Authors:Biao Zeng; Zhenhai Liu; Stanisław Migórski Abstract: In this paper we investigate the convergence behavior of the solutions to the time-dependent variational–hemivariational inequalities with respect to the data. First, we give an existence and uniqueness result for the problem, and then, deliver a continuous dependence result when all the data are subjected to perturbations. A semipermeability problem is given to illustrate our main results. PubDate: 2018-06-04 DOI: 10.1007/s00033-018-0980-3 Issue No:Vol. 69, No. 3 (2018)

Authors:Luca Battaglia; Jean Van Schaftingen Abstract: We consider a nonlinear Choquard equation $$\begin{aligned} -\Delta u+u= (V * u ^p ) u ^{p-2}u \qquad \text {in }\mathbb {R}^N, \end{aligned}$$ when the self-interaction potential V is unbounded from below. Under some assumptions on \(V\) and on \(p\) , covering \(p =2\) and \(V\) being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution \(u\in H^1 (\mathbb {R}^N){\setminus }\{0\}\) by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation. PubDate: 2018-05-31 DOI: 10.1007/s00033-018-0975-0 Issue No:Vol. 69, No. 3 (2018)

Authors:Huaqiao Wang Abstract: This paper studies the two-dimensional stochastic magnetohydrodynamic equations which are used to describe the turbulent flows in magnetohydrodynamics. The exponential behavior and the exponential mean square stability of the weak solutions are proved by the application of energy method. Furthermore, we establish the pathwise exponential stability by using the exponential mean square stability. When the stochastic perturbations satisfy certain additional hypotheses, we can also obtain pathwise exponential stability results without using the mean square stability. PubDate: 2018-05-29 DOI: 10.1007/s00033-018-0978-x Issue No:Vol. 69, No. 3 (2018)

Authors:Ashkan Golgoon; Arash Yavari Abstract: In this paper, we present some analytical solutions for the stress fields of nonlinear anisotropic solids with distributed line and point defects. In particular, we determine the stress fields of (i) a parallel cylindrically symmetric distribution of screw dislocations in infinite orthotropic and monoclinic media, (ii) a cylindrically symmetric distribution of parallel wedge disclinations in an infinite orthotropic medium, (iii) a distribution of edge dislocations in an orthotropic medium, and (iv) a spherically symmetric distribution of point defects in a transversely isotropic spherical ball. PubDate: 2018-05-29 DOI: 10.1007/s00033-018-0973-2 Issue No:Vol. 69, No. 3 (2018)

Authors:J. Kaplunov; D. Prikazchikov; L. Sultanova Abstract: Two-parametric asymptotic analysis of the equilibrium of an elastic half-space coated by a thin soft layer is developed. The initial scaling is motivated by the exact solution of the plane problem for a vertical harmonic load. It is established that the Winkler–Fuss hypothesis is valid only for a sufficiently high contrast in the stiffnesses of the layer and the half-space. As an alternative, a uniformly valid non-local approximation is proposed. Higher-order corrections to the Winkler–Fuss formulation, such as the Pasternak model, are also studied. PubDate: 2018-05-28 DOI: 10.1007/s00033-018-0974-1 Issue No:Vol. 69, No. 3 (2018)

Authors:Marco A. Fontelos; Georgy Kitavtsev; Roman M. Taranets Abstract: For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account of surface tension at the free surface, we show exponential \((H^1,\,L^2)\) asymptotic decay to the flat profile of its solutions considered with general initial data. Additionally, by transforming the system to Lagrangian coordinates we show that the minimal thickness of the sheet stays positive for all times. This result proves the conjecture formally accepted in the physical literature (cf. Eggers and Fontelos in Singularities: formation, structure, and propagation. Cambridge Texts in Applied Mathematics, Cambridge, 2015), that a viscous sheet cannot rupture in finite time in the absence of external forcing. Moreover, in the absence of surface tension we find a special class of initial data for which the Lagrangian solution exhibits \(L^2\) -exponential decay to the flat profile. PubDate: 2018-05-28 DOI: 10.1007/s00033-018-0969-y Issue No:Vol. 69, No. 3 (2018)

Authors:Yang Li Abstract: In this paper, we consider the initial-boundary value problem to the one-dimensional compressible heat-conductive model for planar non-resistive magnetohydrodynamics. By making full use of the effective viscous flux and an analogue, together with the structure of the equations, global existence and uniqueness of strong solutions are obtained on condition that the initial density is bounded below away from vacuum and the heat conductivity coefficient \(\kappa \) satisfies the growth condition $$\begin{aligned} \kappa _1(1+\vartheta ^{\alpha })\le \kappa (\vartheta )\le \kappa _2(1+\vartheta ^{\alpha }),\quad \text { for some }0< \alpha < \infty , \end{aligned}$$ with \(\kappa _1,\kappa _2\) being positive constants. Moreover, global solvability of strong solutions is shown with the initial vacuum. The results are obtained without any smallness restriction to the initial data. PubDate: 2018-05-25 DOI: 10.1007/s00033-018-0970-5 Issue No:Vol. 69, No. 3 (2018)

