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 Archive for Rational Mechanics and Analysis   [SJR: 4.091]   [H-I: 66]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0673 - ISSN (Online) 0003-9527    Published by Springer-Verlag  [2345 journals]
• Homogenization of the Brush Problem with a Source Term in L 1
• Authors: Antonio Gaudiello; Olivier Guibé; François Murat
Pages: 1 - 64
Abstract: We consider a domain which has the form of a brush in 3D or the form of a comb in 2D, i.e. an open set which is composed of cylindrical vertical teeth distributed over a fixed basis. All the teeth have a similar fixed height; their cross sections can vary from one tooth to another and are not supposed to be smooth; moreover the teeth can be adjacent, i.e. they can share parts of their boundaries. The diameter of every tooth is supposed to be less than or equal to $${\varepsilon}$$ , and the asymptotic volume fraction of the teeth (as $${\varepsilon}$$ tends to zero) is supposed to be bounded from below away from zero, but no periodicity is assumed on the distribution of the teeth. In this domain we study the asymptotic behavior (as $${\varepsilon}$$ tends to zero) of the solution of a second order elliptic equation with a zeroth order term which is bounded from below away from zero, when the homogeneous Neumann boundary condition is satisfied on the whole of the boundary. First, we revisit the problem where the source term belongs to L 2. This is a classical problem, but our homogenization result takes place in a geometry which is more general that the ones which have been considered before. Moreover we prove a corrector result which is new. Then, we study the case where the source term belongs to L 1. Working in the framework of renormalized solutions and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved (such a definition is new), we identify the limit problem and prove a corrector result.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1079-2
Issue No: Vol. 225, No. 1 (2017)

• A Measure and Orientation Preserving Homeomorphism with Approximate
Jacobian Equal −1 Almost Everywhere
• Authors: Paweł Goldstein; Piotr Hajłasz
Pages: 65 - 88
Abstract: We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit n-dimensional cube onto itself, whose Jacobian is equal to −1 almost everywhere. Moreover, we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1085-4
Issue No: Vol. 225, No. 1 (2017)

• The Elastic Trefoil is the Doubly Covered Circle
• Authors: Henryk Gerlach; Philipp Reiter; Heiko von der Mosel
Pages: 89 - 139
Abstract: To describe the behavior of knotted loops of springy wire with an elementary mathematical model we minimize the integral of squared curvature, $${E = \int \varkappa^2}$$ , together with a small multiple of ropelength $${\mathcal{R}}$$ = length/thickness in order to penalize selfintersection. Our main objective is to characterize all limit configurations of energy minimizers of the total energy $${E_{\vartheta} \equiv E + \vartheta \mathcal{R}}$$ as $${\vartheta}$$ tends to zero. For short, these limit configurations will be referred to as elastic knots. The elastic unknot turns out to be the once covered circle with squared curvature energy $${(2\pi)^2}$$ . For all (non-trivial) knot classes for which the natural lower bound $${(4\pi)^2}$$ on E is sharp, the respective elastic knot is the doubly covered circle. We also derive a new characterization of two-bridge torus knots in terms of E, proving that the only knot classes for which the lower bound $${(4\pi)^2}$$ on E is sharp are the $${(2,b)}$$ -torus knots for odd b with $${ b \ge 3}$$ (containing the trefoil knot class). In particular, the elastic trefoil knot is the doubly covered circle.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1100-9
Issue No: Vol. 225, No. 1 (2017)

• Global Existence and Asymptotic Behavior of Affine Motion of 3D Ideal
Fluids Surrounded by Vacuum
• Authors: Thomas C. Sideris
Pages: 141 - 176
Abstract: The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $${{\rm GL}^+(3, \mathbb{R})}$$ . The evolution of the fluid domain is described by a family of ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank r = 1, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along 3−r of its principal axes. In the compressible case, the asymptotic limit has rank r = 3, and asymptotic completeness holds, when the adiabatic index $${\gamma}$$ satisfies $${4/3 < \gamma < 2}$$ . The number of possible degeneracies, 3−r, increases with the value of the adiabatic index $${\gamma}$$ . In the incompressible case, affine motion reduces to geodesic flow in $${{\rm SL}(3, \mathbb{R})}$$ with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1106-3
Issue No: Vol. 225, No. 1 (2017)

