Authors:Nicola Garofalo; Arshak Petrosyan; Camelia A. Pop; Mariana Smit Vega Garcia Pages: 533 - 570 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Nicola Garofalo, Arshak Petrosyan, Camelia A. Pop, Mariana Smit Vega Garcia We establish the C 1 + γ -Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.

Authors:El Haj Laamri; Michel Pierre Pages: 571 - 591 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): El Haj Laamri, Michel Pierre We prove here global existence in time of weak solutions for some reaction–diffusion systems with natural structure conditions on the nonlinear reactive terms which provide positivity of the solutions and uniform control of the total mass. The diffusion operators are nonlinear, in particular operators of the porous media type u i ↦ − d i Δ u i m i . Global existence is proved under the assumption that the reactive terms are bounded in L 1 . This extends previous similar results obtained in the semilinear case when the diffusion operators are linear of type u i ↦ − d i Δ u i .

Authors:Paolo Baroni; Casimir Lindfors Pages: 593 - 624 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Paolo Baroni, Casimir Lindfors We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy–Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary p-Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself.

Authors:Nicola Soave; Alessandro Zilio Pages: 625 - 654 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Nicola Soave, Alessandro Zilio We consider a family of positive solutions to the system of k components − Δ u i , β = f ( x , u i , β ) − β u i , β ∑ j ≠ i a i j u j , β 2 in Ω , where Ω ⊂ R N with N ≥ 2 . It is known that uniform bounds in L ∞ of { u β } imply convergence of the densities to a segregated configuration, as the competition parameter β diverges to +∞. In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of u β in terms of entire solutions to the limit system Δ U i = U i ∑ j ≠ i a i j U j 2 . Moreover, we develop a uniform-in-β regularity theory for the interfaces.

Authors:Damião J. Araújo; Gleydson C. Ricarte; Eduardo V. Teixeira Pages: 655 - 678 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Damião J. Araújo, Gleydson C. Ricarte, Eduardo V. Teixeira This work is devoted to the study of nonvariational, singularly perturbed elliptic equations of degenerate type. The governing operator is anisotropic and ellipticity degenerates along the set of critical points. The singular behavior is of order O ( 1 ϵ ) along ϵ-level layers { u ϵ ∼ ϵ } , and a non-homogeneous source acts in the noncoincidence region { u ϵ > ϵ } . We obtain the precise geometric behavior of solutions near ϵ-level surfaces, by means of optimal regularity and sharp geometric nondegeneracy. We further investigate Hausdorff measure properties of ϵ-level surfaces. The analysis of the asymptotic limits as the ϵ parameter goes to zero is also carried out. The results obtained are new even if restricted to the uniformly elliptic, isotropic setting.

Authors:Alessandro Fonda; Antonio J. Ureña Pages: 679 - 698 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Alessandro Fonda, Antonio J. Ureña We propose an extension to higher dimensions of the Poincaré–Birkhoff Theorem which applies to Poincaré time-maps of Hamiltonian systems. Examples of applications to pendulum-type systems and weakly-coupled superlinear systems are also given.

Authors:Stephan Fackler Pages: 699 - 709 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Stephan Fackler An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous symmetric forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We give a negative answer to this question and discuss the minimal regularity needed for positive results.

Authors:Philip Isett; Sung-Jin Oh Pages: 711 - 730 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Philip Isett, Sung-Jin Oh In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L t ∞ B 3 , ∞ 1 / 3 due to low regularity of the energy profile. The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class C t , x 1 / 5 − ϵ . The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [1].

Authors:V. Sciacca; M.E. Schonbek; M. Sammartino Pages: 731 - 757 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): V. Sciacca, M.E. Schonbek, M. Sammartino We consider the two-dimensional shallow water model derived in [29], describing the motion of an incompressible fluid, confined in a shallow basin, with varying bottom topography. We construct the approximate inertial manifolds for the associated dynamical system and estimate its order. Finally, working in the whole space R 2 , under suitable conditions on the time dependent forcing term, we prove the L 2 asymptotic decay of the weak solutions.

