Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

Global existence and nonexistence for diffusive polytropic filtration equations with nonlinear boundary conditions

In this paper, we deal with the global existence and nonexistence of solutions to a diffusive polytropic filtration system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions, which extend the recent results of Li et al. (Z Angew Math Phys 60:284–298, 2009) and Wang et al. (Nonlinear Anal 71:2134–2140, 2009) to more general equations and simplify their proofs slightly.     

Stability for the Timoshenko Beam System with Local Kelvin-Voigt Damping

In this paper, we consider a vibrating beam with one segment made of viscoelastic material of a Kelvin-Voigt （shorted as K-V） type and other parts made of elastic material by means of the Timoshenko model. We have deduced mathematical equations modelling its vibration and studied the stability of the semigroup associated with the equation system. We obtain the exponential stability under certain hypotheses of the smoothness and structural condition of the coefficients of the system, and obtain the strong asymptotic stability under weaker hypotheses of the coefficients.     

Author’s response to Dr. Wadee and Yiatros’ discussion of my paper: “On interactive buckling in a sandwich structure” (ZAMP, vol. 61, pp. 565–577, 2010)

In this note I examine a number of statement made in Wadee and Yiatros (Z. Angew. Math. Phys. 61:565–577 2010) in relation to my paper mentioned in the title. Most of the claims by Drs. Wadee and Yiatros are re-examined here in the proper context and it is shown that the results of the title paper are free of any errors.     

Gauging magnetorotational instability

Previously (Z. Angew. Math. Phys. 57:615–622, 2006), we examined the axisymmetric stability of viscous resistive magnetized Couette flow with emphasis on flows that would be hydrodynamically stable according to Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. A uniform axial magnetic field permeates the fluid. In this regime, magnetorotational instability (MRI) may occur. It was proved that MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions, when a term multipling the magnetic Prandtl number is neglected. Likewise, in the current work, including this term, when the magnetic resistivity is sufficiently large, MRI is suppressed. This shows conclusively that small magnetic dissipation is a feature of this instability for all magnetic Prandtl numbers. A criterion is provided for the onset of MRI.     

Discussion: “On interactive buckling in a sandwich structure” by C. D. Coman

A recent article by Coman (Z Angew Math Phys 2009) on the response of compression sandwich struts made some claims on the quality of the simplified version of the interactive buckling model presented in Hunt and Wadee (Proc R Soc A 454(1972):1197–1216, 1998). Some of these claims are examined in detail herein; it is concluded that great care must be exercised when performing parametric studies with equations that have been derived from simplifying a mechanical model. This is because the resulting system of equations does not necessarily describe the original mechanical system in full, since the key assumptions necessarily change.     

Regularity and uniqueness for the stationary large eddy simulation model

In the note we are concerned with higher regularity and uniqueness of solutions to the stationary problem arising from the large eddy simulation of turbulent ows. The system of equations contains a nonlocal nonlinear term, which prevents straightforward application of a difference quotients method. The existence of weak solutions was shown in A. Świerczewska: Large eddy simulation. Existence of stationary solutions to the dynamical model, ZAMM, Z. Angew. Math. Mech. 85 (2005), 593–604 and P. Gwiazda, A. Świerczewska: Large eddy simulation turbulence model with Young measures, Appl. Math. Lett. 18 (2005), 923–929.     

Symmetry and monotonicity of least energy solutions

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Mariş (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brézis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincaré Anal Non Linéaire 1:223–283, 1984).     

Comment on: “Fundamental flows with nonlinear slip conditions: exact solutions”, by R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888

In their article (Fundamental flows with nonlinear slip conditions: exact solutions, R. Ellahi, T. Hayat, F. M. Mahomed and A. Zeeshan, Z. Angew. Math. Phys. 61 (2010) 877–888.), the authors considered three simple cases of the steady flow of a third grade fluid between parallel plates with slip conditions; namely, Couette flow, Poiseuille flow, and generalized Couette flow. They obtained exact solutions, which were utilized in a way that did not lead to useful results. Their conclusion that the Couette flow cannot be obtained from the generalized Couette flow, by dropping the pressure gradient, is incorrect. Meaningful results based on their solution are herein presented.     

On Ulm’s method using divided differences of order one

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv Akad Nauk Est SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.     

The Boundary Effects and Zero Angular and Micro-rotational Viscosities Limits of the Micropolar Fluid Equations

In this paper, we consider an initial-value problem to the two-dimensional incompressible micropolar fluid equations. Our main purpose is to study the boundary layer effects as the angular and micro-rotational viscosities go to zero. It is also shown that the boundary layer thickness is of the order \(O(\gamma^{\beta })\) with \((0<\beta <\frac{2}{3})\). In contrast with Chen et al. (Z. Angew. Math. Phys. 65:687–710, 2014), the BL-thickness we got is thinner than that in Chen et al. (Z. Angew. Math. Phys. 65:687–710, 2014). In addition, the convergence rates are also improved.     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

Nonlinear stability for advective systems

We consider partial differential equations of “advective” type, in the sense defined in Shvydkoy (Comm Math Phys 265:507–545, 2006). We establish rigorous results that allow to infer nonlinear stability for small perturbations from linearized stability results.     

The linear flows in the space of Krichever–Lax matrices over an algebraic curve

Krichever (Commun Math Phys 229(2):229–269, 2002) invented the space of matrices parametrizing the cotangent bundle of moduli space of stable vector bundles over a compact Riemann surface, which is named as the Hitchin system after the investigation (Hitchin, Duke Math J 54(1):91–114, 1987). We study a necessary and sufficient condition for the linearity of flows on the space of Krichever–Lax matrices in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work by Griffiths (Am J Math 107(6):1445–1484, 1985).      

Lorentz Gas with Thermostatted Walls

In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity (Chernov and Dolgopyat in J Am Math Soc 22:821–858, 2009; Phys Rev Lett 99, paper 030601, 2007). To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle’s speed to be constant. Then an electric current sets in and one can prove Ohm’s law and the Einstein relation (Chernov and Dolgopyat in Russian Math Surv 64:73–124, 2009; Chernov et al. Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993). However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the ‘walls’). We rederive all physically interesting facts proven for the Gaussian thermostat in Chernov, Dolgopyat (Russian Math Surv 64:73–124, 2009) and Chernov et al. (Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993), including Ohm’s law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.     

Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves

Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow of closed curves in \mathbbR^d, d 3 2{\mathbb{R}^d, d\geq2} . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.     

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin–Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.     

Finite time blow-up for a reaction-diffusion system in bounded domain

This paper mainly considers the coupled parabolic system in a bounded domain: u _t = Δu + u ^α v ^p , v _t = Δv + u ^q v ^β in Ω × (0, T) with null Dirichlet boundary value condition which had been discussed by Wang in (Z Angew Math Phys 51:160–167, 2000). The aim of this paper is to solve the open problem mentioned in the Remark of Wang (Z Angew Math Phys 51:160–167, 2000).     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).