Evolution Equations (evolution + equation)

Distribution by Scientific Domains


Selected Abstracts


Modeling fluid saturated porous media under frost attack

GAMM - MITTEILUNGEN, Issue 1 2010
Tim Ricken
Abstract Freezing and thawing are important processes in civil engineering. On the one hand frost damage of porous building materials like road pavements and concrete in regions with periodical freezing is well known. On the other hand, artificial freezing techniques are widely used, e.g. for tunneling in non-cohesive soils and other underground constructions as well as for the protection of excavation and compartmentalization of contaminated tracts. Ice formation in porous media results from a coupled heat and mass transport and is accompanied by the ice expansion. The volume increase in space and time is assigned to the moving freezing front inside the porous solid. In this paper, a macroscopic ternary model is presented within the framework of the Theory of Porous Media (TPM) in view of the description of phase transition. For the mass exchange between ice and water an evolution equation based on the local balance of the heat flux vector is used. Examples illustrate the application of the model for saturated porous solids under thermal loading (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Shearing flows of a dry granular material,hypoplastic constitutive theory and numerical simulations

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 14 2006
Chung Fang
Abstract In the present study, the Goodman,Cowin theory is extended to incorporate plastic features to construct an elasto-visco-plastic constitutive model for flowing dry granular materials. A thermodynamic analysis, based on the Müller,Liu entropy principle, is performed to derive the equilibrium expressions of the constitutive variables. Non-equilibrium responses are proposed by use of a quasi-linear theory, in particular a hypoplastic-type relation is introduced to model the internal friction and plastic effects. It is illustrated that the Goodman,Cowin theory can appropriately be extended to include frictional effects into the evolution equation of the volume fraction (i.e. the so-called balance of equilibrated force) and the equilibrium expression of the Cauchy stress tensor. The implemented model is applied to investigate conventional steady isothermal granular flows with incompressible grains, namely simple plane shear, inclined gravity-driven and vertical channel-flows, respectively. Numerical results show that the hypoplastic effect plays a significant role in the behaviour of a flowing granular material. The obtained profiles of the velocity and the volume fraction with hypoplastic features are usually sharper and the shear-thinning effect is more significant than that without such plastic effects. This points at the possible wide applicability of the present model in the fields of granular materials and soil mechanics. In addition, the present paper also provides a framework for a possible extension of the hypoplastic theories which can be further undertaken. Copyright © 2006 John Wiley & Sons, Ltd. [source]


A simple robust numerical integration algorithm for a power-law visco-plastic model under both high and low rate-sensitivity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2004
E. A. de Souza Neto
Abstract This note describes a simple and extremely robust algorithm for numerical integration of the power-law-type elasto-viscoplastic constitutive model discussed by Peri, (Int. J. Num. Meth. Eng. 1993; 36: 1365,1393). As the rate-independent limit is approached with increasing exponents, the evolution equations of power-law-type models are known to become stiff. Under such conditions, the solution of the implicitly discretized viscoplastic evolution equation cannot be easily obtained by standard root-finding algorithms. Here, a procedure which proves to be remarkably robust under stiff conditions is obtained by means of a simple logarithmic mapping of the basic backward Euler time-discrete equation for the incremental plastic multiplier. The logarithm-transformed equation is solved by the standard Newton,Raphson scheme combined with a simple bisection procedure which ensures that the iterative guesses for the equation unknown (the incremental equivalent plastic strain) remain within the domain where the transformed equation makes sense. The resulting implementation can handle small and large (up to order 106) power-law exponents equally. This allows its effective use under any situation of practical interest, ranging from high rate-sensitivity to virtually rate-independent conditions. The robustness of the proposed scheme is demonstrated by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd. [source]


Energy,momentum consistent finite element discretization of dynamic finite viscoelasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
M. Groß
Abstract This paper is concerned with energy,momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy,momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three-field formulation, in which the deformation field, the velocity field and a strain-like viscous internal variable field are treated as independent quantities. The new non-linear viscous evolution equation satisfies a non-negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy,momentum consistency is based on the collocation property and an enhanced second Piola,Kirchhoff stress tensor. The obtained coupled non-linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time-integration algorithm in comparison with the ordinary midpoint rule. Both the quasi-rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd. [source]


