Authors:Satoshi Masaki; Jun-ichi Segata Pages: 283 - 326 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Satoshi Masaki, Jun-ichi Segata In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical L ˆ r space where L ˆ r = { f ∈ S ′ ( R ) ‖ f ‖ L ˆ r = ‖ f ˆ ‖ L r ′ < ∞ } . We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to L ˆ r -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

Authors:B. Dacorogna; W. Gangbo; O. Kneuss Pages: 327 - 356 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): B. Dacorogna, W. Gangbo, O. Kneuss This manuscript identifies a maximal system of equations which renders the classical Darboux problem elliptic, thereby providing a selection criterion for its well posedness. Let f be a symplectic form close enough to ω m , the standard symplectic form on R 2 m . We prove existence of a diffeomorphism φ, with optimal regularity, satisfying φ ⁎ ( ω m ) = f and 〈 d φ ♭ ; ω m 〉 = 0 . We establish uniqueness of φ when the system is coupled with a Dirichlet datum. As a byproduct, we obtain, what we term symplectic factorization of vector fields, that any map u, satisfying appropriate assumptions, can be factored as: u = χ ∘ ψ with ψ ⁎ ( ω m ) = ω m , 〈 d χ ♭ ; ω m 〉 = 0 and ∇ χ + ( ∇ χ ) t > 0 ; moreover there exists a closed 2-form Φ such that χ = ( δ Φ ⌟ ω m ) ♯ . Here, ♯ is the musical isomorphism and ♭ its inverse. We connect the above result to an L 2 -projection problem.

Authors:Ciprian G. Gal Pages: 357 - 392 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Ciprian G. Gal We consider a doubly nonlocal nonlinear parabolic equation which describes phase-segregation of a two-component material in a bounded domain. This model is a more general version than the recent nonlocal Cahn–Hilliard equation proposed by Giacomin and Lebowitz [26], such that it reduces to the latter under certain conditions. We establish well-posedness results along with regularity and long-time results in the case when the interaction between the two levels of nonlocality is strong-to-weak.

Authors:Thomas Chen; Younghun Hong; Nataša Pavlović Pages: 393 - 416 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Thomas Chen, Younghun Hong, Nataša Pavlović In this paper, we study the dynamics of a system of infinitely many fermions in dimensions d ≥ 3 near thermal equilibrium and prove scattering in the case of small perturbation around equilibrium in a certain generalized Sobolev space of density operators. This work is a continuation of our previous paper [11], and extends the important recent result of M. Lewin and J. Sabin in [19] of a similar type for dimension d = 2 . In the work at hand, we establish new, improved Strichartz estimates that allow us to control the case d ≥ 3 .

Authors:Isabeau Birindelli; Giulio Galise; Hitoshi Ishii Pages: 417 - 441 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Isabeau Birindelli, Giulio Galise, Hitoshi Ishii In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.

Authors:Yves Achdou; Alessio Porretta Pages: 443 - 480 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Yves Achdou, Alessio Porretta We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton–Jacobi equation and a forward Kolmogorov equation both posed in ( 0 , T ) × ( R N / Z N ) . Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T.

Authors:Andrew Lorent; Guanying Peng Pages: 481 - 516 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Andrew Lorent, Guanying Peng Let Ω ⊂ R 2 be a bounded simply-connected domain. The Eikonal equation ∇ u = 1 for a function u : Ω ⊂ R 2 → R has very little regularity, examples with singularities of the gradient existing on a set of positive H 1 measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ∇u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if (1) ∇ ⋅ ( Σ ˜ e 1 e 2 ( ∇ u ⊥ ) ) = 0 and ∇ ⋅ ( Σ ˜ ϵ 1 ϵ 2 ( ∇ u ⊥ ) ) = 0 distributionally in Ω , where Σ ˜ e 1 e 2 and Σ ˜ ϵ 1 ϵ 2 are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ∇u is locally Lipschitz continuous outside a locally finite set. Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if lim n → ∞ I ϵ n ( u n ) = 0 for some sequence u n ∈ W 0 2 , 2 ( Ω ) and u = lim n → ∞ ... PubDate: 2018-02-25T15:19:02Z DOI: 10.1016/j.anihpc.2017.06.002

Authors:Robin Ming Chen; Samuel Walsh; Miles H. Wheeler Pages: 517 - 576 Abstract: Publication date: March–April 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 35, Issue 2 Author(s): Robin Ming Chen, Samuel Walsh, Miles H. Wheeler This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation. We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.