Authors:Pakhapoom Sarapat; Duangkamon Baowan; James M. Hill Abstract: The interaction energy of a fullerene symmetrically situated inside a carbon nanotorus is studied. For these non-bonded molecules, the main interaction originates from the van der Waals energy which is modelled by the 6–12 Lennard-Jones potential. Upon utilising the continuum approximation which assumes that there are infinitely many atoms that are uniformly distributed over the surfaces of the molecules, the total interaction energy between the two structures is obtained as a surface integral over the spherical and the toroidal surfaces. This analytical energy is employed to determine the most stable configuration of the torus encapsulating the fullerene. The results show that a torus with major radius around 20–22 Å and minor radius greater than 6.31 Å gives rise to the most stable arrangement. This study will pave the way for future developments in biomolecules design and drug delivery system. PubDate: 2018-05-24 DOI: 10.1007/s00033-018-0972-3 Issue No:Vol. 69, No. 3 (2018)

Authors:A. Merzon; P. Zhevandrov; M. I. Romero Rodríguez; J. E. De la Paz Méndez Abstract: Exact solutions describing the Rayleigh–Bloch waves for the two-dimensional Helmholtz equation are constructed in the case when the refractive index is a sum of a constant and a small amplitude function which is periodic in one direction and of finite support in the other. These solutions are quasiperiodic along the structure and exponentially decay in the orthogonal direction. A simple formula for the dispersion relation of these waves is obtained. PubDate: 2018-05-23 DOI: 10.1007/s00033-018-0953-6 Issue No:Vol. 69, No. 3 (2018)

Authors:Gelson G. dos Santos; Giovany M. Figueiredo Abstract: In this paper, we study the existence of nonegative solutions to a class of nonlinear boundary value problems of the Kirchhoff type. We prove existence results when the problem has discontinuous nonlinearity and critical Caffarelli–Kohn–Nirenberg growth. PubDate: 2018-05-23 DOI: 10.1007/s00033-018-0966-1 Issue No:Vol. 69, No. 3 (2018)

Authors:Jianwei Yang Abstract: In this paper, we consider the quasi-neutral limit of a three-dimensional Euler–Poisson system of compressible fluids coupled to a magnetic field. We prove that, as Debye length tends to zero, periodic initial-value problems of the model have unique smooth solutions existing in the time interval where the ideal incompressible magnetohydrodynamic equations has smooth solution. Meanwhile, it is proved that smooth solutions converge to solutions of incompressible magnetohydrodynamic equations with a sharp convergence rate in the process of quasi-neutral limit. PubDate: 2018-05-23 DOI: 10.1007/s00033-018-0957-2 Issue No:Vol. 69, No. 3 (2018)

Authors:Marcio V. Ferreira; Jaime E. Muñoz Rivera; Fredy M. S. Suárez Abstract: In this article, we make a comparative analysis of the stabilizing effect of the frictional dissipation with the dissipation produced by viscous materials of Kelvin–Voigt type both located in a part of a Mindlin–Timoshenko plate. We model these dissipative mechanisms through transmission problems and show that localized frictional damping, when effective over a strategic component of the plate, produces exponential stability of the corresponding semigroup. On the other hand, although the dissipation of Kelvin–Voigt is considered a strong dissipation, we prove that it loses its uniform stabilizing properties when localized over a component of the material and provides only a slower polynomial decay. PubDate: 2018-05-23 DOI: 10.1007/s00033-018-0971-4 Issue No:Vol. 69, No. 3 (2018)

Authors:Christian Rohde; Christoph Zeiler Abstract: We consider a sharp interface approach for the inviscid isothermal dynamics of compressible two-phase flow that accounts for phase transition and surface tension effects. Kinetic relations are frequently used to fix the mass exchange and entropy dissipation rate across the interface. The complete unidirectional dynamics can then be understood by solving generalized two-phase Riemann problems. We present new well-posedness theorems for the Riemann problem and corresponding computable Riemann solvers that cover quite general equations of state, metastable input data and curvature effects. The new Riemann solver is used to validate different kinetic relations on physically relevant problems including a comparison with experimental data. Riemann solvers are building blocks for many numerical schemes that are used to track interfaces in two-phase flow. It is shown that the new Riemann solver enables reliable and efficient computations for physical situations that could not be treated before. PubDate: 2018-05-23 DOI: 10.1007/s00033-018-0958-1 Issue No:Vol. 69, No. 3 (2018)