Instabilities during batch sedimentation in geometries containing obstacles: A numerical and experimental study,

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2007
Rekha R. Rao
Abstract Batch sedimentation of non-colloidal particle suspensions is studied with nuclear magnetic resonance flow visualization and continuum-level numerical modelling of particle migration. The experimental method gives particle volume fraction as a function of time and position, which then provides validation data for the numerical model. A finite element method is used to discretize the equations of motion, including an evolution equation for the particle volume fraction and a generalized Newtonian viscosity dependent on local particle concentration. The diffusive-flux equation is based on the Phillips model (Phys. Fluids A 1992; 4:30,40) and includes sedimentation terms described by Zhang and Acrivos (Int. J. Multiphase Flow 1994; 20:579,591). The model and experiments are utilized in three distinct geometries with particles that are heavier and lighter than the suspending fluid, depending on the experiment: (1) sedimentation in a cylinder with a contraction; (2) particle flotation in a horizontal cylinder with a horizontal rod; and (3) flotation around a rectangular inclusion. Secondary flows appear in both the experiments and the simulations when a region of higher density fluid is above a lower density fluid. The secondary flows result in particle inhomogeneities, Rayleigh,Taylor-like instabilities, and remixing, though the effect in the simulations is more pronounced than in the experiments. Published in 2007 by John Wiley & Sons, Ltd. [source]


High-order filtering for control volume flow simulation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2001
G. De Stefano
Abstract A general methodology is presented in order to obtain a hierarchy of high-order filter functions, starting from the standard top-hat filter, naturally linked to control volumes flow simulations. The goal is to have a new filtered variable better represented in its high resolved wavenumber components by using a suitable deconvolution. The proposed formulation is applied to the integral momentum equation, that is the evolution equation for the top-hat filtered variable, by performing a spatial reconstruction based on the approximate inversion of the averaging operator. A theoretical analysis for the Burgers' model equation is presented, demonstrating that the local de-averaging is an effective tool to obtain a higher-order accuracy. It is also shown that the subgrid-scale term, to be modeled in the deconvolved balance equation, has a smaller absolute importance in the resolved wavenumber range for increasing deconvolution order. A numerical analysis of the procedure is presented, based on high-order upwind and central fluxes reconstruction, leading to congruent control volume schemes. Finally, the features of the present high-order conservative formulation are tested in the numerical simulation of a sample turbulent flow: the flow behind a backward-facing step. Copyright © 2001 John Wiley & Sons, Ltd. [source]


Generation of Gevrey class semigroup by non-selfadjoint Euler,Bernoulli beam model

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2006
Marianna A. Shubov
Abstract Asymptotic and spectral properties of a non-selfadjoint operator that is a dynamics generator for the Euler,Bernoulli beam model of a finite length are studied in this paper. The hyperbolic equation, which governs the vibrations of the Euler,Bernoulli beam model, is supplied with a one-parameter family of physically meaningful boundary conditions containing damping terms. The initial boundary-value problem is equivalent to the evolution equation that generates a strongly continuous semigroup in the state space of the system. It is found that the semigroup, being non-analytic, belongs to Gevrey class semigroups. This means that the differentiability of such semigroup is slightly weaker than that of an analytic semigroup. In the forthcoming works, the results of the present paper will be applied (a) to the solution of the exact controllability problem for Euler,Bernoulli beam and (b) to spectral analysis of a planar network of serially connected Euler,Bernoulli beams modelling ,flying wing configurations' in aeronautic engineering. Copyright © 2006 John Wiley & Sons, Ltd. [source]


Diffusion in poro-plastic media

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2004
R. E. Showalter
Abstract A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial-boundary-value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto-visco-plastic type. The variational form of this problem in Hilbert space is a non-linear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi-static momentum equation. The essential sufficient conditions for the well-posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid. Copyright © 2004 John Wiley & Sons, Ltd. [source]