Authors:Jérôme Droniou; Kyle S. Talbot Pages: 1 - 25 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Jérôme Droniou, Kyle S. Talbot This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium, with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with L 2 wells (Amirat and Ziani, 2004 [1]). We give a novel treatment of the diffusion–dispersion term, which requires delicate use of the Aubin–Simon lemma to ensure the strong convergence of the pressure gradient, owing to the troublesome lower-order terms introduced by the localisation procedure.

Authors:Manuel Friedrich; Francesco Solombrino Pages: 27 - 64 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Manuel Friedrich, Francesco Solombrino In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation ( G S B D ). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the G S B D setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.

Authors:Seonghak Kim; Baisheng Yan Pages: 65 - 100 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Seonghak Kim, Baisheng Yan We investigate the existence and properties of Lipschitz solutions for some forward–backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Under this framework, we study two important cases of forward–backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona–Malik model in image processing and the other that of Höllig's model related to the Clausius–Duhem inequality in the second law of thermodynamics.

Authors:Thomas Kappeler; Jan-Cornelius Molnar Pages: 101 - 160 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Thomas Kappeler, Jan-Cornelius Molnar In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on H s the KdV2 equation is C 0 -wellposed if s ⩾ 0 and illposed (in a strong sense) if s < 0 .

Authors:Myoungjean Bae; Shangkun Weng Pages: 161 - 186 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Myoungjean Bae, Shangkun Weng We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (= angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler–Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis r = 0 .

Authors:Manuel del Pino; Konstantinos T. Gkikas Pages: 187 - 215 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Manuel del Pino, Konstantinos T. Gkikas We consider the parabolic Allen–Cahn equation in R n , n ≥ 2 , u t = Δ u + ( 1 − u 2 ) u in R n × ( − ∞ , 0 ] . We construct an ancient radially symmetric solution u ( x , t ) with any given number k of transition layers between −1 and +1. At main order they consist of k time-traveling copies of w with spherical interfaces distant O ( log t ) one to each other as t → − ∞ . These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: x = − 2 ( n − 1 ) t . More precisely, if w ( s ) denotes the heteroclinic 1-dimensional solution of w ″ + ( 1 − w 2 ) w = 0 w ( ± ∞ ) = ± 1 given by w ( s ) = tanh ( s 2 ) we have u ( x , t ) ≈ ∑ j = 1 k ( − 1 ) j − 1 w ( x − ρ j ( t ) ) − 1 2 ( 1 + ( − 1 ) k ) as t → − ∞ where ρ j ( t ) = − 2 ( n − 1 ) t + 1 2 ( j − k + 1 2 ) log ( t log PubDate: 2018-02-05T13:32:55Z DOI: 10.1016/j.anihpc.2017.03.005

Authors:Jiaqi Liu; Peter A. Perry; Catherine Sulem Pages: 217 - 265 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Jiaqi Liu, Peter A. Perry, Catherine Sulem The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.

Authors:Yury Grabovsky; Davit Harutyunyan Pages: 267 - 282 Abstract: Publication date: January–February 2018 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 35, Issue 1 Author(s): Yury Grabovsky, Davit Harutyunyan We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.

Authors:Luc Molinet; Didier Pilod; Stéphane Vento Abstract: Publication date: Available online 21 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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 ∂ t u − D x α ∂ x u = ∂ x ( u 2 ) , 0 < α ≤ 1 , is locally well-posed in H s ( R ) when s > s α : = 3 2 − 5 α 4 . As a consequence, we obtain global well-posedness in the energy space H α 2 ( R ) as soon as α 2 > s α , i.e. α > 6 7 .

