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Abstract: We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our well-posedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. PubDate: 2019-03-08 DOI: 10.1007/s00332-019-09536-5

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Abstract: We investigate the effects of structural perturbations on the networks ability to synchronize. We establish a classification of directed links according to their impact on synchronizability. We focus on adding directed links in weakly connected networks having a strongly connected component acting as driver. When the connectivity of the driver is not stronger than the connectivity of the slave component, we can always make the network strongly connected while hindering synchronization. On the other hand, we prove the existence of a perturbation which makes the network strongly connected while increasing the synchronizability. Under additional conditions, there is a node in the driving component such that adding a single link starting at an arbitrary node of the driven component and ending at this node increases the synchronizability. PubDate: 2019-03-06 DOI: 10.1007/s00332-019-09534-7

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Abstract: We study localization occurring during high-speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the competition between Hadamard instability (caused by softening response) and the stabilizing effects of strain rate hardening. We consider a hyperbolic–parabolic system that expresses the above mechanism and construct self-similar solutions of localizing type that arise as the outcome of the above competition. The existence of self-similar solutions is turned, via a series of transformations, into a problem of constructing a heteroclinic orbit for an induced dynamical system. The dynamical system is in four dimensions but has a fast–slow structure with respect to a small parameter capturing the strength of strain rate hardening. Geometric singular perturbation theory is applied to construct the heteroclinic orbit as a transversal intersection of two invariant manifolds in the phase space. PubDate: 2019-03-04 DOI: 10.1007/s00332-019-09538-3

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Abstract: In this paper, we present a formal analysis of the long-time asymptotics of a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which have the form \(f\left( x,v,t\right) =g\left( v-L\left( t\right) x,t\right) \) where \(L\left( t\right) =A\left( I+tA\right) ^{-1}\) with the matrix A describing a shear flow or a dilatation or a combination of both. We began this study in James et al. (Arch Ration Mech Anal 231(2):787–843, 2019). Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In James et al. (2019), it has been proved rigorously the existence of self-similar solutions which describe the change of the average energy of the particles of the system in the case in which there is a balance between the hyperbolic and the collision term. In this paper, we focus in homoenergetic solutions for which the collision term is much larger than the hyperbolic term (collision-dominated behavior). In this case, the long-time asymptotics for the distribution of velocities is given by a time-dependent Maxwellian distribution with changing temperature. PubDate: 2019-03-02 DOI: 10.1007/s00332-019-09535-6

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Abstract: We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in \({\mathbb {R}}^3\) , using a definition of symmetry that arises naturally when calculating the equilibrium probability of a cluster of spheres in the sticky-sphere limit. We define the sticky symmetry group of the cluster as the set of permutations and inversions of the spheres which preserve adjacency and can be realized by continuous deformations of the cluster that do not change the set of contacts or cause particles to overlap. The symmetry number is the size of the sticky symmetry group. We introduce a numerical algorithm to compute the sticky symmetry group and symmetry number, and show it works well on several test cases. Furthermore, we show that once the sticky symmetry group has been calculated for indistinguishable spheres, the symmetry group for partially distinguishable spheres (those with nonidentical interactions) can be efficiently obtained without repeating the laborious parts of the computations. We use our algorithm to calculate the partition functions of every possible connected cluster of six identical sticky spheres, generating data that may be used to design interactions between spheres so they self-assemble into a desired structure. PubDate: 2019-02-22 DOI: 10.1007/s00332-019-09537-4

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Abstract: We study models of dilute rigid rod-like polymer solutions. We establish the global well-posedness of the Doi model for large data and for arbitrarily large viscous stress parameter. The main ingredient in the proof is the fact that the viscous stress adds dissipation to high derivatives of velocity. PubDate: 2019-02-06 DOI: 10.1007/s00332-019-09533-8

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Abstract: We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory, the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material is added to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion. PubDate: 2019-02-06 DOI: 10.1007/s00332-019-09531-w

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Abstract: In this paper, we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents. Moreover, we discuss a novel dissipation structure for such currents, namely the flat distance, that will serve to drive the evolution of the dislocation clusters. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9488-4

