Authors:Zdzisław Brzeźniak; Ben Goldys; Terence Jegaraj Pages: 497 - 558 Abstract: Abstract We study a stochastic Landau–Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is the compactness, or weak to strong continuity, of the solution map for a deterministic Landau–Lifschitz equation when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1117-0 Issue No:Vol. 226, No. 2 (2017)

Authors:Vieri Benci; Donato Fortunato; Filippo Gazzola Pages: 559 - 585 Abstract: Abstract This paper studies the existence of solitons, namely stable solitary waves, in an idealized suspension bridge. The bridge is modeled as an unbounded degenerate plate, that is, a central beam with cross sections, and displays two degrees of freedom: the vertical displacement of the beam and the torsional angles of the cross sections. Under fairly general assumptions, we prove the existence of solitons. Under the additional assumption of large tension in the sustaining cables, we prove that these solitons have a nontrivial torsional component. This appears relevant for security since several suspension bridges collapsed due to torsional oscillations. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1138-8 Issue No:Vol. 226, No. 2 (2017)

Authors:Zhigang Wu; Weike Wang Pages: 587 - 638 Abstract: Abstract The Cauchy problem of the bipolar Navier–Stokes–Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit a generalized Huygens’ principle as the Navier–Stokes system. This phenomenon is the most important difference from the unipolar Navier–Stokes–Poisson system. Due to the non-conservative structure of the system (1.1) and the interplay of two carriers which counteract the influence of the electric field (a nonlocal term), some new observations are essential for the proof. We fully use the conservative structure of the system for the total density and total momentum, and the mechanism of the linearized unipolar Navier–Stokes–Poisson system together with the special form of the nonlinear terms in the system for the difference of densities and the difference of momentums. Lastly, as a byproduct, we extend the usual \({L^2({\mathbb{R}}^3)}\) -decay rate to the \({L^p({\mathbb{R}}^3)}\) -decay rate with \({p > 1}\) and also improve former decay rates. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1140-1 Issue No:Vol. 226, No. 2 (2017)

Authors:Blanche Buet; Gian Paolo Leonardi; Simon Masnou Pages: 639 - 694 Abstract: Abstract We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold, in particular, for sequences of discrete varifolds associated with point clouds or pixel/voxel-type discretizations of d-surfaces in the Euclidean n-space, without restrictions on dimension and codimension. The variational nature of the approach also allows us to consider surfaces with singularities, and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface. A series of numerical tests are provided in order to illustrate the effectiveness and generality of the method. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1141-0 Issue No:Vol. 226, No. 2 (2017)

Authors:Scott Armstrong; Tuomo Kuusi; Jean-Christophe Mourrat; Christophe Prange Pages: 695 - 741 Abstract: Abstract We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1142-z Issue No:Vol. 226, No. 2 (2017)

Authors:Davit Harutyunyan Pages: 743 - 766 Abstract: Abstract In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn’s first (linear geometric rigidity estimate) and second inequalities on that kind of shell for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like h, and if the Gaussian curvature is negative, then the Korn constant scales like h 4/3, where h is the thickness of the shell. These results have a classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke et al. for plates in Arch Ration Mech Anal 180(2):183–236, 2006 (where they show that the Korn constant in the nonlinear Korn’s first inequality scales like h 2), extended to shells with nonzero curvature. We also recover the uniform Korn–Poincaré inequality proven for “boundary-less” shells by Lewicka and Müller in Annales de l’Institute Henri Poincare (C) Non Linear Anal 28(3):443–469, 2011 in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents 1 and 4/3 in the present work appear for the first time in any sharp geometric rigidity estimate. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1143-y Issue No:Vol. 226, No. 2 (2017)

Authors:Yuan Chen; Yong Yu Pages: 767 - 808 Abstract: Abstract In this article we construct a global solution of the simplified Ericksen-Leslie system. We show that the velocity of the solution can be decomposed into the sum of three parts. The main flow is governed by the Oseen vortex with the same circulation Reynolds number as the initial fluid. The secondary flow has finite kinetic energy and decay in the speed (1 + t)−2 as \({t \rightarrow \infty}\) . The third part is a minor flow whose kinetic energy decays faster than the secondary flow. As for the orientation variable, our solution has a phase function which diverges logarithmically to \({\infty}\) as \({t \rightarrow \infty}\) . This indicates that the orientation variable will keep rotating around the z-axis while \({t \rightarrow \infty}\) . This phenomenon results from a non-trivial coupling between the orientation variable and a fluid with a non-zero circulation Reynolds number. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1144-x Issue No:Vol. 226, No. 2 (2017)

Authors:U. S. Fjordholm; S. Lanthaler; S. Mishra Pages: 809 - 849 Abstract: Abstract We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on p-integrable functions. To do so, we prove the equivalence between probability measures on L p spaces and infinite families of correlation measures. Each member of this family, termed a correlation marginal, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a L p space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions, even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting entropy statistical solutions exist, are unique and are stable with respect to the 1-Wasserstein metric on probability measures on L 1. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1145-9 Issue No:Vol. 226, No. 2 (2017)