Authors:José A. Gálvez; Asun Jiménez; Pablo Mira Abstract: Publication date: Available online 15 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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.

Authors:Timothy Candy; Sebastian Herr Abstract: Publication date: Available online 9 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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.

Authors:Emanuele Caglioti; François Golse; Mikaela Iacobelli Abstract: Publication date: Available online 2 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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 strictly 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.

Authors:P. Jameson Graber; Alpár R. Mészáros Abstract: Publication date: Available online 1 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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.

Authors:Chao Liang; Karina Marin; Jiagang Yang Abstract: Publication date: Available online 1 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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 C r open and there exists a C r open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.

Authors:Nicolas Ginoux; Olaf Müller Abstract: Publication date: Available online 1 February 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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.

Authors:Eduard Feireisl; Antonín Novotný Abstract: Publication date: Available online 31 January 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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.

Authors:Paulo Abstract: Publication date: Available online 10 January 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(s): Paulo Brandão We 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.

Authors:Hannes Luiro; Mikko Parviainen Abstract: Publication date: Available online 4 January 2018 Source:Annales de l'Institut Henri Poincaré C, Analyse non linéaire Author(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.

Authors:Sun-Sig Byun; Jihoon Ok; Jung-Tae Park Pages: 1639 - 1667 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Sun-Sig Byun, Jihoon Ok, Jung-Tae Park We investigate a quasilinear elliptic equation with variable growth in a bounded nonsmooth domain involving a signed Radon measure. We obtain an optimal global Calderón–Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal function of the associated measure, up to a correct power, under minimal regularity requirements on the nonlinearity, the variable exponent and the boundary of the domain.

Authors:Tomoyuki Miyaji; Yoshio Tsutsumi Pages: 1707 - 1725 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Tomoyuki Miyaji, Yoshio Tsutsumi We show the existence of global solution and the global attractor in L 2 ( T ) for the third order Lugiato–Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the L 2 ( T ) norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the L 2 norm conservation is useful for the third order Lugiato–Lefever equation unlike the KdV and the cubic NLS equations. Therefore, it seems important and natural to construct the global attractor in L 2 ( T ) . For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.

Authors:Ricardo Alonso; Thierry Goudon; Arthur Vavasseur Pages: 1727 - 1758 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Ricardo Alonso, Thierry Goudon, Arthur Vavasseur We investigate the large time behavior of the solutions of a Vlasov–Fokker–Planck equation where particles are subjected to a confining external potential and a self-consistent potential intended to describe the interaction of the particles with their environment. The environment is seen as a medium vibrating in a direction transverse to particles' motion. We identify equilibrium states of the model and justify the asymptotic trend to equilibrium. The analysis relies on hypocoercivity techniques.

Authors:Lassaad Aloui; Moez Khenissi; Luc Robbiano Pages: 1759 - 1792 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Lassaad Aloui, Moez Khenissi, Luc Robbiano We prove, under the exterior geometric control condition, the Kato smoothing effect for solutions of an inhomogeneous and damped Schrödinger equation on exterior domains.

Authors:Huy Quang Nguyen Pages: 1793 - 1836 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Huy Quang Nguyen This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. Local well-posedness for this problem with Lipschitz initial velocity was established by Alazard–Burq–Zuily [1]. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for initial data 1 4 -derivative less regular than the aforementioned threshold, which corresponds to the gain of Hölder regularity of the semi-classical Strichartz estimate for the fully nonlinear system. In order to obtain this Cauchy theory, we establish global, quantitative results for the paracomposition theory of Alinhac [5].

Authors:Christian Seis Pages: 1837 - 1850 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Christian Seis In this work, we provide stability estimates for the continuity equation with Sobolev vector fields. The results are inferred from contraction estimates for certain logarithmic Kantorovich–Rubinstein distances. As a by-product, we obtain a new proof of uniqueness in the DiPerna–Lions setting. The novelty in the proof lies in the fact that it is not based on the theory of renormalized solutions.

