Blowup of solutions for a class of non‐linear evolution equations with non‐linear damping and source terms

We consider the blowup of solutions of the initial boundary value problem for a class of non‐linear evolution equations with non‐linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, α}, where m, α and p are non‐negative real numbers and m+1, α+1, p+1 are, respectively, the growth orders of the non‐linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above‐mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non‐linear terms, the states of the initial energy and the existence and non‐existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

A relaxed model and its homogenization for nematic liquid crystals in composite materials

We analyse a model for equilibrium configurations of composite systems of nematic liquid crystal with polymer inclusions, in the presence of an external magnetic field. We assume that the system has a periodic structure, and consider the relaxed problem on the unit length constraint of the nematic director field. The relaxation of the Oseen–Frank energy functional is carried out by including bulk as well as surface energy penalty terms, rendering the problem fully non‐linear. We employ two‐scale convergence methods to obtain effective configurations of the system, as the size of the polymeric inclusions tends to zero. We discuss the minimizers of the effective energies for, both, the constrained as well as the unconstrained models. Copyright © 2004 John Wiley & Sons, Ltd.     

Notes on the peripheral volume of hyperbolic 3‐manifolds

We consider orientable hyperbolic 3‐manifolds with either non‐empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion can only be maximal in some combinatorially extremal configurations. The techniques we employ are very elementary but the result is in our opinion of some interest.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

First and second moments for self‐similar couplings and Wasserstein distances

We study aspects of the Wasserstein distance in the context of self‐similar measures. Computing this distance between two measures involves minimising certain moment integrals over the space of couplings, which are measures on the product space with the original measures as prescribed marginals. We focus our attention on self‐similar measures associated to equicontractive iterated function systems consisting of two maps on the unit interval and satisfying the open set condition. We are particularly interested in understanding the restricted family of self‐similar couplings and our main achievement is the explicit computation of the 1st and 2nd moment integrals for such couplings. We show that this family is enough to yield an explicit formula for the 1st Wasserstein distance and provide non‐trivial upper and lower bounds for the 2nd Wasserstein distance for these self‐similar measures.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

Global non‐existence of solutions of a class of wave equations with non‐linear damping and source terms

In this paper we consider the non‐linear wave equation a,b>0, associated with initial and Dirichlet boundary conditions. We prove, under suitable conditions on α,β,m,p and for negative initial energy, a global non‐existence theorem. This improves a result by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833), who requires that the initial energy be sufficiently negative and relates the global non‐existence of solutions to the size of Ω. Copyright © 2004 John Wiley & Sons, Ltd.     

Single dopant diffusion in semiconductor technology

The paper deals with the analysis of pair diffusion models in semiconductor technology. The underlying model contains reaction‐drift‐diffusion equations for the mobile point defects and dopant‐defect pairs as well as reaction equations for immobile dopants which are coupled with a non‐linear Poisson equation for the chemical potential of the electrons. For homogeneous structures we present an existence and uniqueness result for strong solutions. Starting with energy estimates we derive further a priori estimates such that fixed point arguments due to Leray–Schauder guarantee the solvability of the model equations. Copyright © 2004 John Wiley & Sons, Ltd.     

A dynamical approach to large eddy simulation of turbulent flows: existence of weak solutions

We consider a system of equations coming from turbulence models using a large eddy simulation (LES) technique. The idea of this approach bases on decomposing the velocity into a part containing large flow structures and a part consisting of small scales. The equations for large‐scale quantities are derived from the Navier–Stokes equations with an additional constitutive relation for the contribution of small eddies. The mathematical difficulties in this paper focus on the non‐linear and non‐local turbulent term. Copyright © 2005 John Wiley & Sons, Ltd.     

I‐graphs and the corresponding configurations

We consider the class of I‐graphs I(n,j,k), which is a generalization over the class of the generalized Petersen graphs. We study different properties of I‐graphs, such as connectedness, girth, and whether they are bipartite or vertex‐transitive. We give an efficient test for isomorphism of I‐graphs and characterize the automorphism groups of I‐graphs. Regular bipartite graphs with girth at least 6 can be considered as Levi graphs of some symmetric combinatorial configurations. We consider configurations that arise from bipartite I‐graphs. Some of them can be realized in the plane as cyclic astral configurations, i.e., as geometric configurations with maximal isometric symmetry. © 2005 Wiley Periodicals, Inc.     