Low-gain adaptive stabilization of semilinear second-order hyperbolic systems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2004
Toshihiro Kobayashi
Abstract In this paper low-gain adaptive stabilization of undamped semilinear second-order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low-gain adaptive velocity feedback. The closed-loop system is governed by a non-linear evolution equation. First, the well-posedness of the closed-loop system is shown. Next, an energy-like function and a multiplier function are introduced and the exponential stability of the closed-loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd. [source]


Adaptive grid based on geometric conservation law level set method for time dependent PDE

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2009
Ali R. Soheili
Abstract A new method for mesh generation is formulated based on the level set functions, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values (Liao et al. J Comput Phys 159 (2000), 103,122; Osher and Sethian J Comput Phys 79 (1988), 12; Sethian, Level set methods and fast marching methods, Cambridge University Press, 1999; Di et al. J Sci Comput 31 (2007), 75,98). The intersection of the level contours of the evolving functions form a new grid at each time. The velocity vector in the evolution equation is chosen according to the Geometric Conservation Law (GCL) method (Cao et al., SIAM J Sci Comput 24 (2002), 118,142.). This method has precise control over the Jacobian of transformation because of using the GCL method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]


From the Rainbow to the Structure of Atoms

PARTICLE & PARTICLE SYSTEMS CHARACTERIZATION, Issue 6 2007
Gérard Gouesbet
Abstract The optical rainbow is nowadays of much use, in the laboratory and in industry, in the field of optical characterization. However, the geometrical optics interpretation of the optical rainbow generates a caustic singularity which is a clue that something is wrong, and that a more fundamental theory (in practice, wave optics, or Maxwell's equations) is required to explain the optical rainbow and to accurately enough interpret experiments. There also exists a mechanical rainbow in classical mechanics, also leading to a singularity, implying that classical mechanics is wrong too. To smooth out this singularity, we have to turn to a wave mechanics. By lifting the Hamilton-Jacobi formulation of classical mechanics to a wave mechanics and looking to a time evolution equation, we may reach Schrödinger's equation. We therefore establish a beautiful connection between the rainbow in the sky, and the structure of atoms, in the sky and on earth. [source]


Creep of Single Crystals , Modelling and Numerical Aspects

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
Ivaylo Vladimirov
A number of constitutive models, utilizing both microstructural and/or phenomenological considerations, have been developed for the simulation of the creep behaviour of nickel-base single crystal superalloys at elevated temperatures. In this work, emphasis is placed on the rate-dependent single crystal plasticity model [1]. A strategy for the identification of the material parameters of the model to fit the results from experiments has been implemented. The parameter fitting methodology rests upon a two-membered evolution strategy. In addition, a proposal is made for the extension of the Cailletaud model [1] by means of an evolution equation for a damage variable which enables the modelling of the tertiary creep stage. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Local and non-local ductile damage and failure modelling at large deformation with applications to engineering

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Bob Svendsen Prof. Dr.
The numerical analysis of ductile damage and failure in engineering materials is often based on the micromechanical model of Gurson [1]. Numerical studies in the context of the finite-element method demonstrate that, as with other such types of local damage models, the numerical simulation of the initiation and propagation of damage zones is strongly mesh-dependent and thus unreliable. The numerical problems concern the global load-displacement response as well as the onset, size and orientation of damage zones. From a mathematical point of view, this problem is caused by the loss of ellipticity of the set of partial di.erential equations determining the (rate of) deformation field. One possible way to overcome these problems with and shortcomings of the local modelling is the application of so-called non-local damage models. In particular, these are based on the introduction of a gradient type evolution equation of the damage variable regarding the spatial distribution of damage. In this work, we investigate the (material) stability behaviour of local Gurson-based damage modelling and a gradient-extension of this modelling at large deformation in order to be able to model the width and other physical aspects of the localization of the damage and failure process in metallic materials. [source]


Ecological processes influencing mortality of juvenile pink salmon (Oncorhynchus gorbuscha) in Prince William Sound, Alaska