Authors:Jürgen Jost; Lei Liu; Miaomiao Zhu Pages: 1851 - 1882 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Jürgen Jost, Lei Liu, Miaomiao Zhu We show the existence of a global weak solution of the heat flow for Dirac-harmonic maps from compact Riemann surfaces with boundary when the energy of the initial map and the L 2 -norm of the boundary values of the spinor are sufficiently small. Dirac-harmonic maps couple a second order harmonic map type system with a first-order Dirac type system. The heat flow that has been introduced in [9] and that we investigate here is novel insofar as we only make the second order part parabolic, but carry the first order part along the resulting flow as an elliptic constraint. Of course, since the equations are coupled, both parts then change along the flow. The solution is unique and regular with the exception of at most finitely many singular times. We also discuss the behavior at the singularities of the flow. As an application, we deduce some existence results for Dirac-harmonic maps. Since we may impose nontrivial boundary conditions also for the spinor part, in the limit, we shall obtain Dirac-harmonic maps with nontrivial spinor part.

Authors:Marta Lewicka; Annie Raoult; Diego Ricciotti Pages: 1883 - 1912 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Marta Lewicka, Annie Raoult, Diego Ricciotti We study the elastic behaviour of incompatibly prestrained thin plates of thickness h whose internal energy E h is governed by an imposed three-dimensional smooth Riemann metric G only depending on the variable in the midsurface ω. It is already known that h − 2 inf E h converges to a finite value c when the metric G restricted to the midsurface has a sufficiently regular immersion, namely W 2 , 2 ( ω , R 3 ) . The obtained limit model generalizes the bending (Kirhchoff) model of Euclidean elasticity. In the present paper, we deal with the case when c equals 0. Then, equivalently, three independent entries of the three-dimensional Riemann curvature tensor associated with G are null. We prove that, in such regime, necessarily inf E h ≤ C h 4 . We identify the Γ-limit of the scaled energies h − 4 E h and show that it consists of a von Kármán-like energy. The unknowns in this energy are the first order incremental displacements with respect to the deformation defined by the bending model and the second order tangential strains. In addition, we prove that when inf h − 4 E h → 0 , then G is realizable and hence min E h = 0 for every h.

Authors:Alessio Porretta; Philippe Souplet Pages: 1913 - 1923 Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Alessio Porretta, Philippe Souplet We consider the diffusive Hamilton–Jacobi equation, with superquadratic Hamiltonian, homogeneous Dirichlet conditions and regular initial data. It is known from [4] (Barles–DaLio, 2004) that the problem admits a unique, continuous, global viscosity solution, which extends the classical solution in case gradient blowup occurs. We study the question of the possible loss of boundary conditions after gradient blowup, which seems to have remained an open problem till now. Our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena. For any smooth bounded domain, we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary, as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup. Actually, we show that the latter possibility is rather exceptional. More generally, we show that the set of the points where boundary conditions are lost, can be prescribed to be arbitrarily close to any given open subset of the boundary.

Authors:Wolansky Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): G. Wolansky The object of this paper is to study estimates of ϵ − q W p ( μ + ϵ ν , μ ) for small ϵ > 0 . Here W p is the Wasserstein metric on positive measures, p > 1 , μ is a probability measure and ν a signed, neutral measure ( ∫ d ν = 0 ). In [16] we proved uniform (in ϵ) estimates for q = 1 provided ∫ ϕ d ν can be controlled in terms of ∫ ∇ ϕ p / ( p − 1 ) d μ , for any smooth function ϕ. In this paper we extend the results to the case where such a control fails. This is the case where, e.g., μ has a disconnected support, or the dimension d of μ (to be defined) is larger or equal to p / ( p − 1 ) . In the latter case we get such an estimate provided 1 / p + 1 / d ≠ 1 for q = min ( 1 , 1 / p + 1 / d ) . If 1 / p + 1 / d = 1 we get a log-Lipschitz estimate. As an application we obtain Hölder estimates in W p for curves of probability measures which are absolutely continuous in the total variation norm. In case the support of μ is disconnected (corresponding to d = ∞ ) we obtain sharp estimates for q = 1 / p (“optimal teleportation”): lim ϵ → 0 ϵ − 1 / p W p ( μ , μ + ϵ ν ) = ‖ ν ‖ μ where ‖ ν ‖ μ is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of μ, and the weights of the measure ν in each connected component of this support.