Authors:Benjamin Dodson; Changxing Miao; Jason Murphy; Jiqiang Zheng Pages: 759 - 787 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Benjamin Dodson, Changxing Miao, Jason Murphy, Jiqiang Zheng We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

Authors:Nicola Zamponi; Ansgar Jüngel Pages: 789 - 792 Abstract: Publication date: May–June 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 3 Author(s): Nicola Zamponi, Ansgar Jüngel This note corrects Lemma 7 in [1] on the positive (semi-)definiteness of a certain matrix product, which yields a priori estimates for the cross-diffusion system.

Authors:Agathe Decaster; Dragoş Iftimie Pages: 277 - 291 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Agathe Decaster, Dragoş Iftimie In this paper, we address the problem of determining the asymptotic behaviour of the solutions of the incompressible stationary Navier–Stokes system in the full space, with a forcing term whose asymptotic behaviour at infinity is homogeneous of degree −3. We identify the asymptotic behaviour at infinity of the solution. We prove that it is homogeneous and that the leading term in the expansion at infinity uniquely solves the homogeneous Navier–Stokes equations with a forcing term which involves an additional Dirac mass. This also applies to the case of an exterior domain.

Authors:Henrik Shahgholian; Karen Yeressian Pages: 293 - 334 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Henrik Shahgholian, Karen Yeressian In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary B ∩ { x 1 = 0 } in the obstacle type problem { div ( x 1 a ∇ u ) = χ { u > 0 } in B + , u = 0 on B ∩ { x 1 = 0 } where a < 1 , B + = B ∩ { x 1 > 0 } , B is the unit ball in R n and n ≥ 2 is an integer. Let Γ = B + ∩ ∂ { u > 0 } be the free boundary and assume that the origin is a contact point, i.e. 0 ∈ Γ ‾ . We prove that the free boundary touches the fixed boundary uniformly tangentially at the origin, near to the origin it is the graph of a C 1 function and there is a uniform modulus of continuity for the derivatives of this function.

Authors:Alberto Bressan; Geng Chen Pages: 335 - 354 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Alberto Bressan, Geng Chen The paper is concerned with conservative solutions to the nonlinear wave equation u t t − c ( u ) ( c ( u ) u x ) x = 0 . For an open dense set of C 3 initial data, we prove that the solution is piecewise smooth in the t–x plane, while the gradient u x can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.

Authors:Verena Bögelein; Frank Duzaar; Paolo Marcellini; Stefano Signoriello Pages: 355 - 379 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Verena Bögelein, Frank Duzaar, Paolo Marcellini, Stefano Signoriello In this paper we establish the existence of Lipschitz-continuous solutions to the Cauchy Dirichlet problem of evolutionary partial differential equations { ∂ t u − div D f ( D u ) = 0 in Ω T , u = u o on ∂ P Ω T . The only assumptions needed are the convexity of the generating function f : R n → R , and the classical bounded slope condition on the initial and the lateral boundary datum u o ∈ W 1 , ∞ ( Ω ) . We emphasize that no growth conditions are assumed on f and that – an example which does not enter in the elliptic case – u o could be any Lipschitz initial and boundary datum, vanishing at the boundary ∂Ω, and the boundary may contain flat parts, for instance Ω could be a rectangle in R n .

Authors:Animikh Biswas; Ciprian Foias; Adam Larios Pages: 381 - 405 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Animikh Biswas, Ciprian Foias, Adam Larios In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier–Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.

Authors:Godofredo Iommi; Thomas Jordan; Mike Todd Pages: 407 - 421 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Godofredo Iommi, Thomas Jordan, Mike Todd We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

Authors:Bernard Helffer; Ayman Kachmar Pages: 423 - 438 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Bernard Helffer, Ayman Kachmar We explore the relationship between two reference functions arising in the analysis of the Ginzburg–Landau functional. The first function describes the distribution of superconductivity in a type II superconductor subjected to a constant magnetic field. The second function describes the distribution of superconductivity in a type II superconductor submitted to a variable magnetic field that vanishes non-degenerately along a smooth curve.

