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:Paulo Amorim; Wladimir Neves; José Francisco Rodrigues Pages: 221 - 248 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Paulo Amorim, Wladimir Neves, José Francisco Rodrigues In this work we introduce the obstacle-mass constraint problem for a multidimensional scalar hyperbolic conservation law. We prove existence of an entropy solution to this problem by a penalization/viscosity method. The mass constraint introduces a nonlocal Lagrange multiplier in the penalized equation, giving rise to a nonlocal parabolic problem. We introduce a compatibility condition relating the initial datum and the obstacle function which ensures global in time existence of solution. This is not a smoothness condition, but relates to the propagation of the support of the initial datum.

Authors:Ilaria Mondello Pages: 249 - 275 Abstract: Publication date: January–February 2017 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 34, Issue 1 Author(s): Ilaria Mondello On a compact stratified space ( X , g ) , a metric of constant scalar curvature exists in the conformal class of g if the scalar curvature S g satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant Y ℓ ( X ) . This latter is a conformal invariant introduced in the recent work of K. Akutagawa, G. Carron and R. Mazzeo. It depends on the local structure of X, in particular on its links, but its explicit value is unknown. We show that if the links satisfy a Ricci positive lower bound, then we can compute Y ℓ ( X ) . In order to achieve this, we prove a lower bound for the spectrum of the Laplacian, by extending a well-known theorem by A. Lichnerowicz, and a Sobolev inequality, inspired by a result due to D. Bakry. A particular stratified space, with one stratum of codimension 2 and cone angle bigger than 2π, must be handled separately – in this case we prove the existence of an Euclidean isoperimetric inequality.

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.

Authors:Anna Bohun; François Bouchut; Gianluca Crippa Pages: 1409 - 1429 Abstract: Publication date: November–December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 33, Issue 6 Author(s): Anna Bohun, François Bouchut, Gianluca Crippa We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov–Poisson equation with measure density. The proof exploits an anisotropic variant of the argument in [14,20] and suitable estimates for the difference quotients in such anisotropic context. In contrast to regularization methods, this approach gives quantitative estimates in terms of the given regularity bounds. From such estimates it is possible to recover the well posedness for the ordinary differential equation and for Lagrangian solutions to the continuity and transport equations.

Authors:Michela Eleuteri; Elisabetta Rocca; Giulio Schimperna Pages: 1431 - 1454 Abstract: Publication date: November–December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 33, Issue 6 Author(s): Michela Eleuteri, Elisabetta Rocca, Giulio Schimperna We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in [11] where existence of weak solutions was proved in three space dimensions. Here, we aim to study the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in [11].

Authors:Gohar Aleksanyan Pages: 1455 - 1471 Abstract: Publication date: November–December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 33, Issue 6 Author(s): Gohar Aleksanyan In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstacles { min ( − Δ u 1 + f 1 , u 1 − u 2 + ψ 1 ) = 0 min ( − Δ u 2 + f 2 , u 2 − u 1 + ψ 2 ) = 0 , arising in the optimal switching problem with two modes. We derive the optimal C 1 , 1 -regularity for the minimal solution under the assumption that the zero loop set L : = { ψ 1 + ψ 2 = 0 } is the closure of its interior. This result is optimal and we provide a counterexample showing that the C 1 , 1 -regularity does not hold without the assumption L = L 0 ‾ .

Authors:Constantin N. Beli; Liviu I. Ignat; Enrique Zuazua Pages: 1473 - 1495 Abstract: Publication date: November–December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 33, Issue 6 Author(s): Constantin N. Beli, Liviu I. Ignat, Enrique Zuazua In this paper we analyze the dispersion for one dimensional wave and Schrödinger equations with BV coefficients. In the case of the wave equation we give a complete answer in terms of the variation of the logarithm of the coefficient showing that dispersion occurs if this variation is small enough but it may fail when the variation goes beyond a sharp threshold. For the Schrödinger equation we prove that the dispersion holds under the same smallness assumption on the variation of the coefficient. But, whether dispersion may fail for larger coefficients is unknown for the Schrödinger equation.

Authors:Yuzhao Wang; Jie Xiao Pages: 1497 - 1507 Abstract: Publication date: November–December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Volume 33, Issue 6 Author(s): Yuzhao Wang, Jie Xiao Grigor'yan–Sun in [6] (with p = 2 ) and Sun in [10] (with p > 1 ) proved that if sup r ≫ 1 vol ( B ( x 0 , r ) ) r p σ p − σ − 1 ( ln r ) p − 1 p − σ − 1 < ∞ then the only non-negative weak solution of Δ p u + u σ ≤ 0 on a complete Riemannian manifold is identically 0; moreover, the powers of r and ln r are sharp. In this note, we present a constructive approach to the sharpness, which is flexible enough to treat the sharpness for Δ p u + f ( u , ∇ u ) ≤ 0 . Our construction is based on a perturbation of the fundamental solution to the p-Laplace equation, and we believe that the ideas introduced here are applicable to other nonlinear differential inequalities on manifolds.

Authors:Tomoyuki Miyaji; Yoshio Tsutsumi Abstract: Publication date: Available online 23 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Wolansky Abstract: Publication date: Available online 15 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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 if, e.g. μ has a disconnected support, or if the dimension of μ , d (to be defined) is larger or equal 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:Charles Baker; Huy The Nguyen Abstract: Publication date: Available online 9 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Charles Baker, Huy The Nguyen We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature smoothly evolve by mean curvature flow to round points.