Authors:Michaela A. C. Vollmer Pages: 851 - 922 Abstract: Abstract We study the bifurcation diagram of the Onsager free-energy functional for liquid crystals with orientation parameter on the sphere. In particular, we concentrate on the bifurcations from the isotropic solution for a general class of two-body interaction potentials including the Onsager kernel. Reformulating the problem as a non-linear eigenvalue problem for the kernel operator, we prove that spherical harmonics are the corresponding eigenfunctions and we present a direct relationship between the coefficients of the Taylor expansion of this class of interaction potentials and their eigenvalues. We find explicit expressions for all bifurcation points corresponding to bifurcations from the isotropic state of the Onsager free-energy functional equipped with the Onsager interaction potential. A substantial amount of our analysis is based on the use of spherical harmonics and a special algorithm for computing expansions of products of spherical harmonics in terms of spherical harmonics is presented. Using a Lyapunov–Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the invariant theory of groups. On the basis of these results, we show that the first bifurcation from the isotropic state of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global properties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser if the bifurcation parameter is high, that it is not a local minimiser if the bifurcation parameter is small, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch. PubDate: 2017-11-01 DOI: 10.1007/s00205-017-1146-8 Issue No:Vol. 226, No. 2 (2017)

Authors:M. Olive; B. Kolev; N. Auffray Pages: 1 - 31 Abstract: Abstract We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1127-y Issue No:Vol. 226, No. 1 (2017)

Authors:Hyeonbae Kang; Mikyoung Lim; Sanghyeon Yu Pages: 83 - 115 Abstract: Abstract The purpose of this paper is to investigate the spectral nature of the Neumann–Poincaré operator on the intersecting disks, which is a domain with the Lipschitz boundary. The complete spectral resolution of the operator is derived, which shows, in particular, that it admits only the absolutely continuous spectrum; no singularly continuous spectrum and no pure point spectrum. We then quantitatively analyze using the spectral resolution of the plasmon resonance at the absolutely continuous spectrum. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1129-9 Issue No:Vol. 226, No. 1 (2017)

Authors:David Chiron; Mihai Mariş Pages: 143 - 242 Abstract: Abstract We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross–Pitaevskii and the so-called “cubic-quintic” equations) in space dimension \({ N \geq 2}\) . We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1131-2 Issue No:Vol. 226, No. 1 (2017)

Authors:Sebastián Ferraro; Manuel de León; Juan Carlos Marrero; David Martín de Diego; Miguel Vaquero Pages: 243 - 302 Abstract: Abstract In this paper we develop a geometric version of the Hamilton–Jacobi equation in the Poisson setting. Specifically, we “geometrize” what is usually called a complete solution of the Hamilton–Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80–88, 1991), Ge (Indiana Univ. Math. J. 39(3):859–876, 1990), Ge and Marsden (Phys Lett A 133(3):134–139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1133-0 Issue No:Vol. 226, No. 1 (2017)

Authors:Antonio Carlos Fernandes; Jaume Llibre; Luis Fernando Mello Pages: 303 - 320 Abstract: Abstract We study the convex central configurations of the 4-body problem assuming that they have two pairs of equal masses located at two adjacent vertices of a convex quadrilateral. Under these assumptions we prove that the isosceles trapezoid is the unique central configuration. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1134-z Issue No:Vol. 226, No. 1 (2017)

Authors:Youcef Amirat; Rachid Touzani Pages: 405 - 440 Abstract: Abstract We consider a three-dimensional time-harmonic eddy current problem formulated in terms of the magnetic field. We prove that in the case of one thin toroidal conductor, eddy current equations have as a limit Kirchhoff’s algebraic equation for circuits. This approximation is valid in the case of small resistivity and voltage. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1136-x Issue No:Vol. 226, No. 1 (2017)

Authors:S. Bianchini; E. Marconi Pages: 441 - 493 Abstract: Abstract We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular, the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C 0-sense up to the degeneracy due to the segments where \({f''=0}\) . We prove also that the initial data is taken in a suitably strong sense and we give some examples which show that these results are sharp. PubDate: 2017-10-01 DOI: 10.1007/s00205-017-1137-9 Issue No:Vol. 226, No. 1 (2017)

Authors:Alberto Bressan; Marta Lewicka Abstract: Abstract We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with \({\mathcal{C}^{2,\alpha}}\) boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion. PubDate: 2017-10-12 DOI: 10.1007/s00205-017-1183-3

Authors:Ibrahim Ekren; H. Mete Soner Abstract: The classical duality theory of Kantorovich (C R (Doklady) Acad Sci URSS (NS) 37:199–201, 1942) and Kellerer (Z Wahrsch Verw Gebiete 67(4):399–432, 1984) for classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice \({\mathcal{X}}\) with an order unit. The problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of \({\mathcal{X}}\) and the dual problem is defined on the bi-dual of \({\mathcal{X}}\) . These results are then applied to several extensions of the classical optimal transport. PubDate: 2017-10-09 DOI: 10.1007/s00205-017-1178-0

Authors:Lihui Chai; Carlos J. García-Cervera; Xu Yang Abstract: Abstract The Schrödinger–Poisson–Landau–Lifshitz–Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials. This system has been used in connection with the study of spin transfer and magnetization reversal in ferromagnetic materials. In this paper, we rigorously prove the existence of weak solutions to SPLLG and derive the Vlasov–Poisson–Landau–Lifshitz–Glibert system as the semiclassical limit. PubDate: 2017-10-09 DOI: 10.1007/s00205-017-1177-1