Authors:Nicola Abatangelo; Louis Dupaigne Pages: 439 - 467 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Nicola Abatangelo, Louis Dupaigne We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak- L 1 theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method. We finally show the existence of large solutions for some power-like nonlinearities.

Authors:François Hamel; Xavier Ros-Oton; Yannick Sire; Enrico Valdinoci Pages: 469 - 482 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): François Hamel, Xavier Ros-Oton, Yannick Sire, Enrico Valdinoci We consider entire solutions to L u = f ( u ) in R 2 , where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L , we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

Authors:Laurent Chupin; Sébastien Martin Pages: 483 - 508 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Laurent Chupin, Sébastien Martin We investigate the influence of the rough boundaries on viscoelastic flows, described by the diffusive Oldroyd model. The fluid domain has a rough wall modeled by roughness patterns of size ε ≪ 1 . We present and rigorously justify an asymptotic expansion with respect to ε, at any order, based upon the definition of elementary problems: Oldroyd-type problems at the global scale defined on a smoothened domain and boundary-layer corrector problems. The resulting analysis guarantees optimality with respect to the truncation error and leads to a numerical algorithm which allows us to build the approximation of the solution at any required precision.

Authors:Joseph Thirouin Pages: 509 - 531 Abstract: Publication date: March–April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 2 Author(s): Joseph Thirouin This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrödinger equation on the torus: i ∂ t u = D α u + u 2 u , u ( 0 , ⋅ ) = u 0 , where α is a real parameter. We show that, apart from the case α = 1 , which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.

Authors:Nicola Zamponi; Ansgar Jüngel Pages: 1 - 29 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Nicola Zamponi, Ansgar Jüngel A class of parabolic cross-diffusion systems modeling the interaction of an arbitrary number of population species is analyzed in a bounded domain with no-flux boundary conditions. The equations are formally derived from a random-walk lattice model in the diffusion limit. Compared to previous results in the literature, the novelty is the combination of general degenerate diffusion and volume-filling effects. Conditions on the nonlinear diffusion coefficients are identified, which yield a formal gradient-flow or entropy structure. This structure allows for the proof of global-in-time existence of bounded weak solutions and the exponential convergence of the solutions to the constant steady state. The existence proof is based on an approximation argument, the entropy inequality, and new nonlinear Aubin–Lions compactness lemmas. The proof of the large-time behavior employs the entropy estimate and convex Sobolev inequalities. Moreover, under simplifying assumptions on the nonlinearities, the uniqueness of weak solutions is shown by using the H − 1 method, the E-monotonicity technique of Gajewski, and the subadditivity of the Fisher information.

Authors:Oscar F. Bandtlow; Wolfram Just; Julia Slipantschuk Pages: 31 - 43 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Oscar F. Bandtlow, Wolfram Just, Julia Slipantschuk We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural representation of the respective adjoint operators.

Authors:Marta Lewicka; L. Mahadevan; Mohammad Reza Pakzad Pages: 45 - 67 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Marta Lewicka, L. Mahadevan, Mohammad Reza Pakzad The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of W 2 , 2 solutions of the Monge–Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometric embeddings of 2d surface in R 3 . In the second part, we rigorously derive the Γ-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness h, where the depth of the shell scales like h α and the applied forces scale like h α + 2 , in the limit when h → 0 . We offer a full analysis of the problem in the parameter range α ∈ ( 1 / 2 , 1 ) . We also complete the analysis in some specific cases for the full range α ∈ ( 0 , 1 ) , applying the results of the first part of the paper.

