Authors:Luca Placidi; Anil Misra; Emilio Barchiesi Abstract: In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed. PubDate: 2018-04-17 DOI: 10.1007/s00033-018-0947-4 Issue No:Vol. 69, No. 3 (2018)

Authors:Tomáš Roubíček; Giuseppe Tomassetti Abstract: A theory of elastic magnets is formulated under possible diffusion and heat flow governed by Fick’s and Fourier’s laws in the deformed (Eulerian) configuration, respectively. The concepts of nonlocal nonsimple materials and viscous Cahn–Hilliard equations are used. The formulation of the problem uses Lagrangian (reference) configuration while the transport processes are pulled back. Except the static problem, the demagnetizing energy is ignored and only local non-self-penetration is considered. The analysis as far as existence of weak solutions of the (thermo) dynamical problem is performed by a careful regularization and approximation by a Galerkin method, suggesting also a numerical strategy. Either ignoring or combining particular aspects, the model has numerous applications as ferro-to-paramagnetic transformation in elastic ferromagnets, diffusion of solvents in polymers possibly accompanied by magnetic effects (magnetic gels), or metal-hydride phase transformation in some intermetallics under diffusion of hydrogen accompanied possibly by magnetic effects (and in particular ferro-to-antiferromagnetic phase transformation), all in the full thermodynamical context under large strains. PubDate: 2018-04-16 DOI: 10.1007/s00033-018-0932-y Issue No:Vol. 69, No. 3 (2018)

Authors:Klemens Fellner; Bao Quoc Tang Abstract: The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied. The considered reaction–diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, i.e. equilibrium states lying on the boundary of \({\mathbb {R}}_+^N\) , then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite-dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite-dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria. PubDate: 2018-04-13 DOI: 10.1007/s00033-018-0948-3 Issue No:Vol. 69, No. 3 (2018)

Authors:N. P. Lazarev; T. S. Popova; G. A. Rogerson Abstract: A two-dimensional model describing the equilibrium state of a cracked inhomogeneous body with a rigid circular inclusion is investigated. The body is assumed to have a crack that reaches the boundary of the rigid inclusion. We assume that the Signorini condition, ensuring non-penetration of the crack faces, is satisfied. We analyze the dependence of solutions on the radius of rigid inclusion. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional, with the radius of inclusion chosen as the control parameter. PubDate: 2018-04-11 DOI: 10.1007/s00033-018-0949-2 Issue No:Vol. 69, No. 3 (2018)

Authors:A. Battista; A. Della Corte; F. dell’Isola; P. Seppecher Abstract: In the present paper we study a natural nonlinear generalization of Timoshenko beam model and show that it can describe the homogenized deformation energy of a 1D continuum with a simple microstructure. We prove the well posedness of the corresponding variational problem in the case of a generic end load, discuss some regularity issues and evaluate the critical load. Moreover, we generalize the model so as to include an additional rotational spring in the microstructure. Finally, some numerical simulations are presented and discussed. PubDate: 2018-04-06 DOI: 10.1007/s00033-018-0946-5 Issue No:Vol. 69, No. 3 (2018)

Authors:Zhong Tan; Leilei Tong Abstract: In this paper, we consider the large time behavior of the compressible Hall magnetohydrodynamic equations with Coulomb force in \(\mathbb {R}^3\) near the non-constant equilibrium state. We derive the global existence provided that the initial perturbation is sufficiently small. Moreover, under the further assumption that the doping profile is of small variation, we obtain the convergence rates by combining the linear \(L^p\) – \(L^q\) decay estimates. PubDate: 2018-04-05 DOI: 10.1007/s00033-018-0944-7 Issue No:Vol. 69, No. 3 (2018)

Authors:Xin Xu Abstract: In the present paper, we investigate the asymptotic behavior of solutions to an electromagnetic fluid system for viscous compressible flow without heat conduction in three spatial dimensions. The global existence and time-decay estimates of classical solution are established when the initial data are small perturbations of some given constant state. The proof is based on some elaborate energy estimates and the decay estimates for the linearized system. PubDate: 2018-04-04 DOI: 10.1007/s00033-018-0945-6 Issue No:Vol. 69, No. 2 (2018)

Authors:Daniel Q. Eckhardt; Isom H. Herron Abstract: Magnetorotational instability (MRI) is an instability that is responsible for accretion, the phenomenon observed in astrophysical disks, for example around black holes. MRI can be modeled using the equations of MHD. A typical linear analysis in the past made use of the small Prandtl number approximation, which results in dropping one term from these equations. Previously, one of the authors (Herron) showed that the small magnetic Prandtl number approximation suppresses MRI in axisymmetric viscous resistive magnetized Taylor–Couette flow when one has no-slip velocity boundary conditions, and insulating magnetic boundary conditions. We follow up here with a proof that MRI is still suppressed with perfectly conducting magnetic boundary conditions on the cylinders. PubDate: 2018-04-02 DOI: 10.1007/s00033-018-0943-8 Issue No:Vol. 69, No. 2 (2018)