Authors:L.J. Díaz; K. Gelfert; M. Rams Abstract: Publication date: Available online 9 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): L.J. Díaz, K. Gelfert, M. Rams We study transitive step skew-product maps modeled over a complete shift of k, k ≥ 2 , symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak⁎ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.

Authors:Andrej Abstract: Publication date: Available online 7 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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 exa mples in periodic media which are non-degenerate in a natural sense, and they also prove a conjecture from [7].

Authors:Sun-Sig Byun; Jihoon Ok; Jung-Tae Park Abstract: Publication date: Available online 7 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis 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:Toshiaki Hishida; Ana Leonor Silvestre; Takéo Takahashi Abstract: Publication date: Available online 6 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Toshiaki Hishida, Ana Leonor Silvestre, Takéo Takahashi Consider a rigid body S ⊂ R 3 immersed in an infinitely extended Navier-Stokes fluid. We are interested in self-propelled motions of S in the steady state regime of the system rigid body-fluid, assuming that the mechanism used by the body to reach such a motion is modeled through a distribution of velocities v ⁎ on ∂ S . If the velocity V of S is given, can we find v ⁎ that generates V? We show that this can be solved as a control problem in which v ⁎ is a six-dimensional control such that either Supp v ⁎ ⊂ Γ , an arbitrary nonempty open subset of ∂Ω, or v ⁎ ⋅ n ∂ Ω = 0 . We also show that one of the self-propelled conditions implies a better summability of the fluid velocity.

Authors:Ting-Ying Chang; Florica C. Cîrstea Abstract: Publication date: Available online 5 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Ting-Ying Chang, Florica C. Cîrstea We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form (0.1) − div ( A ( x ) ∇ u p − 2 ∇ u ) + b ( x ) h ( u ) = 0 in B 1 ∖ { 0 } , where B r denotes the open ball with radius r > 0 centred at 0 in R N ( N ≥ 2 ) . We assume that A ∈ C 1 ( 0 , 1 ] , b ∈ C ( B 1 ‾ ∖ { 0 } ) and h ∈ C [ 0 , ∞ ) are positive functions associated with regularly varying functions of index ϑ, σ and q at 0, 0 and ∞ respectively, satisfying q > p − 1 > 0 and ϑ − σ < p < N + ϑ . We prove that the condition b ( x ) h ( Φ ) ∉ L 1 ( B 1 / 2 ) is sharp for the removability of all singularities at 0 for the positive solutions of (0.1), where Φ denotes the “fundamental solution” of − div ( A ( x ) ∇ u p − 2 ∇ u ) = δ 0 (the Dirac mass at 0) in B 1 , subject to Φ ∂ B 1 = 0 . If b ( x ) h ( Φ ) ∈ L 1 ( B 1 / 2 ) PubDate: 2016-12-12T09:15:15Z DOI: 10.1016/j.anihpc.2016.12.001

Authors:Kari Astala; Albert Clop; Daniel Faraco; Jarmo Jääskeläinen; Aleksis Koski Abstract: Publication date: Available online 5 December 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Kari Astala, Albert Clop, Daniel Faraco, Jarmo Jääskeläinen, Aleksis Koski We provide Schauder estimates for nonlinear Beltrami equations and lower bounds of the Jacobians for homeomorphic solutions. The results were announced in [1] but here we give detailed proofs.

Authors:Chenjie Fan Abstract: Publication date: Available online 21 November 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Chenjie Fan We study the focusing mass-critical nonlinear Schrödinger equation, and construct certain solutions which blow up at exactly m points according to the log–log law.

Authors:Jianfeng Cheng; Lili Du; Yongfu Wang Abstract: Publication date: Available online 17 November 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Jianfeng Cheng, Lili Du, Yongfu Wang The free streamline theory in hydrodynamics is an important and difficult issue not only in fluid mechanics but also in mathematics. The major purpose in this paper is to establish the well-posedness of the impinging jets in steady incompressible, rotational, plane flows. More precisely, given a mass flux and a vorticity of the incoming flows in the inlet of the nozzle, there exists a unique smooth impinging plane jet. Moreover, there exists a smooth free streamline, which goes to infinity and initiates at the endpoint of the nozzle smoothly. In addition, asymptotic behavior in upstream and downstream, uniform direction and other properties of the impinging jet are also obtained. The main ingredients of the mathematic analysis in this paper are based on the modified variational method developed by H. W. Alt, L. A. Caffarelli and A. Friedman in the elegant works [1,17], which has been shown to be powerful to deal with the steady irrotational flows with free streamlines.

Authors:Serena Dipierro; Enrico Valdinoci Abstract: Publication date: Available online 16 November 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Serena Dipierro, Enrico Valdinoci We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.

Authors:Pierre Berger; Alejandro Kocsard Abstract: Publication date: Available online 8 November 2016 Source:Annales de l'Institut Henri Poincare (C) Non Linear Analysis Author(s): Pierre Berger, Alejandro Kocsard We prove that every endomorphism which satisfies Axiom A and the strong transversality conditions is C 1 -inverse limit structurally stable. These conditions were conjectured to be necessary and sufficient. This result is applied to the study of unfolding of some homoclinic tangencies. This also achieves a characterization of C 1 -inverse limit structurally stable covering maps.