FISHERIES OCEANOGRAPHY, Issue 2001
T. M. Willette
Abstract Our collaborative work focused on understanding the system of mechanisms influencing the mortality of juvenile pink salmon (Oncorhynchus gorbuscha) in Prince William Sound, Alaska. Coordinated field studies, data analysis and numerical modelling projects were used to identify and explain the mechanisms and their roles in juvenile mortality. In particular, project studies addressed the identification of major fish and bird predators consuming juvenile salmon and the evaluation of three hypotheses linking these losses to (i) alternative prey for predators (prey-switching hypothesis); (ii) salmon foraging behaviour (refuge-dispersion hypothesis); and (iii) salmon size and growth (size-refuge hypothesis). Two facultative planktivorous fishes, Pacific herring (Clupea pallasi) and walleye pollock (Theragra chalcogramma), probably consumed the most juvenile pink salmon each year, although other gadids were also important. Our prey-switching hypothesis was supported by data indicating that herring and pollock switched to alternative nekton prey, including juvenile salmon, when the biomass of large copepods declined below about 0.2 g m,3. Model simulations were consistent with these findings, but simulations suggested that a June pteropod bloom also sheltered juvenile salmon from predation. Our refuge-dispersion hypothesis was supported by data indicating a five-fold increase in predation losses of juvenile salmon when salmon dispersed from nearshore habitats as the biomass of large copepods declined. Our size-refuge hypothesis was supported by data indicating that size- and growth-dependent vulnerabilities of salmon to predators were a function of predator and prey sizes and the timing of predation events. Our model simulations offered support for the efficacy of representing ecological processes affecting juvenile fishes as systems of coupled evolution equations representing both spatial distribution and physiological status. Simulations wherein model dimensionality was limited through construction of composite trophic groups reproduced the dominant patterns in salmon survival data. In our study, these composite trophic groups were six key zooplankton taxonomic groups, two categories of adult pelagic fishes, and from six to 12 groups for tagged hatchery-reared juvenile salmon. Model simulations also suggested the importance of salmon density and predator size as important factors modifying the predation process. [source]


Analysis of a microcrack model and constitutive equations for time-dependent dilatancy of rocks

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2003
Zuan Chen
SUMMARY Based on experimental observations and theoretical analyses, the author introduces an ideal microcrack model in which an array of cracks with the same shape and initial size is distributed evenly in rocks. The mechanism of creep dilatancy for rocks is analysed theoretically. Initiation, propagation and linkage of pre-existing microcracks during creep are well described. Also, the relationship between the velocity of microcrack growth and the duration of the creep process is derived numerically. The relationship agrees well with the character of typical experimental creep curves, and includes three stages of creep. Then the damage constitutive equations and damage evolution equations, which describe the dilatant behaviour of rocks, are presented. Because the dilatant estimated value is taken as the damage variable, the relationship between the microscopic model and the macroscopic constitutive equations is established. In this way the mechanical behaviour of rocks can be predicted. [source]


Integration and calibration of a plasticity model for granular materials

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2002
L. Jacobsson
Abstract A new macroscopic constitutive model for non-cohesive granular materials, with the focus on coarse-sized materials (railway ballast), is presented. The model is based on the concepts of rate-independent isotropic plasticity. The Backward Euler rule is used for integrating the pertinent evolution equations. The resulting incremental relations are solved in the strain space that is extended with the internal (hardening) variables. The model is calibrated using data from Conventional Triaxial Compression (CTC) tests, carried out at the University of Colorado at Boulder. A function evaluation method is used for the optimization, whereby a ,multi-vector' strategy for choosing the appropriate start vector is proposed. Copyright © 2002 John Wiley & Sons, Ltd. [source]