Authors:Moshe Marcus; Phuoc-Tai Nguyen Pages: 69 - 88 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Moshe Marcus, Phuoc-Tai Nguyen Let Ω be a bounded smooth domain in R N . We study positive solutions of equation (E) − L μ u + u q = 0 in Ω where L μ = Δ + μ δ 2 , 0 < μ , q > 1 and δ ( x ) = dist ( x , ∂ Ω ) . A positive solution of (E) is moderate if it is dominated by an L μ -harmonic function. If μ < C H ( Ω ) (the Hardy constant for Ω) every positive L μ -harmonic function can be represented in terms of a finite measure on ∂Ω via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, 1 < q < q μ , c . (The critical value depends only on N and μ.) For q ≥ q μ , c there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator L μ . These results form the basis for the study of the nonlinear problem.

Authors:Juhana Siljander; Changyou Wang; Yuan Zhou Pages: 119 - 138 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Juhana Siljander, Changyou Wang, Yuan Zhou We show the everywhere differentiability of viscosity solutions to a class of Aronsson equations in R n for n ≥ 2 , where the coefficient matrices A are assumed to be uniformly elliptic and C 1 , 1 . Our result extends an earlier important theorem by Evans and Smart [18] who have studied the case A = I n which correspond to the ∞-Laplace equation. We also show that every point is a Lebesgue point for the gradient. In the process of proving the results we improve some of the gradient estimates obtained for the infinity harmonic functions. The lack of suitable gradient estimates has been a major obstacle for solving the C 1 , α problem in this setting, and we aim to take a step towards better understanding of this problem, too. A key tool in our approach is to study the problem in a suitable intrinsic geometry induced by the coefficient matrix A. Heuristically, this corresponds to considering the question on a Riemannian manifold whose the metric is given by the matrix A.

Authors:Moon-Jin Kang; Alexis F. Vasseur Pages: 139 - 156 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Moon-Jin Kang, Alexis F. Vasseur We consider the L 2 -contraction up to a shift for viscous shocks of scalar viscous conservation laws with strictly convex fluxes in one space dimension. In the case of a flux which is a small perturbation of the quadratic Burgers flux, we show that any viscous shock induces a contraction in L 2 , up to a shift. That is, the L 2 norm of the difference of any solution of the viscous conservation law, with an appropriate shift of the shock wave, does not increase in time. If, in addition, the difference between the initial value of the solution and the shock wave is also bounded in L 1 , the L 2 norm of the difference converges at the optimal rate t − 1 / 4 . Both results do not involve any smallness condition on the initial value, nor on the size of the shock. In this context of small perturbations of the quadratic Burgers flux, the result improves the Choi and Vasseur's result in [7]. However, we show that the L 2 -contraction up to a shift does not hold for every convex flux. We construct a smooth strictly convex flux, for which the L 2 -contraction does not hold any more even along any Lipschitz shift.

Authors:Andrew Comech; Tuoc Van Phan; Atanas Stefanov Pages: 157 - 196 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Andrew Comech, Tuoc Van Phan, Atanas Stefanov For the nonlinear Dirac equation in ( 1 + 1 ) D with scalar self-interaction (Gross–Neveu model), with quintic and higher order nonlinearities (and within certain range of the parameters), we prove that solitary wave solutions are asymptotically stable in the “even” subspace of perturbations (to ignore translations and eigenvalues ± 2 ω i ). The asymptotic stability is proved for initial data in H 1 . The approach is based on the spectral information about the linearization at solitary waves which we justify by numerical simulations. For the proof, we develop the spectral theory for the linearized operators and obtain appropriate estimates in mixed Lebesgue spaces, with and without weights.