Unconditional non‐linear stability for a fluid of third grade

The thermal convection in a layer of a third grade fluid is investigated, with viscosity being a general function of temperature. We develop a non‐linear stability analysis and prove that unconditional non‐linear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a fluid of third grade are preferable to those for a fluid of second grade or a dipolar fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

Reverse task as a designing of mechatronic vibrating branched systems

In this paper a new approach for designing mechatronic vibrating branched structures has been presented. Mechatronic structures have been built from mechanical discrete systems connected to piezoelectric actuator and externalL_xR_xC_x network, with different configurations. Modeling simplification has been performed by use of non dimensional transformations and retransformations. In each case reverse task has been solved by distribution into partial fraction method in respect to required dynamic properties in form of frequency spectrum: resonant and anti resonant frequencies. Furthermore, different configurations of final L_xR_xC_x network have been presented. These considerations have been supported by calculation examples, and all results have been presented in the graphical form. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Data structures with dynamical random transitions

We present a (non‐standard) probabilistic analysis of dynamic data structures whose sizes are considered dynamic random walks. The basic operations (insertion, deletion, positive and negative queries, batched insertion, lazy deletion, etc.) are time‐dependent random variables. This model is a (small) step toward the analysis of these structures when the distribution of the set of histories is not uniform. As an illustration, we focus on list structures (linear lists, priority queues, and dictionaries) but the technique is applicable as well to more advanced data structures. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Viscous compressible barotropic symmetric flows with free boundary under general mass force Part I: Uniform‐in‐time bounds and stabilization

We consider symmetric flows of a viscous compressible barotropic fluid with a free boundary, under a general mass force depending both on the Eulerian and Lagrangian co‐ordinates, with arbitrarily large initial data. For a general non‐monotone state function p, we prove uniform‐in‐time energy bound and the uniform bounds for the density ρ, together with the stabilization as t → ∞ of the kinetic and potential energies. We also obtain H¹‐stabilization of the velocity v to zero provided that the second viscosity is zero. For either increasing or non‐decreasing p, we study the L^λ‐stabilization of ρ and the stabilization of the free boundary together with the corresponding ω‐limit set in the general case of non‐unique stationary solution possibly with zones of vacuum. In the case of increasing p and stationary densities ρ_S separated from zero, we establish the uniform‐in‐time H¹‐bounds and the uniform stabilization for ρ and v. All these results are stated and mainly proved in the Eulerian co‐ordinates. They are supplemented with the corresponding stabilization results in the Lagrangian co‐ordinates in the case of ρ_S separated from zero. Copyright © 2005 John Wiley & Sons, Ltd.     

Regularity and solvability of linear differential operators in Gevrey spaces

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.     

Linear and non‐linear stability thresholds for thermal convection in a box

We analyse the problem of finding instability thresholds and global non‐linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non‐uniform temperature field in steady state. This problem was previously analysed by Georgescu and Mansutti (Int. J. Non‐Linear Mech. 1999; 34 :603–613). In our work we determine the linear instability threshold to be well above the global stability boundary found by an energy method. Since the perturbed system is not symmetric we expect this to be the case, and our analysis yields a parameter region where possible sub‐critical instabilities may be found. Copyright © 2006 John Wiley & Sons, Ltd.     

Clifford algebras and Maxwell's equations in Lipschitz domains

We present a simple, Clifford algebra‐based approach to several key results in the theory of Maxwell's equations in non‐smooth subdomains of ℝ^m. Among other things, we give new proofs to the boundary energy estimates of Rellich type for Maxwell's equations in Lipschitz domains from [20, 10], discuss radiation conditions and the case of variable wave number. Copyright © 1999 John Wiley & Sons, Ltd.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Ferromagnets with biquadratic exchange coupling energy. Global existence of weak solutions

A global existence theorem is proved for the Landau–Lifshitz–Gilbert equations with biquadratic exchange coupling energy acting on the interfaces of a material composed by two ferromagnetic layers separated by a nonmagnetic one. This energy is not convex. The magnetization M satisfies on the interfaces a coupled non‐linear Neumann boundary condition with cubic growth. We use several regularizations, in particular for the traces of the magnetization at the interfaces, to obtain global weak solutions of the problem with finite energy. Copyright © 2005 John Wiley & Sons, Ltd.     

Asymptotic computation for transient heat conduction performance of periodic porous materials in curvilinear coordinates by the second‐order two‐scale method

A novel second‐order two‐scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non‐periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non‐periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.