A simple robust numerical integration algorithm for a power-law visco-plastic model under both high and low rate-sensitivity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2004
E. A. de Souza Neto
Abstract This note describes a simple and extremely robust algorithm for numerical integration of the power-law-type elasto-viscoplastic constitutive model discussed by Peri, (Int. J. Num. Meth. Eng. 1993; 36: 1365,1393). As the rate-independent limit is approached with increasing exponents, the evolution equations of power-law-type models are known to become stiff. Under such conditions, the solution of the implicitly discretized viscoplastic evolution equation cannot be easily obtained by standard root-finding algorithms. Here, a procedure which proves to be remarkably robust under stiff conditions is obtained by means of a simple logarithmic mapping of the basic backward Euler time-discrete equation for the incremental plastic multiplier. The logarithm-transformed equation is solved by the standard Newton,Raphson scheme combined with a simple bisection procedure which ensures that the iterative guesses for the equation unknown (the incremental equivalent plastic strain) remain within the domain where the transformed equation makes sense. The resulting implementation can handle small and large (up to order 106) power-law exponents equally. This allows its effective use under any situation of practical interest, ranging from high rate-sensitivity to virtually rate-independent conditions. The robustness of the proposed scheme is demonstrated by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd. [source]


Combining interface damage and friction in a cohesive-zone model

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2006
Giulio Alfano
Abstract A new method to combine interface damage and friction in a cohesive-zone model is proposed. Starting from the mesomechanical assumption, typically made in a damage-mechanics approach, whereby a representative elementary area of the interface can be additively decomposed into an undamaged and a fully damaged part, the main idea consists of assuming that friction occurs only on the fully damaged part. The gradual increase of the friction effect is then a natural outcome of the gradual increase of the interface damage from the initial undamaged state to the complete decohesion. Suitable kinematic and static hypotheses are made in order to develop the interface model whereas no special assumptions are required on the damage evolution equations and on the friction law. Here, the Crisfield's interface model is used for the damage evolution and a simple Coulomb friction relationship is adopted. Numerical and analytical results for two types of constitutive problem show the effectiveness of the model to capture all the main features of the combined effect of interface damage and friction. A finite-step interface law has then been derived and implemented in a finite-element code via interface elements. The results of the simulations made for a fibre push-out test and a masonry wall loaded in compression and shear are then presented and compared with available experimental results. They show the effectiveness of the proposed model to predict the failure mechanisms and the overall structural response for the analysed problems. Copyright © 2006 John Wiley & Sons, Ltd. [source]


An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2007
Christopher P. Kent
Abstract An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave,body interaction problem into body and free-surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free-surface problem satisfies modified nonlinear free-surface boundary conditions. It is then shown that the nonlinear free-surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free-surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo-spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd. [source]


Analysis of a class of potential Korteweg-de Vries-like equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2010
R. M. Edelstein
Abstract We analyze a class of third-order evolution equations, i.e. ut = f(x, ux, uxx) uxxx+g(x, ux, uxx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd. [source]


Almost-periodic solutions to quasilinear evolution equations with a nonlocal nonlinearity

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2006
Albert Milani
Abstract We prove an existence and uniqueness result for almost-periodic solutions to the quasilinear evolution equations (1) and (5). Copyright © 2006 John Wiley & Sons, Ltd. [source]


Exact solutions of space,time dependent non-linear Schrödinger equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2004
Hang-yu Ruan
Abstract Using a general symmetry approach we establish transformations between different non-linear space,time dependent evolution equations of Schrödinger type and their respective solutions. As a special case we study the transformation of the standard non-linear Schrödinger equation (NLS)-equation to a NLS-equation with a dispersion coefficient which decreases exponentially with increasing distance along the fiber. By this transformation we construct from well known solutions of the standard NLS-equation some new exact solutions of the NLS-equation with dispersion. Copyright 2004 John Wiley & Sons, Ltd. [source]


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

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2002
Yang Zhijian
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, ,}, where m, , and p are non-negative real numbers and m+1, ,+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. [source]


Delayed quasilinear evolution equations with application to heat flow

MATHEMATISCHE NACHRICHTEN, Issue 5 2010
BártaArticle first published online: 15 MAR 2010
Abstract In this paper we show local and global existence for a class of (hyperbolic) quasilinear equations perturbed by bounded delay operators. In the last section, the abstract results are applied to a heat conduction model (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Polynomial stability of operator semigroups