Authors:Philippe Laurençot; Noriko Mizoguchi Pages: 197 - 220 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Philippe Laurençot, Noriko Mizoguchi The present paper is concerned with the parabolic–parabolic Keller–Segel system ∂ t u = div ( ∇ u q + 1 − u ∇ v ) , t > 0 , x ∈ Ω , ∂ t v = Δ v − α v + u , t > 0 , x ∈ Ω , ( u , v ) ( 0 ) = ( u 0 , v 0 ) ≥ 0 , x ∈ Ω , with degenerate critical diffusion q = q ⋆ : = ( N − 2 ) / N in space dimension N ≥ 3 , the underlying domain Ω being either Ω = R N or the open ball Ω = B R ( 0 ) of R N with suitable boundary conditions. It has remained open whether there exist solutions blowing up in finite time, the existence of such solutions being known for the parabolic–elliptic reduction with the second equation replaced by 0 = Δ v − α v + u . Assuming that N = 3 , 4 and α > 0 , we prove that radially symmetric solutions with negative initial energy blow up in finite time in Ω = R N and in Ω = B R ( 0 ) under mixed Neumann–Dirichlet boundary conditions. Moreover, if Ω = B R ( 0 ) and Neumann boundary conditions are imposed on both u and v, we show the existence of a positive constant C depending only on N, Ω, and the mass of u 0 such that radially symmetric solutions blow up in finite time if the initial energy does not exceed −C. The criterion for finite time blowup is satisfied by a large class of initial data.

Authors:Satoshi Masaki; Jun-ichi Segata Abstract: Publication date: Available online 20 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Jiaqi Liu; Peter A. Perry; Catherine Sulem Abstract: Publication date: Available online 20 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Yujin Guo; Xiaoyu Zeng Abstract: Publication date: Available online 19 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Yujin Guo, Xiaoyu Zeng We consider ground states of pseudo-relativistic boson stars with a self-interacting potential K ( x ) in R 3 , which can be described by minimizers of the pseudo-relativistic Hartree energy functional. Under some assumptions on K ( x ) , minimizers exist if the stellar mass N satisfies 0 < N < N ⁎ , and there is no minimizer if N > N ⁎ , where N ⁎ is called the critical stellar mass. In contrast to the case of the Coulomb-type potential where K ( x ) ≡ 1 , we prove that the existence of minimizers may occur at N = N ⁎ , depending on the local profile of K ( x ) near the origin. When there is no minimizer at N = N ⁎ , we also present a detailed analysis of the behavior of minimizers as N approaches N ⁎ from below, for which the stellar mass concentrates at a unique point.

Authors:Manuel del Pino; Konstantinos T. Gkikas Abstract: Publication date: Available online 4 April 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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 t ) + O ( PubDate: 2017-04-11T15:39:24Z DOI: 10.1016/j.anihpc.2017.03.005

Authors:Jérôme Droniou; Kyle S. Talbot Abstract: Publication date: Available online 30 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Thomas Kappeler; Jan-Cornelius Molnar Abstract: Publication date: Available online 28 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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 Abstract: Publication date: Available online 28 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Alessio Porretta; Philippe Souplet Abstract: Publication date: Available online 21 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Manuel Friedrich; Francesco Solombrino Abstract: Publication date: Available online 21 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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 Abstract: Publication date: Available online 18 March 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Marta Lewicka; Annie Raoult; Diego Ricciotti Abstract: Publication date: Available online 15 February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Jürgen Jost; Lei Liu; Miaomiao Zhu Abstract: Publication date: Available online 11 February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Huy Quang Nguyen Abstract: Publication date: Available online 30 January 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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 Abstract: Publication date: Available online 27 January 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Lassaad Aloui; Moez Khenissi; Luc Robbiano Abstract: Publication date: Available online 27 January 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Ricardo Alonso; Thierry Goudon; Arthur Vavasseur Abstract: Publication date: Available online 9 January 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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.

Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Benoît Mésognon-Gireau This paper shows that the long time existence of solutions to the Water Waves equations remains true with a large topography in presence of surface tension. More precisely, the dimensionless equations depend strongly on three parameters ε , μ , β measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations respectively. In [2], the local existence of solutions to this problem is proved on a time interval of size 1 max ( β , ε ) and uniformly with respect to μ. In presence of large bathymetric variations (typically β ≫ ε ), the existence time is therefore considerably reduced. We remove here this restriction and prove the local existence on a time interval of size 1 ε under the constraint that the surface tension parameter must be at the same order as the shallowness parameter μ. We also show that the result of [5] dealing with large bathymetric variations for the Shallow Water equations can be viewed as a particular endpoint case of our result.