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Abstract: We establish Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results are facilitated by a rigorous version of the Ott–Mann–Gawȩdzki relation, sometimes described as a “Lagrangian analogue of the 4 / 5-law.” In particular, we prove that for weak solutions of the Euler equations, the Lagrangian forward/backward dispersion measure matches onto the energy defect (Onsager in Nuovo Cimento (Supplemento) 6:279–287, 1949; Duchon and Robert in Nonlinearity 13(1):249–255, 2000) in the sense of distributions. For strong limits of \(d\ge 3\) -dimensional Navier–Stokes solutions, the defect distribution coincides with the viscous dissipation anomaly. The Lagrangian formula shows that particles released into a 3d turbulent flow will initially disperse faster backward in time than forward, in agreement with recent theoretical predictions of Jucha et al. (Phys Rev Lett 113(5):054501, 2014). In two dimensions, we consider strong limits of solutions of the forced Euler equations with increasingly high-wave number forcing as a model of an ideal inverse cascade regime. We show that the same Lagrangian dispersion measure matches onto the anomalous input from the infinite-frequency force. As forcing typically acts as an energy source, this leads to the prediction that particles in 2d typically disperse faster forward in time than backward, which is opposite to that which occurs in 3d. Time asymmetry of the Lagrangian dispersion is thereby closely tied to the direction of the turbulent cascade, downscale in \(d\ge 3\) and upscale in \(d=2\) . These conclusions lend support to the conjecture of Eyink and Drivas (J Stat Phys 158(2):386–432, 2015) that a similar connection holds for time asymmetry of Richardson two-particle dispersion and cascade direction. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9476-8

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Abstract: The Onsager’s conjecture has two parts: conservation of energy, if the exponent is larger than 1 / 3, and the possibility of dissipative Euler solutions, if the exponent is less than or equal to 1 / 3. The paper proves half of the conjecture, the conservation part, in bounded domains. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9483-9

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Abstract: An important aspect of understanding FPU chains is the existence of invariant manifolds (called “bushes”) in FPU chains. We will focus on the classical periodic FPU chain and on the FPU chain with alternating masses where we show that also in the alternating case nested manifolds (related to bushes) exist. The use of symmetries leads to the emergence of systems of n particles as invariant manifolds of systems with a multiple of n particles. This analysis is followed by examples of existence and stability of special invariant manifolds and phase-space dynamics in the case of 4 and 8 particles. These examples are typical for periodic FPU chains with 4n or 8n particles. It turns out that in the alternating case the dynamics is strongly affected by the choice of the alternating mass m. Normal form calculations help to identify quasi-trapping regions leading to delay of recurrence. The results suggest that equipartition of energy near stable equilibrium is improbable. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9482-x

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Abstract: This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9479-5

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Abstract: We motivate and analyze a simple model for the formation of banded vegetation patterns. The model incorporates a minimal number of ingredients for vegetation growth in semiarid landscapes. It allows for comprehensive analysis and sheds new light onto phenomena such as the migration of vegetation bands and the interplay between their upper and lower edges. The key ingredient is the formulation as a closed reaction–diffusion system, thus introducing a conservation law that both allows for analysis and provides ready intuition and understanding through analogies with characteristic speeds of propagation and shock waves. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9486-6

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Abstract: While studying many wave turbulence (WT) phenomena, owing to their inherent complexity, one frequently encounters the necessity of uncertainty quantification (UQ) in high dimension. The implementation of many existing algorithms for the UQ under these circumstances demands a vast amount of computation and, as a consequence, the straightforward approach tends to be infeasible or impractical. One effort to circumvent this obstacle in high dimension is to find an effective dimension reduction procedure for the probability distribution of the dynamical system model. One of the methodologies to do this is to replace the true Markovian model with a simple stochastic model that is significantly more amenable to the UQ than the underlying system. The procedure can be carried out via approximating the original equation for each Fourier mode by an independent and analytically tractable stochastic differential equation. In this work, we introduce a new approach for the so-called reduced-order model strategy within the context of the Majda–McLaughlin–Tabak model. Our framework makes use of a detailed analysis of the one-dimensional WT prototype to build a family of simplified models. Furthermore, the adaptive parameters are tuned without performing a direct numerical simulation of the true dynamical system model. PubDate: 2019-02-01 DOI: 10.1007/s00332-019-09532-9