Authors:Andrej Abstract: Publication date: December 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 7 Author(s): Andrej Zlatoš We study reaction–diffusion equations in one spatial dimension and with general (space- or time-) inhomogeneous mixed bistable–ignition reactions. For those satisfying a simple quantitative hypothesis, we prove existence and uniqueness of transition fronts, as well as convergence of “typical” solutions to the unique transition front (the existence part even extends to mixed bistable–ignition–monostable reactions). These results also hold for all pure ignition reactions without the extra hypothesis, but not for all pure bistable reactions. In fact, we find examples of either spatially or temporally periodic pure bistable reactions (independent of the other space–time variable) for which we can prove non-existence of transition fronts. The latter are the first such examples in periodic media which are non-degenerate in a natural sense, and they also prove a conjecture from [7].

Authors:Haïm Brezis; Petru Mironescu Abstract: Publication date: Available online 23 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Haïm Brezis, Petru Mironescu We investigate the validity of the Gagliardo–Nirenberg type inequality (1) ‖ f ‖ W s , p ( Ω ) ≲ ‖ f ‖ W s 1 , p 1 ( Ω ) θ ‖ f ‖ W s 2 , p 2 ( Ω ) 1 − θ , with Ω ⊂ R N . Here, 0 ≤ s 1 ≤ s ≤ s 2 are non negative numbers (not necessarily integers), 1 ≤ p 1 , p , p 2 ≤ ∞ , and we assume the standard relations s = θ s 1 + ( 1 − θ ) s 2 , 1 / p = θ / p 1 + ( 1 − θ ) / p 2 for some θ ∈ ( 0 , 1 ) . By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s 1 , s 2 , s are integers. It turns out that (1) holds for “most” of values of s 1 , … , p 2 , but not for all of them. We present an explicit condition on s 1 , s 2 , p 1 , p 2 which allows to decide whether (1) holds or fails.

Authors:Eduard Feireisl; Václav Mácha; Šárka Nečasová; Marius Tucsnak Abstract: Publication date: Available online 21 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Eduard Feireisl, Václav Mácha, Šárka Nečasová, Marius Tucsnak We 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 → ∞ .

Authors:Mitia Duerinckx; Julian Fischer Abstract: Publication date: Available online 16 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Mitia Duerinckx, Julian Fischer We 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.

Authors:A. Avila; P. Hubert Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): A. Avila, P. Hubert In this paper, we give a geometric criterion ensuring the recurrence of the vertical flow on Z d -covers of compact translation surfaces ( d ≥ 2 ). We prove that the linear flow in the windtree model is recurrent for every pair of parameters and almost every direction.

Authors:Andrei V. Faminskii Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Andrei V. Faminskii Initial-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.

Authors:Inwon Kim; Olga Turanova Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Inwon Kim, Olga Turanova We 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.

Authors:Denis Serre Abstract: Publication date: Available online 15 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Denis Serre We 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 ( det A ) 1 d − 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 ρ 1 n p , where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.

Authors:Dongxiang Chen; Yuxi Wang; Zhifei Zhang Abstract: Publication date: Available online 14 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Dongxiang Chen, Yuxi Wang, Zhifei Zhang In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class 2 − θ for any θ > 0 . This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like e k t for the linearized Prandtl equation around a non-monotonic shear flow.

Authors:Christoph Scheven; Thomas Schmidt Abstract: Publication date: Available online 14 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Christoph Scheven, Thomas Schmidt We 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.

Authors:Andrea Giorgini; Maurizio Grasselli Hao Abstract: Publication date: Available online 6 November 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Andrea Giorgini, Maurizio Grasselli, Hao Wu The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.