Authors:Peng-Fei Hou; Yang Zhang Abstract: This paper presents a refined approach of the electro-elastic fields through the 2D Green’s functions under a tangential line load. The structure of piezoelectric devices is composed of a piezoelectric substrate and an elastic coating. When arbitrary distributed load is applied, the components can be obtained by superposition principle. This method has high stability, efficiency and computational precision, compared with finite element method. And the conclusions provide meaningful value for the design of layered structure in engineering. PubDate: 2018-03-30 DOI: 10.1007/s00033-018-0941-x Issue No:Vol. 69, No. 2 (2018)

Authors:Rinaldo M. Colombo; Graziano Guerra Abstract: Consider two hyperbolic systems of conservation laws in one space dimension with the same eigenvalues and (right) eigenvectors. We prove that solutions to Cauchy problems with the same initial data differ at third order in the total variation of the initial datum. As a first application, relying on the classical Glimm–Lax result (Glimm and Lax in Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, No. 101. American Mathematical Society, Providence, 1970), we obtain estimates improving those in Saint-Raymond (Arch Ration Mech Anal 155(3):171–199, 2000) on the distance between solutions to the isentropic and non-isentropic inviscid compressible Euler equations, under general equations of state. Further applications are to the general scalar case, where rather precise estimates are obtained, to an approximation by Di Perna of the p-system and to a traffic model. PubDate: 2018-03-27 DOI: 10.1007/s00033-018-0942-9 Issue No:Vol. 69, No. 2 (2018)

Authors:Sun-Hye Park Abstract: In this paper, we consider suspension bridge equations with time delay of the form $$\begin{aligned} u_{tt}(x,t) + \Delta ^2 u (x,t) + k u^+ (x,t) + a_0 u_t (x,t) + a_1 u_t (x, t- \tau ) + f(u(x,t)) = g(x). \end{aligned}$$ Many researchers have studied well-posedness, decay rates of energy, and existence of attractors for suspension bridge equations without delay effects. But, as far as we know, there is no work about suspension equations with time delay. In addition, there are not many studies on attractors for other delayed systems. Thus we first provide well-posedness for suspension equations with time delay. And then show the existence of global attractors and the finite dimensionality of the attractors by establishing energy functionals which are related to the norm of the phase space to our problem. PubDate: 2018-03-24 DOI: 10.1007/s00033-018-0934-9 Issue No:Vol. 69, No. 2 (2018)

Authors:Paolo Antonelli; Alessandro Michelangeli; Raffaele Scandone Abstract: We prove the existence of weak solutions in the space of energy for a class of nonlinear Schrödinger equations in the presence of a external, rough, time-dependent magnetic potential. Under our assumptions, it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularisation, and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem. PubDate: 2018-03-24 DOI: 10.1007/s00033-018-0938-5 Issue No:Vol. 69, No. 2 (2018)

Authors:E. I. Saad; M. S. Faltas Abstract: An expression for electrophoretic apparent velocity slip in the time-dependent flow of an electrolyte solution saturated in a charged porous medium within an electric double layer adjacent to a dielectric plate under the influence of a tangential uniform electric field is derived. The velocity slip is used as a boundary condition to solve the electrophoretic motion of an impermeable dielectric spherical particle embedded in an electrolyte solution saturated in porous medium under the unsteady Darcy–Brinkman model. Throughout the system, a uniform electric field is applied and maintains with constant strength. Two cases are considered, when the electric double layer enclosing the particle is thin, but finite and when of a particle with a thick double layer. Expressions for the electrophoretic mobility of the particle as functions of the relevant parameters are found. Our results indicate that the time scale for the growth of mobility is significant and small for high permeability. Generally, the effect of the relaxation time for starting electrophoresis is negligible, irrespective of the thickness of the double layer and permeability of the medium. The effects of the elapsed time, permeability, mass density and Debye length parameters on the fluid velocity, the electrophoretic mobility and the acceleration are shown graphically. PubDate: 2018-03-21 DOI: 10.1007/s00033-018-0939-4 Issue No:Vol. 69, No. 2 (2018)

Authors:Zhiqiang Shao Abstract: The relativistic full Euler system with generalized Chaplygin proper energy density–pressure relation is studied. The Riemann problem is solved constructively. The delta shock wave arises in the Riemann solutions, provided that the initial data satisfy some certain conditions, although the system is strictly hyperbolic and the first and third characteristic fields are genuinely nonlinear, while the second one is linearly degenerate. There are five kinds of Riemann solutions, in which four only consist of a shock wave and a centered rarefaction wave or two shock waves or two centered rarefaction waves, and a contact discontinuity between the constant states (precisely speaking, the solutions consist in general of three waves), and the other involves delta shocks on which both the rest mass density and the proper energy density simultaneously contain the Dirac delta function. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine–Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the generalized Rankine–Hugoniot relation and entropy condition, we establish the existence and uniqueness of solutions involving delta shocks for the Riemann problem. PubDate: 2018-03-21 DOI: 10.1007/s00033-018-0937-6 Issue No:Vol. 69, No. 2 (2018)

