approximate problems, in a first step achieving some spatially global but weak initial regularity information which in a series of spatially localized arguments is thereafter successively improved.To the best of our knowledge, this is the first result on global existence of solutions to the two-dimensional version of the full original system (⋆) for arbitrarily large values of χ.

Abstract: Publication date: Available online 30 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Weiwei Ding, Yihong Du, Xing Liang This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space–time periodic media. In Part 1 (see [2]) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger [15], [16] and others [6], [7], [8], [9], [10] to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space–time periodic media considered in our work here.

Abstract: Publication date: Available online 25 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Xiaosen Han, Gabriella Tarantello In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs (N×N)-system:Δui=λ(∑j=1N∑k=1NKkjKjieujeuk−∑j=1NKjieuj)+4π∑j=1niδpij,i=1,…,N; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of SU(N+1), (see (1.2) below). Here, λ>0 is the coupling parameter, δp is the Dirac measure with pole at p and ni∈N, for i=1,…,N. When N=1,2 many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for N≥3, only recently in [27] the authors managed to obtain the existence of one doubly periodic solution via a minimisation procedure, in the spirit of [46] . Our main contribution in this paper is to show (as in [46]) that actually the given system admits a second doubly periodic solutions of “Mountain-pass” type, provided that 3≤N≤5. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chern–Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a “compactness” property encompassed by the so called Palais–Smale condition for the corresponding “action” functional, whose validity remains still open for N≥6.

Abstract: Publication date: Available online 25 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Miroslav Bulíček, Jan Burczak, Sebastian Schwarzacher We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system∂tu−div(ν( ∇u )∇u)=−divf with a given strictly positive bounded function ν, such that limk→∞ν(k)=ν∞ and f∈Lq with q∈(1,∞). The existence, uniqueness and regularity results for q≥2 are by now standard. However, even if a priori estimates are available, the existence in case q∈(1,2) was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range q∈(1,∞).Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted Lq spaces.

Abstract: Publication date: Available online 25 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Tobias Breiten, Karl Kunisch, Laurent Pfeiffer A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton-Jacobi-Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker-Planck equation is also provided.

Abstract: Publication date: Available online 25 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Iwona Chlebicka, Piotr Gwiazda, Anna Zatorska–Goldstein We study a general nonlinear parabolic equation on a Lipschitz bounded domain in RN,{∂tu−divA(t,x,∇u)=f(t,x)inΩT,u(t,x)=0on(0,T)×∂Ω,u(0,x)=u0(x)inΩ, with f∈L∞(ΩT) and u0∈L∞(Ω). The growth of the monotone vector field A is controlled by a generalized fully anisotropic N-function M:[0,T)×Ω×RN→[0,∞) inhomogeneous in time and space, and under no growth restrictions on the last variable. It results in the need of the integration by parts formula which has to be formulated in an advanced way. Existence and uniqueness of solutions are proven when the Musielak-Orlicz space is reflexive OR in absence of Lavrentiev's phenomenon. To ensure approximation properties of the space we impose natural assumption that the asymptotic behaviour of the modular function is sufficiently balanced. Its instances are log-Hölder continuity of variable exponent or optimal closeness condition for powers in double phase spaces.The noticeable challenge of this paper is cosidering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time.

Abstract: Publication date: Available online 22 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Masayuki Hayashi We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger equation (DNLS). It is known that DNLS has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui [15], the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.

Abstract: Publication date: Available online 9 January 2019Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Sigmund Selberg We consider the initial value problem for the Dirac-Klein-Gordon equations in two space dimensions. Global regularity for C∞ data was proved by Grünrock and Pecher. Here we consider analytic data, proving that if the initial radius of analyticity is σ0>0, then for later times t>0 the radius of analyticity obeys a lower bound σ(t)≥σ0exp(−At). This provides information about the possible dynamics of the complex singularities of the holomorphic extension of the solution at time t. The proof relies on an analytic version of Bourgain's Fourier restriction norm method, multilinear space-time estimates of null form type and an approximate conservation of charge.

Abstract: Publication date: Available online 12 December 2018Source: Annales de l'Institut Henri Poincaré C, Analyse non linéaireAuthor(s): Benjamin Obando, Takéo Takahashi We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.

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: 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 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): 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.