• A Temperature-Dependent Phase-Field Model for Phase Separation and Damage
• Authors: Christian Heinemann; Christiane Kraus; Elisabetta Rocca; Riccarda Rossi
Pages: 177 - 247
Abstract: In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature concerning phase separation and damage processes in elastic media, in our model we encompass thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More particularly, we prove the existence of “entropic weak solutions”, resorting to a solvability concept first introduced in Feireisl (Comput Math Appl 53:461–490, 2007) in the framework of Fourier–Navier–Stokes systems and then recently employed in Feireisl et al. (Math Methods Appl Sci 32:1345–1369, 2009) and Rocca and Rossi (Math Models Methods Appl Sci 24:1265–1341, 2014) for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1102-7
Issue No: Vol. 225, No. 1 (2017)

• A Condition for the Hölder Regularity of Local Minimizers of a Nonlinear
Elastic Energy in Two Dimensions
• Authors: Jonathan J. Bevan
Pages: 249 - 285
Abstract: We prove the local Hölder continuity of strong local minimizers of the stored energy functional $$E(u)=\int_{\Omega}\lambda \nabla u ^{2}+h({\rm det} \nabla u)\,{\rm d}x$$ subject to a condition of ‘positive twist’. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as $${s\to 0+}$$ , linearly as $${s \to +\infty}$$ , and satisfies $${h(s)=+\infty}$$ if $${s \leqq 0}$$ . These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed Hölder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term $${\int_{\Omega} h({\rm det} \nabla u)\,{\rm d}x}$$ can have by analysing the regularity of local minimizers in the class of ‘shear maps’. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are Hölder continuous.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1104-5
Issue No: Vol. 225, No. 1 (2017)

• On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk
• Authors: Yasunori Maekawa
Pages: 287 - 374
Abstract: We study the stability of some exact stationary solutions to the two-dimensional Navier–Stokes equations in an exterior domain to the unit disk. These stationary solutions are known as a simple model of the flow around a rotating obstacle, while their stability has been open due to the difficulty arising from their spatial decay in a scale-critical order. In this paper we affirmatively settle this problem for small solutions. That is, we will show that if these exact solutions are small enough then they are asymptotically stable with respect to small L 2 perturbations.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1105-4
Issue No: Vol. 225, No. 1 (2017)

• Global Well-Posedness of the Boltzmann Equation with Large Amplitude
Initial Data
• Authors: Renjun Duan; Feimin Huang; Yong Wang; Tong Yang
Pages: 375 - 424
Abstract: The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new $${L^\infty_xL^1_{v}\cap L^\infty_{x,v}}$$ approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted $${L^\infty}$$ norm under some smallness condition on the $${L^1_xL^\infty_v}$$ norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in the $${L^\infty_{x,v}}$$ norm with explicit rates of convergence are also studied.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1107-2
Issue No: Vol. 225, No. 1 (2017)

• A Derivation of Linearized Griffith Energies from Nonlinear Models
• Authors: Manuel Friedrich
Pages: 425 - 467
Abstract: We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via $${\Gamma}$$ -convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1108-1
Issue No: Vol. 225, No. 1 (2017)

• Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger
Equation on $${\mathbb{R}^3}$$ R 3
• Authors: Rowan Killip; Tadahiro Oh; Oana Pocovnicu; Monica Vişan
Pages: 469 - 548
Abstract: We consider the cubic–quintic nonlinear Schrödinger equation: $$i\partial_t u = -\Delta u - u ^2u + u ^4u.$$ In the first part of the paper, we analyze the one-parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region $${{\mathcal{R}}}$$ of the mass/energy plane where the virial is positive. We show that this region is partially bounded by solitons also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1109-0
Issue No: Vol. 225, No. 1 (2017)