Authors:Jian Zhang; Zhenluo Lou; Yanju Ji; Wei Shao Abstract: In this paper, we study multiplicity of solutions of the critical bi-harmonic equation $$\begin{aligned} \varepsilon ^4 \Delta ^2 u +V(x) u =h(x) f(u)+g(x) u^{2_*-1}\ \ \mathrm{in } \ \ \mathbb R^N, \end{aligned}$$ where \(2_*=\frac{2N}{N-4}\) is the critical exponent. When \(\varepsilon >0\) is small, we establish the relationship between the number of solutions and the profile of V, h, g. Also, without the restriction on \(\varepsilon \) , we obtain a multiplicity result. PubDate: 2018-03-19 DOI: 10.1007/s00033-018-0940-y Issue No:Vol. 69, No. 2 (2018)

Authors:Chiun-Chang Lee; Rolf J. Ryham Abstract: This article addresses the boundary asymptotics of the electrostatic potential in non-neutral electrochemistry models with small Debye length in bounded domains. Under standard physical assumptions motivated by non-electroneutral phenomena in oxidation–reduction reactions, we show that the electrostatic potential asymptotically blows up at boundary points with respect to the bulk reference potential as the scaled Debye length tends to zero. The analysis gives a lower bound for the blow-up rate with respect to the model parameters. Moreover, the maximum potential difference over any compact subset of the physical domain vanishes exponentially in the zero-Debye-length limit. The results mathematically confirm the physical description that electrolyte solutions are electrically neutral in the bulk and are strongly electrically non-neutral near charged surfaces. PubDate: 2018-03-16 DOI: 10.1007/s00033-018-0931-z Issue No:Vol. 69, No. 2 (2018)

Authors:Michael Winkler Abstract: We consider radially symmetric solutions of the Keller–Segel system with generalized logistic source given by $$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + \lambda u - \mu u^\kappa , \\ 0 = \Delta v - v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ under homogeneous Neumann boundary conditions in the ball \(\Omega =B_R(0) \subset \mathbb {R}^n\) for \(n\ge 3\) and \(R>0\) , where \(\lambda \in \mathbb {R}, \mu >0\) and \(\kappa >1\) . Under the assumption that $$\begin{aligned} \kappa < \left\{ \begin{array}{ll} \frac{7}{6} &{}\quad \text {if } n\in \{3,4\}, \\ 1+ \frac{1}{2(n-1)} &{}\quad \text {if } n \ge 5, \end{array} \right. \end{aligned}$$ a condition on the initial data is derived which is seen to be sufficient to ensure the occurrence of finite-time blow-up for the corresponding solution of ( \(\star \) ). Moreover, this criterion is shown to be mild enough so as to allow for the conclusion that in fact any positive continuous radial function on \(\overline{\Omega }\) is the limit in \(L^1(\Omega )\) of a sequence \((u_{0k})_{k\in \mathbb {N}}\) of continuous radial initial data which are such that for each \(k\in \mathbb {N}\) the associated initial-boundary value problem for ( \(\star \) ) exhibits a finite-time explosion phenomenon in the above sense. In particular, this apparently provides the first rigorous detection of blow-up in a superlinearly dampened but otherwise essentially original Keller–Segel system in the physically relevant three-dimensional case. PubDate: 2018-03-07 DOI: 10.1007/s00033-018-0935-8 Issue No:Vol. 69, No. 2 (2018)

Authors:Panxiao Li; Shi-Liang Wu Abstract: This paper is concerned with a time-periodic and delayed nonlocal reaction–diffusion population model with monostable nonlinearity. Under quasi-monotone or non-quasi-monotone assumptions, it is known that there exists a critical wave speed \(c_*>0\) such that a periodic traveling wave exists if and only if the wave speed is above \(c_*\) . In this paper, we first prove the uniqueness of non-critical periodic traveling waves regardless of whether the model is quasi-monotone or not. Further, in the quasi-monotone case, we establish the exponential stability of non-critical periodic traveling fronts. Finally, we illustrate the main results by discussing two types of death and birth functions arising from population biology. PubDate: 2018-03-05 DOI: 10.1007/s00033-018-0936-7 Issue No:Vol. 69, No. 2 (2018)

Authors:Julieta Bollati; Domingo A. Tarzia Abstract: Recently, in Tarzia (Thermal Sci 21A:1–11, 2017) for the classical two-phase Lamé–Clapeyron–Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction was obtained. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in Zhou et al. (J Eng Math 2017. https://doi.org/10.1007/s10665-017-9921-y). PubDate: 2018-03-02 DOI: 10.1007/s00033-018-0923-z Issue No:Vol. 69, No. 2 (2018)

Authors:Jiten C. Kalita; Sougata Biswas; Swapnendu Panda Abstract: Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by diagnosing the assumptions of the existing theorems on such vortices. We provide our own set of proofs for establishing the finiteness of the sequence of corner vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a nonzero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. Making use of some elementary concepts of mathematical analysis and our own construction of diametric disks, we conclude that the sequence of corner vortices is finite. PubDate: 2018-03-02 DOI: 10.1007/s00033-018-0933-x Issue No:Vol. 69, No. 2 (2018)