Authors:Feimin Huang; Tianhong Li; Huimin Yu; Difan Yuan Abstract: We are concerned with the global existence and large time behavior of entropy solutions to the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler–Poisson equations in a bounded interval. In this paper, we first prove the global existence of entropy solution by vanishing viscosity and compensated compactness framework. In particular, the solutions are uniformly bounded with respect to space and time variables by introducing modified Riemann invariants and the theory of invariant region. Based on the uniform estimates of density, we further show that the entropy solution converges to the corresponding unique stationary solution exponentially in time. No any smallness condition is assumed on the initial data and doping profile. Moreover, the novelty in this paper is about the unform bound with respect to time for the weak solutions of the isentropic Euler–Poisson system. PubDate: 2018-05-22 DOI: 10.1007/s00033-018-0968-z Issue No:Vol. 69, No. 3 (2018)

Authors:Bair V. Budaev; David B. Bogy Abstract: We extend the statistical analysis of equilibrium systems to systems with a constant heat flux. This extension leads to natural generalizations of Maxwell–Boltzmann’s and Planck’s equilibrium energy distributions to energy distributions of systems with a net heat flux. This development provides a long needed foundation for addressing problems of nanoscale heat transport by a systematic method based on a few fundamental principles. As an example, we consider the computation of the radiative heat flux between narrowly spaced half-spaces maintained at different temperatures. PubDate: 2018-05-22 DOI: 10.1007/s00033-018-0950-9 Issue No:Vol. 69, No. 3 (2018)

Authors:Manuel Friedrich; Ulisse Stefanelli Abstract: Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling up often occurs when boundaries are not fixed. We address this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two- and three-body terms. As a consequence, we prove that all ground-state deformations are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes. PubDate: 2018-05-22 DOI: 10.1007/s00033-018-0965-2 Issue No:Vol. 69, No. 3 (2018)

Authors:Benoît Perthame; Weiran Sun; Min Tang Abstract: Kinetic-transport equations that take into account the intracellular pathways are now considered as the correct description of bacterial chemotaxis by run and tumble. Recent mathematical studies have shown their interest and their relations to more standard models. Macroscopic equations of Keller–Segel type have been derived using parabolic scaling. Due to the randomness of receptor methylation or intracellular chemical reactions, noise occurs in the signaling pathways and affects the tumbling rate. Then comes the question to understand the role of an internal noise on the behavior of the full population. In this paper we consider a kinetic model for chemotaxis which includes biochemical pathway with noises. We show that under proper scaling and conditions on the tumbling frequency as well as the form of noise, fractional diffusion can arise in the macroscopic limits of the kinetic equation. This gives a new mathematical theory about how long jumps can be due to the internal noise of the bacteria. PubDate: 2018-05-21 DOI: 10.1007/s00033-018-0964-3 Issue No:Vol. 69, No. 3 (2018)

Abstract: In this paper, we deal with the uniform stabilization to the mixed problem for a nonlinear wave equation and acoustic boundary conditions on a non-locally reacting boundary. The main purpose is to study the stability when the internal damping acts only over a subset \(\omega \) of the domain \(\Omega \) and the boundary damping is of the viscoelastic type. PubDate: 2018-05-30 DOI: 10.1007/s00033-018-0977-y

Abstract: We study the asymptotic behavior of the effective thermal conductivity of a periodic two-phase dilute composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material, each of them of size proportional to a positive parameter \(\epsilon \) . We assume a perfect thermal contact at constituent interfaces, i.e., a continuity of the normal component of the heat flux and of the temperature. For \(\epsilon \) small, we prove that the effective conductivity can be represented as a convergent power series in \(\epsilon \) and we determine the coefficients in terms of the solutions of explicit systems of integral equations. PubDate: 2018-05-30 DOI: 10.1007/s00033-018-0976-z

Abstract: The paper studies the physical-constraints-preserving (PCP) schemes for multi-dimensional special relativistic magnetohydrodynamics with a general equation of state (EOS) on more general meshes. It is an extension of the work (Wu and Tang in Math. Models Methods Appl. Sci. 27:1871–1928, 2017) which focuses on the ideal EOS and uniform Cartesian meshes. The general EOS without a special expression poses some additional difficulties in discussing the mathematical properties of admissible state set with the physical constraints on the fluid velocity, density and pressure. Rigorous analyses are provided for the PCP property of finite volume or discontinuous Galerkin schemes with the Lax–Friedrichs (LxF)-type flux on a general mesh with non-self-intersecting polytopes. Those are built on a more general form of generalized LxF splitting property and a different convex decomposition technique. It is shown in theory that the PCP property is closely connected with a discrete divergence-free condition, which is proposed on the general mesh and milder than that in Wu and Tang (2017). PubDate: 2018-05-30 DOI: 10.1007/s00033-018-0979-9