• On the Liouville Type Theorems for Self-Similar Solutions to the
Navier–Stokes Equations
• Authors: Dongho Chae; Jörg Wolf
Pages: 549 - 572
Abstract: We prove Liouville type theorems for the self-similar solutions to the Navier–Stokes equations. One of our results generalizes the previous ones by Nečas–Ru̇žička–Šverák and Tsai. Using a Liouville type theorem, we also remove a scenario of asymptotically self-similar blow-up for the Navier–Stokes equations with the profile belonging to $${L^{p, \infty} (\mathbb{R}^3)}$$ with $${p > \frac{3}{2}}$$ .
PubDate: 2017-07-01
DOI: 10.1007/s00205-017-1110-7
Issue No: Vol. 225, No. 1 (2017)

• A Partial Differential Equation for the Rank One Convex Envelope
• Authors: Adam M. Oberman; Yuanlong Ruan
Pages: 955 - 984
Abstract: A partial differential equation (PDE) for the rank one convex envelope is introduced. The existence and uniqueness of viscosity solutions to the PDE is established. Elliptic finite difference schemes are constructed and convergence of finite difference solutions to the viscosity solution of the PDE is proven. Computational results are presented and laminates are computed from the envelopes. Results include the Kohn–Strang example, the classical four gradient example, and an example with eight gradients which produces nontrivial laminates.
PubDate: 2017-06-01
DOI: 10.1007/s00205-017-1092-5
Issue No: Vol. 224, No. 3 (2017)

• Traveling Waves for Nonlinear Schrödinger Equations with Nonzero
Conditions at Infinity
• Authors: David Chiron; Mihai Mariş
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: 2017-06-19
DOI: 10.1007/s00205-017-1131-2

• Pointwise Estimates for Bipolar Compressible Navier–Stokes–Poisson
System in Dimension Three
• Authors: Zhigang Wu; Weike Wang
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: 2017-06-10
DOI: 10.1007/s00205-017-1140-1

• On the Geometry of the Hamilton–Jacobi Equation and Generating
Functions
• Authors: Sebastián Ferraro; Manuel de León; Juan Carlos Marrero; David Martín de Diego; Miguel Vaquero
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: 2017-06-09
DOI: 10.1007/s00205-017-1133-0

• On the Structure of $${L^\infty}$$ L ∞ -Entropy Solutions to Scalar
Conservation Laws in One-Space Dimension
• Authors: S. Bianchini; E. Marconi
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: 2017-06-07
DOI: 10.1007/s00205-017-1137-9

• Existence of Torsional Solitons in a Beam Model of Suspension Bridge
• Authors: Vieri Benci; Donato Fortunato; Filippo Gazzola
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: 2017-06-07
DOI: 10.1007/s00205-017-1138-8

• Convex Central Configurations of the 4-Body Problem with Two Pairs of
• Authors: Antonio Carlos Fernandes; Jaume Llibre; Luis Fernando Mello
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: 2017-06-06
DOI: 10.1007/s00205-017-1134-z

• Homogenization Near Resonances and Artificial Magnetism in Three
Dimensional Dielectric Metamaterials
• Authors: Guy Bouchitté; Christophe Bourel; Didier Felbacq
Abstract: It is now well established that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a (possibly) negative frequency-dependent effective permeability. However this result based on a two-dimensional approach holds merely in the case of linearly polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this paper we consider a dielectric structure placed in a bounded domain of $${\mathbb{R}^3}$$ and perform a full three dimensional asymptotic analysis. The main ingredient is a new averaging method for characterizing the bulk effective magnetic field in the vanishing-period limit. We give evidence of a vectorial spectral problem on the periodic cell which determines micro-resonances and encodes the oscillating behavior of the magnetic field from which artificial magnetism arises. At a macroscopic level we deduce an effective permeability tensor that we can make explicit as a function of the frequency. As far as sign-changing permeability is sought after, we may foresee that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry. Part of these results have been announced in Bouchitté et al. (C R Math Acad Sci Paris 347(9–10):571–576, 2009).
PubDate: 2017-06-05
DOI: 10.1007/s00205-017-1132-1

• A Circuit Equation as a Limit of Eddy Current Equations
• Authors: Youcef Amirat; Rachid Touzani
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: 2017-06-02
DOI: 10.1007/s00205-017-1136-x

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