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Abstract: We show that the ill-posedness observed in backward parabolic equation, or cross-diffusion systems, can be interpreted as a limiting Turing instability for a corresponding semi-linear parabolic system. Our analysis is based on the, now well established, derivation of nonlinear parabolic and cross-diffusion systems from semi-linear reaction–diffusion systems with fast reaction rates. We illustrate our observation with two generic examples for \(2\times 2\) and \(4\times 4\) reaction–diffusion systems. For these examples, we prove that backward parabolicity in cross-diffusion systems is equivalent to Turing instability for fast reaction rates. In one dimension, the Turing patterns are periodic solutions which have frequencies which increase with the reaction rate. Furthermore, in some specific cases, the structure of the equations at hand involves classical entropy/Lyapunov functions which lead to a priori estimates allowing to pass rigorously to the fast reaction limit in the absence of Turing instabilities. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9480-z

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Abstract: Using the scale invariance of the Navier–Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the 3D Navier–Stokes equations, on a periodic domain \(\mathcal {V} =[0,\,L]^{3}\) an infinite ‘chessboard’ of estimates for these inverse length scales is displayed in terms of labels \((n,\,m)\) corresponding to n derivatives of the velocity field in \(L^{2m}(\mathcal {V})\) . The \((1,\,1)\) position corresponds to the inverse Kolmogorov length \(Re^{3/4}\) . These estimates ultimately converge to a finite limit, \(Re^3\) , as \(n,\,m\rightarrow \infty \) , although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by \((n,\,m)\) . In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by \((n,\,m)\) , the only difference being a factor of 2 in the exponent. This appears to be a generalization of the Prodi–Serrin conditions for \(n\ge 1\) . PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9484-8

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Abstract: In this paper, we discuss the dynamics of a predator–prey model with Beddington–DeAngelis functional response and nonselective harvesting. By using the Lyapunov–Schmidt reduction, we obtain the existence of spatially nonhomogeneous steady-state solution. The stability and existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the change of a specific parameter are investigated by analyzing the distribution of the eigenvalues. We also get an algorithm for determining the bifurcation direction of the Hopf bifurcating periodic solutions near the nonhomogeneous steady-state solution. Finally, we show some numerical simulations to verify our analytical results. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9487-5

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Abstract: A vector-borne disease is caused by a range of pathogens and transmitted to hosts through vectors. To investigate the multiple effects of the spatial heterogeneity, the temperature sensitivity of extrinsic incubation period and intrinsic incubation period, and the seasonality on disease transmission, we propose a nonlocal reaction–diffusion model of vector-borne disease with periodic delays. We introduce the basic reproduction number \(\mathfrak {R}_0\) for this model and then establish a threshold-type result on its global dynamics in terms of \(\mathfrak {R}_0\) . In the case where all the coefficients are constants, we also prove the global attractivity of the positive constant steady state when \(\mathfrak {R}_0>1\) . Numerically, we study the malaria transmission in Maputo Province, Mozambique. PubDate: 2019-02-01 DOI: 10.1007/s00332-018-9475-9

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Authors:Chunxia Li; Shi-Hao Li Abstract: This paper mainly talks about the Cauchy two-matrix model and its corresponding integrable hierarchy with the help of orthogonal polynomial theory and Toda-type equations. Starting from the symmetric reduction in Cauchy biorthogonal polynomials, we derive the Toda equation of CKP type (or the C-Toda lattice) as well as its Lax pair by introducing time flows. Then, matrix integral solutions to the C-Toda lattice are extended to give solutions to the CKP hierarchy which reveals the time-dependent partition function of the Cauchy two-matrix model is nothing but the \(\tau \) -function of the CKP hierarchy. At last, the connection between the Cauchy two-matrix model and Bures ensemble is established from the point of view of integrable systems. PubDate: 2018-06-01 DOI: 10.1007/s00332-018-9474-x