MATHEMATISCHE NACHRICHTEN, Issue 13-14 2006
András Bátkai
Abstract We investigate polynomial decay of classical solutions of linear evolution equations. For bounded strongly continuous semigroups on a Banach space this property is closely related to polynomial growth estimates of the resolvent of the generator. For systems of commuting normal operators polynomial decay is characterized in terms of the location of the generator spectrum. The results are applied to systems of coupled wave-type equations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


On the spectrum of a weak class of operator pencils of waveguide type

MATHEMATISCHE NACHRICHTEN, Issue 8 2006
M. Hasanov
Abstract The paper introduces a new class of two parameter non-overdamped operator pencils arising from evolution equations. We investigate spectral properties, including variational principles for "interior" points of the spectrum. Examples leading to pencils of the new class are given. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Kinetics of coupled ordering and segregation in antiphase domains

PHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 1 2009
K. Gumennyk
Abstract We study a multi-domain ordering kinetics in solid solutions under simultaneous diffusion of solute atoms. By the example of a binary bcc alloy a system of kinetic equations is derived, describing the coupled relaxation of occupancies of the two sublattices, building a bcc lattice, by A and B atomic species. Such an approach supplemented by the simplest mean-field approximation proves sufficient to describe both the establishing of long-range order and the segregation processes occurring in antiphase domains. An interaction and interrelation between evolution of the conserved and non-conserved order parameter fields are elucidated. Asymptotical and numerical analysis of the obtained evolution equations reveals a multi-stage scenario of the alloy relaxation: first comes the quick development of long-range order which is then followed by the slow redistribution of local alloy concentration, so that the majority atoms segregate towards the region of an antiphase boundary. The alloy exhibits a distinct tendency to form a multi-domain structure out of a solitary long-range order parameter fluctuation of a certain sign. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Four-dimensional variational assimilation in the unstable subspace and the optimal subspace dimension

THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 647 2010
Anna Trevisan
Abstract Key apriori information used in 4D-Var is the knowledge of the system's evolution equations. In this article we propose a method for taking full advantage of the knowledge of the system's dynamical instabilities in order to improve the quality of the analysis. We present an algorithm for four-dimensional variational assimilation in the unstable subspace (4D-Var , AUS), which consists of confining in this subspace the increment of the control variable. The existence of an optimal subspace dimension for this confinement is hypothesized. Theoretical arguments in favour of the present approach are supported by numerical experiments in a simple perfect nonlinear model scenario. It is found that the RMS analysis error is a function of the dimension N of the subspace where the analysis is confined and is a minimum for N approximately equal to the dimension of the unstable and neutral manifold. For all assimilation windows, from 1 to 5 d, 4D-Var , AUS performs better than standard 4D-Var. In the presence of observational noise, the 4D-Var solution, while being closer to the observations, is farther away from the truth. The implementation of 4D-Var , AUS does not require the adjoint integration. Copyright © 2010 Royal Meteorological Society [source]


Semiclassical expansion of quantum characteristics for many-body potential scattering problem

ANNALEN DER PHYSIK, Issue 9 2007
M.I. Krivoruchenko
Abstract In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in ,. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into the semiclassical transport models. We formulate the principle of stationary action for quantum Hamilton's equations and give quantum-mechanical extensions of the Liouville theorem on conservation of the phase-space volume and the Poincaré theorem on conservation of 2p -forms. The lowest order quantum corrections to the Kepler periodic orbits are constructed. These corrections show the resonance behavior. [source]


Matrix perturbation theory for driven three-level systems with damping

ANNALEN DER PHYSIK, Issue 10 2004
B.N. Sanchez
Abstract We investigate the dynamics of the , system driven by two resonant laser fields in presence of dissipation for coupling strengths where the rotating-wave approximation starts to break down. This regime is characterised by Rabi frequencies being approximately equal or smaller than the field frequencies. A systematic procedure to obtain an expansion for the solution of the Bloch evolution equations of the system is presented. The lowest contribution results to be the well-known rotating-wave approximation. The method is based on a semi-classical treatment of the problem, and its predictions are interpreted fully quantum mechanically. The theory is illustrated by a detailed study of the disappearance of coherent population trapping as the intensities of the fields increase. [source]